Help for package RTMBdist

Type:

Package

Title:

Distributions Compatible with Automatic Differentiation by 'RTMB'

Version:

1.0.0

Description:

Extends the functionality of the 'RTMB' https://kaskr.r-universe.dev/RTMB package by providing a collection of non-standard probability distributions compatible with automatic differentiation (AD). While 'RTMB' enables flexible and efficient modelling, including random effects, its built-in support is limited to standard distributions. The package adds additional AD-compatible distributions, broadening the range of models that can be implemented and estimated using 'RTMB'. Automatic differentiation and Laplace approximation are described in Kristensen et al. (2016) <doi:10.18637/jss.v070.i05>.

License:

MIT + file LICENSE

Encoding:

UTF-8

Imports:

stats, gamlss.dist, circular, sn, statmod, movMF

Depends:

R (≥ 3.5.0), RTMB (≥ 1.7.0)

RoxygenNote:

7.3.3

Suggests:

knitr, rmarkdown, testthat (≥ 3.0.0), LaMa (≥ 2.0.6), Matrix, TMB, gamlss.data, mvtnorm

Config/testthat/edition:

URL:

https://janoleko.github.io/RTMBdist/

VignetteBuilder:

knitr

NeedsCompilation:

Packaged:

2026-01-13 11:24:31 UTC; jan-ole

Author:

Jan-Ole Koslik

[aut, cre]

Maintainer:

Jan-Ole Koslik <jan-ole.koslik@uni-bielefeld.de>

Repository:

CRAN

Date/Publication:

2026-01-13 11:50:17 UTC

Smooth approximation to the absolute value function

Description

Smooth approximation to the absolute value function

Usage

abs_smooth(x, epsilon = 1e-06)

Arguments

x

vector of evaluation points

epsilon

smoothing constant

Details

We approximate the absolute value here as

\vert x \vert \approx \sqrt{x^2 + \epsilon}

Value

Smooth absolute value of x.

Examples

abs(0)
abs_smooth(0, 1e-4)

Box–Cox Cole and Green distribution (BCCG)

Description

Density, distribution function, quantile function, and random generation for the Box–Cox Cole and Green distribution.

Usage

dbccg(x, mu = 1, sigma = 0.1, nu = 1, log = FALSE)

pbccg(q, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)

qbccg(p, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)

rbccg(n, mu = 1, sigma = 0.1, nu = 1)

Arguments

x, q

vector of quantiles

mu

location parameter, must be positive.

sigma

scale parameter, must be positive.

nu

skewness parameter (real).

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE (default), probabilities are P[X \le x], otherwise P[X > x].

p

vector of probabilities

n

number of random values to return

Details

This implementation of dbccg and pbccg allows for automatic differentiation with RTMB while the other functions are imported from gamlss.dist package. See gamlss.dist::BCCG for more details.

Value

dbccg gives the density, pbccg gives the distribution function, qbccg gives the quantile function, and rbccg generates random deviates.

References

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Examples

x <- rbccg(5, mu = 10, sigma = 0.2, nu = 0.5)
d <- dbccg(x, mu = 10, sigma = 0.2, nu = 0.5)
p <- pbccg(x, mu = 10, sigma = 0.2, nu = 0.5)
q <- qbccg(p, mu = 10, sigma = 0.2, nu = 0.5)

Box-Cox Power Exponential distribution (BCPE)

Description

Density, distribution function, quantile function, and random generation for the Box-Cox Power Exponential distribution.

Usage

dbcpe(x, mu = 5, sigma = 0.1, nu = 1, tau = 2, log = FALSE)

pbcpe(q, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)

qbcpe(p, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)

rbcpe(n, mu = 5, sigma = 0.1, nu = 1, tau = 2)

Arguments

x, q

vector of quantiles

mu

location parameter, must be positive.

sigma

scale parameter, must be positive.

nu

vector of nu parameter values.

tau

vector of tau parameter values, must be positive.

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE (default), probabilities are P[X \le x], otherwise P[X > x].

p

vector of probabilities

n

number of random values to return

Details

This implementation of dbcpe and pbcpe allows for automatic differentiation with RTMB while the other functions are imported from gamlss.dist package. See gamlss.dist::BCPE for more details.

Value

dbcpe gives the density, pbcpe gives the distribution function, qbcpe gives the quantile function, and rbcpe generates random deviates.

References

Examples

x <- rbcpe(1, mu = 5, sigma = 0.1, nu = 1, tau = 1)
d <- dbcpe(x, mu = 5, sigma = 0.1, nu = 1, tau = 1)
p <- pbcpe(x, mu = 5, sigma = 0.1, nu = 1, tau = 1)
q <- qbcpe(p, mu = 5, sigma = 0.1, nu = 1, tau = 1)

Box–Cox t distribution (BCT)

Description

Density, distribution function, quantile function, and random generation for the Box–Cox t distribution.

Usage

dbct(x, mu = 5, sigma = 0.1, nu = 1, tau = 2, log = FALSE)

pbct(q, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)

qbct(p, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)

rbct(n, mu = 5, sigma = 0.1, nu = 1, tau = 2)

Arguments

x, q

vector of quantiles

mu

location parameter, must be positive.

sigma

scale parameter, must be positive.

nu

skewness parameter (real).

tau

degrees of freedom, must be positive.

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE (default), probabilities are P[X \le x], otherwise P[X > x].

p

vector of probabilities

n

number of random values to return

Details

This implementation of dbct and pbct allows for automatic differentiation with RTMB while the other functions are imported from gamlss.dist package. See gamlss.dist::BCT for more details.

Value

dbct gives the density, pbct gives the distribution function, qbct gives the quantile function, and rbct generates random deviates.

References

Examples

x <- rbct(1, mu = 10, sigma = 0.2, nu = 0.5, tau = 4)
d <- dbct(x, mu = 10, sigma = 0.2, nu = 0.5, tau = 4)
p <- pbct(x, mu = 10, sigma = 0.2, nu = 0.5, tau = 4)
q <- qbct(p, mu = 10, sigma = 0.2, nu = 0.5, tau = 4)

Reparameterised beta distribution

Description

Density, distribution function, quantile function, and random generation for the beta distribution reparameterised in terms of mean and concentration.

Usage

dbeta(x, shape1, shape2, log = FALSE, eps = 0)

dbeta2(x, mu, phi, log = FALSE, eps = 0)

pbeta2(q, mu, phi, lower.tail = TRUE, log.p = FALSE)

qbeta2(p, mu, phi, lower.tail = TRUE, log.p = FALSE)

rbeta2(n, mu, phi)

Arguments

x, q

vector of quantiles

shape1, shape2

non-negative parameters

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

eps

for internal use only, don't change.

mu

mean parameter, must be in the interval from 0 to 1.

phi

concentration parameter, must be positive.

lower.tail

logical; if TRUE (default), probabilities are P[X \le x], otherwise P[X > x].

p

vector of probabilities

n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

Currently, dbeta masks RTMB::dbeta because the latter has a numerically unstable gradient.

Value

dbeta2 gives the density, pbeta2 gives the distribution function, qbeta2 gives the quantile function, and rbeta2 generates random deviates.

Examples

set.seed(123)
x <- rbeta2(1, 0.5, 1)
d <- dbeta2(x, 0.5, 1)
p <- pbeta2(x, 0.5, 1)
q <- qbeta2(p, 0.5, 1)

Beta-binomial distribution

Description

Density and random generation for the beta-binomial distribution.

Usage

dbetabinom(x, size, shape1, shape2, log = FALSE)

rbetabinom(n, size, shape1, shape2)

Arguments

x

vector of non-negative counts.

size

vector of total counts (number of trials). Needs to be >= x.

shape1

positive shape parameter 1 of the Beta prior.

shape2

positive shape parameter 2 of the Beta prior.

log

logical; if TRUE, densities are returned on the log scale.

n

number of random values to return (for rbetabinom).

Details

This implementation of dbetabinom allows for automatic differentiation with RTMB.

Value

dbetabinom gives the density and rbetabinom generates random samples.

Examples

set.seed(123)
x <- rbetabinom(1, 10, 2, 5)
d <- dbetabinom(x, 10, 2, 5)

Clayton copula constructors

Description

Construct a function that computes the log density or CDF of the bivariate Clayton copula, intended to be used with dcopula.

Usage

cclayton(theta)

Cclayton(theta)

Arguments

theta

Positive dependence parameter (\theta > 0).

Details

The Clayton copula density is

c(u,v;\theta) = (1+\theta) (uv)^{-(1+\theta)} \left( u^{-\theta} + v^{-\theta} - 1 \right)^{-(2\theta+1)/\theta}, \quad \theta > 0.

Value

A function of two arguments (u,v) returning log copula density (cclayton) or copula CDF (Cclayton).

Examples

x <- c(0.5, 1); y <- c(0.2, 0.8)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dbeta(y, 2, 1, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pbeta(y, 2, 1)
dcopula(d1, d2, p1, p2, copula = cclayton(2), log = TRUE)

# CDF version (for discrete copulas)
Cclayton(1.5)(0.5, 0.4)

Frank copula constructor

Description

Returns a function computing the log density of the bivariate Frank copula, intended to be used with dcopula.

Usage

cfrank(theta)

Cfrank(theta)

Arguments

theta

Dependence parameter (\theta \neq 0).

Details

The Frank copula density is

c(u,v;\theta) = \frac{\theta (1-e^{-\theta}) e^{-\theta(u+v)}} {\left[(e^{-\theta u}-1)(e^{-\theta v}-1) + (1 - e^{-\theta}) \right]^2}, \quad \theta \ne 0.

Value

A function of two arguments (u, v) returning either the log copula density (cfrank) or the copula CDF (Cfrank).

Examples

x <- c(0.5, 1); y <- c(1, 2)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dexp(y, 2, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pexp(y, 2)
dcopula(d1, d2, p1, p2, copula = cfrank(2), log = TRUE)

Gaussian copula constructor

Description

Returns a function computing the log density of the bivariate Gaussian copula, intended to be used with dcopula.

Usage

cgaussian(rho = 0)

Arguments

rho

Correlation parameter (-1 < rho < 1).

Value

Function of two arguments (u,v) returning log copula density.

The Gaussian copula density is

c(u,v;\rho) = \frac{1}{\sqrt{1-\rho^2}} \exp\left\{-\frac{1}{2(1-\rho^2)} (z_1^2 - 2 \rho z_1 z_2 + z_2^2) + \frac{1}{2}(z_1^2 + z_2^2) \right\},

where z_1 = \Phi^{-1}(u), z_2 = \Phi^{-1}(v), and -1 < \rho < 1.

Examples

x <- c(0.5, 1); y <- c(1, 2)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dexp(y, 2, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pexp(y, 2)
dcopula(d1, d2, p1, p2, copula = cgaussian(0.5), log = TRUE)

Multivariate Gaussian copula constructor parameterised by inverse correlation matrix

Description

Returns a function computing the log density of the multivariate Gaussian copula, parameterised by the inverse correlation matrix.

Usage

cgmrf(Q)

Arguments

Q

Inverse of a positive definite correlation matrix with unit diagonal. Can either be sparse or dense matrix.

Details

Caution: Parameterising the inverse correlation directly is difficult, as inverting it needs to yield a positive definite matrix with unit diagonal. Hence we still advise parameterising the correaltion matrix R and computing its inverse. This function is useful when you need access to the precision (i.e. inverse correlation) in your likelihood function.

Value

Function with matrix argument U returning log copula density.

Examples

x <- c(0.5, 1); y <- c(1, 2); z <- c(0.2, 0.8)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dexp(y, 2, log = TRUE); d3 <- dbeta(z, 2, 1, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pexp(y, 2); p3 <- pbeta(z, 2, 1)
R <- matrix(c(1,0.5,0.3,0.5,1,0.4,0.3,0.4,1), nrow = 3)

## Based on correlation matrix
dmvcopula(cbind(d1, d2, d3), cbind(p1, p2, p3), copula = cmvgauss(R), log = TRUE)

## Based on precision matrix
Q <- solve(R)
dmvcopula(cbind(d1, d2, d3), cbind(p1, p2, p3), copula = cgmrf(Q), log = TRUE)

## Parameterisation inside a model
# using RTMB::unstructured to get a valid correlation matrix
library(RTMB)
d <- 5 # dimension
cor_func <- unstructured(d)
npar <- length(cor_func$parms())
R <- cor_func$corr(rep(0.1, npar))

Gumbel copula constructors

Description

Construct functions that compute either the log density or the CDF of the bivariate Gumbel copula, intended for use with dcopula.

Usage

cgumbel(theta)

Cgumbel(theta)

Arguments

theta

Dependence parameter (\theta >= 1).

Details

The Gumbel copula density

c(u,v;\theta) = \exp\Big[-\big((-\log u)^\theta + (-\log v)^\theta\big)^{1/\theta}\Big] \cdot h(u,v;\theta),

where h(u,v;\theta) contains the derivative terms ensuring the function is a density.

Value

A function of two arguments (u, v) returning either the log copula density (cgumbel) or the copula CDF (Cgumbel).

Examples

x <- c(0.5, 1); y <- c(0.2, 0.4)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dbeta(y, 2, 1, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pbeta(y, 2, 1)
dcopula(d1, d2, p1, p2, copula = cgumbel(1.5), log = TRUE)

# CDF version (for discrete copulas)
Cgumbel(1.5)(0.5, 0.4)

Multivariate Gaussian copula constructor

Description

Returns a function computing the log density of the multivariate Gaussian copula, intended to be used with dmvcopula.

Usage

cmvgauss(R)

Arguments

R

Positive definite correlation matrix (unit diagonal)

Value

Function with matrix argument U returning log copula density.

Examples

x <- c(0.5, 1); y <- c(1, 2); z <- c(0.2, 0.8)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dexp(y, 2, log = TRUE); d3 <- dbeta(z, 2, 1, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pexp(y, 2); p3 <- pbeta(z, 2, 1)
R <- matrix(c(1,0.5,0.3,0.5,1,0.4,0.3,0.4,1), nrow = 3)

## Based on correlation matrix
dmvcopula(cbind(d1, d2, d3), cbind(p1, p2, p3), copula = cmvgauss(R), log = TRUE)

## Based on precision matrix
Q <- solve(R)
dmvcopula(cbind(d1, d2, d3), cbind(p1, p2, p3), copula = cgmrf(Q), log = TRUE)

## Parameterisation inside a model
# using RTMB::unstructured to get a valid correlation matrix
library(RTMB)
d <- 5 # dimension
cor_func <- unstructured(d)
npar <- length(cor_func$parms())
R <- cor_func$corr(rep(0.1, npar))

Joint density under a bivariate copula

Description

Computes the joint density (or log-density) of a bivariate distribution constructed from two arbitrary margins combined with a specified copula.

Usage

dcopula(d1, d2, p1, p2, copula = cgaussian(0), log = FALSE)

Arguments

d1, d2

Marginal density values. If log = TRUE, supply the log-density. If log = FALSE, supply the raw density.

p1, p2

Marginal CDF values. Need not be supplied on log scale.

copula

A function of two arguments (u, v) returning the log copula density \log c(u,v). You can either construct this yourself or use the copula constructors available (see details)

log

Logical; if TRUE, return the log joint density. In this case, d1 and d2 must be on the log scale.

Details

The joint density is

f(x,y) = c(F_1(x), F_2(y)) \, f_1(x) f_2(y),

where F_i are the marginal CDFs, f_i are the marginal densities, and c is the copula density.

The marginal densities d1, d2 and CDFs p1, p2 must be differentiable for automatic differentiation (AD) to work.

Available copula constructors are:

cgaussian (Gaussian copula)
cclayton (Clayton copula)
cgumbel (Gumbel copula)
cfrank (Frank copula)

Value

Joint density (or log-density) under the bivariate copula.

Examples

# Normal + Exponential margins with Gaussian copula
x <- c(0.5, 1); y <- c(1, 2)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dexp(y, 2, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pexp(y, 2)
dcopula(d1, d2, p1, p2, copula = cgaussian(0.5), log = TRUE)

# Normal + Beta margins with Clayton copula
x <- c(0.5, 1); y <- c(0.2, 0.8)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dbeta(y, 2, 1, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pbeta(y, 2, 1)
dcopula(d1, d2, p1, p2, copula = cclayton(2), log = TRUE)

# Normal + Beta margins with Gumbel copula
x <- c(0.5, 1); y <- c(0.2, 0.4)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dbeta(y, 2, 1, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pbeta(y, 2, 1)
dcopula(d1, d2, p1, p2, copula = cgumbel(1.5), log = TRUE)

# Normal + Exponential margins with Frank copula
x <- c(0.5, 1); y <- c(1, 2)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dexp(y, 2, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pexp(y, 2)
dcopula(d1, d2, p1, p2, copula = cfrank(2), log = TRUE)

Joint probability under a discrete bivariate copula

Description

Computes the joint probability mass function of two discrete margins combined with a copula CDF.

Usage

ddcopula(d1, d2, p1, p2, Copula, log = FALSE)

Arguments

d1, d2

Marginal p.m.f. values at the observed points, not on log-scale.

p1, p2

Marginal CDF values at the observed points.

Copula

A function of two arguments returning the copula CDF.

log

Logical; if TRUE, return the log joint density. In this case, d1 and d2 must be on the log scale.

Details

The joint probability mass function for two discrete margins is

\Pr(Y_1 = y_1, Y_2 = y_2) = C(F_1(y_1), F_2(y_2)) - C(F_1(y_1-1), F_2(y_2)) - C(F_1(y_1), F_2(y_2-1)) + C(F_1(y_1-1), F_2(y_2-1)),

where F_i are the marginal CDFs, and C is the copula CDF.

Available copula CDF constructors are:

Cclayton (Clayton copula)
Cgumbel (Gumbel copula)
Cfrank (Frank copula)

Value

Joint probability (or log-probability) under chosen copula

Examples

x <- c(3,5); y <- c(2,4)
d1 <- dpois(x, 4); d2 <- dpois(y, 3)
p1 <- ppois(x, 4); p2 <- ppois(y, 3)
ddcopula(d1, d2, p1, p2, Copula = Cclayton(2), log = FALSE)

Dirichlet distribution

Description

Density and and random generation for the Dirichlet distribution.

Usage

ddirichlet(x, alpha, log = FALSE)

rdirichlet(n, alpha)

Arguments

x

vector or matrix of quantiles. If x is a vector, it needs to sum to one. If x is a matrix, each row should sum to one.

alpha

vector or matrix of positive shape parameters

log

logical; if TRUE, densities p are returned as \log(p).

n

number of random values to return.

Details

This implementation of ddirichlet allows for automatic differentiation with RTMB.

Value

ddirichlet gives the density, rdirichlet generates random deviates.

Examples

# single alpha
alpha <- c(1,2,3)
x <- rdirichlet(1, alpha)
d <- ddirichlet(x, alpha)
# vectorised over alpha
alpha <- rbind(alpha, 2*alpha)
x <- rdirichlet(2, alpha)

Dirichlet-multinomial distribution

Description

Density and and random generation for the Dirichlet-multinomial distribution.

Usage

ddirmult(x, size, alpha, log = FALSE)

rdirmult(n, size, alpha)

Arguments

x

vector or matrix of non-negative counts, where rows are observations and columns are categories.

size

vector of total counts for each observation. Needs to match the row sums of x.

alpha

vector or matrix of positive shape parameters

log

logical; if TRUE, densities p are returned as \log(p).

n

number of random values to return.

Details

This implementation of ddirmult allows for automatic differentiation with RTMB.

Value

ddirmult gives the density and rdirmult generates random samples.

Examples

# single alpha
alpha <- c(1,2,3)
size <- 10
x <- rdirmult(1, size, alpha)
d <- ddirmult(x, size, alpha)
# vectorised over alpha and size
alpha <- rbind(alpha, 2*alpha)
size <- c(size, 3*size)
x <- rdirmult(2, size, alpha)

Joint density under a multivariate copula

Description

Computes the joint density (or log-density) of a distribution constructed from any number of arbitrary margins combined with a specified copula.

Usage

dmvcopula(D, P, copula = cmvgauss(diag(ncol(D))), log = FALSE)

Arguments

D

Matrix of marginal density values of with rows corresponding to observations and columns corresponding to dimensions. If log = TRUE, supply the log-densities. If log = FALSE, supply the raw densities

P

Matrix of marginal CDF values of the same dimension as D. Need not be supplied on log scale.

copula

A function of a matrix argument U returning the log copula density \log c(u_1, ... u_d). The columns of U correspond to dimensions. You can either construct this yourself or use the copula constructors available (see details)

log

Logical; if TRUE, return the log joint density. In this case, D must be on the log scale.

Details

The joint density is

f(x_1, \dots, x_d) = c(F_1(x_1), \dots, F_d(x_d)) \, f_1(x_1) \dots f_d(x_d),

where F_i are the marginal CDFs, f_i are the marginal densities, and c is the copula density.

The marginal densities d_1, ..., d_d and CDFs p_1, ..., p_d must be differentiable for automatic differentiation (AD) to work.

Available multivariate copula constructors are:

cmvgauss (Multivariate Gaussian copula)
cgmrf (Multivariate Gaussian copula parameterised by precision (inverse correlation) matrix)

Value

Joint density (or log-density) under the chosen copula.

Examples

x <- c(0.5, 1); y <- c(1, 2); z <- c(0.2, 0.8)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dexp(y, 2, log = TRUE); d3 <- dbeta(z, 2, 1, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pexp(y, 2); p3 <- pbeta(z, 2, 1)
R <- matrix(c(1,0.5,0.3,0.5,1,0.4,0.3,0.4,1), nrow = 3)

### Multivariate Gaussian copula
## Based on correlation matrix
dmvcopula(cbind(d1, d2, d3), cbind(p1, p2, p3), copula = cmvgauss(R), log = TRUE)

## Based on precision matrix
Q <- solve(R)
dmvcopula(cbind(d1, d2, d3), cbind(p1, p2, p3), copula = cgmrf(Q), log = TRUE)

## Parameterisation inside a model
# using RTMB::unstructured to get a valid correlation matrix
library(RTMB)
d <- 5 # dimension
cor_func <- unstructured(d)
npar <- length(cor_func$parms())
R <- cor_func$corr(rep(0.1, npar))

AD-compatible error function and complementary error function

Description

AD-compatible error function and complementary error function

Usage

erf(x)

erfc(x)

Arguments

x

vector of evaluation points

Value

erf(x) returns the error function and erfc(x) returns the complementary error function.

Examples

erf(1)
erfc(1)

Exponentially modified Gaussian distribution

Description

Density, distribution function, quantile function, and random generation for the exponentially modified Gaussian distribution.

Usage

dexgauss(x, mu = 0, sigma = 1, lambda = 1, log = FALSE)

pexgauss(q, mu = 0, sigma = 1, lambda = 1, lower.tail = TRUE, log.p = FALSE)

qexgauss(p, mu = 0, sigma = 1, lambda = 1, lower.tail = TRUE, log.p = FALSE)

rexgauss(n, mu = 0, sigma = 1, lambda = 1)

Arguments

x, q

vector of quantiles

mu

mean parameter of the Gaussian part

sigma

standard deviation parameter of the Gaussian part, must be positive.

lambda

rate parameter of the exponential part, must be positive.

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

p

vector of probabilities

n

number of random values to return

Details

This implementation of dexgauss and pexgauss allows for automatic differentiation with RTMB. qexgauss and rexgauss are reparameterised imports from gamlss.dist::exGAUS.

If X \sim N(\mu, \sigma^2) and Y \sim \text{Exp}(\lambda), then Z = X + Y follows the exponentially modified Gaussian distribution with parameters \mu, \sigma, and \lambda.

Value

dexgauss gives the density, pexgauss gives the distribution function, qexgauss gives the quantile function, and rexgauss generates random deviates.

References

Examples

x <- rexgauss(1, 1, 2, 2)
d <- dexgauss(x, 1, 2, 2)
p <- pexgauss(x, 1, 2, 2)
q <- qexgauss(p, 1, 2, 2)

Folded normal distribution

Description

Density, distribution function, and random generation for the folded normal distribution.

Usage

dfoldnorm(x, mu = 0, sigma = 1, log = FALSE)

pfoldnorm(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rfoldnorm(n, mu = 0, sigma = 1)

Arguments

x, q

vector of quantiles

mu

location parameter

sigma

scale parameter, must be positive.

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

n

number of random values to return

p

vector of probabilities

Details

This implementation of dfoldnorm allows for automatic differentiation with RTMB.

Value

dfoldnorm gives the density, pfoldnorm gives the distribution function, and rfoldnorm generates random deviates.

Examples

x <- rfoldnorm(1, 1, 2)
d <- dfoldnorm(x, 1, 2)
p <- pfoldnorm(x, 1, 2)

Reparameterised gamma distribution

Description

Density, distribution function, quantile function, and random generation for the gamma distribution reparameterised in terms of mean and standard deviation.

Usage

dgamma2(x, mean = 1, sd = 1, log = FALSE)

pgamma2(q, mean = 1, sd = 1, lower.tail = TRUE, log.p = FALSE)

qgamma2(p, mean = 1, sd = 1, lower.tail = TRUE, log.p = FALSE)

rgamma2(n, mean = 1, sd = 1)

Arguments

x, q

vector of quantiles

mean

mean parameter, must be positive.

sd

standard deviation parameter, must be positive.

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

p

vector of probabilities

n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

Value

dgamma2 gives the density, pgamma2 gives the distribution function, qgamma2 gives the quantile function, and rgamma2 generates random deviates.

Examples

x <- rgamma2(1)
d <- dgamma2(x)
p <- pgamma2(x)
q <- qgamma2(p)

Generalised Poisson distribution

Description

Probability mass function, distribution function, and random generation for the generalised Poisson distribution.

Usage

dgenpois(x, lambda = 1, phi = 1, log = FALSE)

pgenpois(q, lambda = 1, phi = 1, lower.tail = TRUE, log.p = FALSE)

qgenpois(p, lambda = 1, phi = 1,
         lower.tail = TRUE, log.p = FALSE, max.value = 10000)

rgenpois(n, lambda = 1, phi = 1, max.value = 10000)

Arguments

x, q

integer vector of counts

lambda

vector of positive means

phi

vector of non-negative dispersion parameters

log, log.p

logical; return log-density if TRUE

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

p

vector of probabilities

max.value

a constant, set to the default value of 10000 for how far the algorithm should look for q.

n

number of random values to return.

Details

This implementation of dgenpois allows for automatic differentiation with RTMB. The other functions are imported from gamlss.dist::GPO.

The distribution has mean \lambda and variance \lambda(1 + \phi \lambda)^2. For \phi = 0 it reduces to the Poisson distribution, however \phi must be strictly positive here.

Value

dgenpois gives the probability mass function, pgenpois gives the distribution function, qgenpois gives the quantile function, and rgenpois generates random deviates.

Examples

set.seed(123)
x <- rgenpois(1, 2, 3)
d <- dgenpois(x, 2, 3)
p <- pgenpois(x, 2, 3)
q <- qgenpois(p, 2, 3)

Gumbel distribution

Description

Density, distribution function, quantile function, and random generation for the Gumbel distribution.

Usage

dgumbel(x, location = 0, scale = 1, log = FALSE)

pgumbel(q, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)

qgumbel(p, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)

rgumbel(n, location = 0, scale = 1)

Arguments

x, q

vector of quantiles

location

location parameter

scale

scale parameter, must be positive.

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

p

vector of probabilities

n

number of random values to return

Details

This implementation of dgumbel allows for automatic differentiation with RTMB.

Value

dgumbel gives the density, pgumbel gives the distribution function, qgumbel gives the quantile function, and rgumbel generates random deviates.

Examples

x <- rgumbel(1, 0.5, 2)
d <- dgumbel(x, 0.5, 2)
p <- pgumbel(x, 0.5, 2)
q <- qgumbel(p, 0.5, 2)

Inverse Gaussian distribution

Description

Density, distribution function, and random generation for the inverse Gaussian distribution.

Usage

dinvgauss(x, mean = 1, shape = 1, log = FALSE)

pinvgauss(q, mean = 1, shape = 1, lower.tail = TRUE, log.p = FALSE)

qinvgauss(p, mean = 1, shape = 1, lower.tail = TRUE, log.p = FALSE, ...)

rinvgauss(n, mean = 1, shape = 1)

Arguments

x, q

vector of quantiles, must be positive.

mean

location parameter

shape

shape parameter, must be positive.

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

p

vector of probabilities

...

additional parameter passed to statmod::qinvgauss for numerical evaluation of the quantile function.

n

number of random values to return

Details

This implementation of dinvgauss allows for automatic differentiation with RTMB. qinvgauss and rinvgauss are imported from the statmod package.

Value

dinvgauss gives the density, pinvgauss gives the distribution function, qinvgauss gives the quantile function, and rinvgauss generates random deviates.

Examples

x <- rinvgauss(1, 1, 0.5)
d <- dinvgauss(x, 1, 0.5)
p <- pinvgauss(x, 1, 0.5)
q <- qinvgauss(p, 1, 0.5)

Laplace distribution

Description

Density, distribution function, quantile function, and random generation for the Laplace distribution.

Usage

dlaplace(x, mu = 0, b = 1, eps = NULL, log = FALSE)

plaplace(q, mu = 0, b = 1, lower.tail = TRUE, log.p = FALSE)

qlaplace(p, mu = 0, b = 1, lower.tail = TRUE, log.p = FALSE)

rlaplace(n, mu = 0, b = 1)

Arguments

x, q

vector of quantiles

mu

location parameter

b

scale parameter, must be positive.

eps

optional smoothing parameter for dlaplace to smooth the absolute value function. See abs_smooth for details. It is recommended to set this to a small constant like 1e-6 for numerical optimisation.

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

p

vector of probabilities

n

number of random values to return

Details

This implementation of dlaplace allows for automatic differentiation with RTMB.

Value

dlaplace gives the density, plaplace gives the distribution function, qlaplace gives the quantile function, and rlaplace generates random deviates.

Examples

x <- rlaplace(1, 1, 1)
d <- dlaplace(x, 1, 1)
p <- plaplace(x, 1, 1)
q <- qlaplace(p, 1, 1)

Multivariate t distribution

Description

Density and and random generation for the multivariate t distribution

Usage

dmvt(x, mu, Sigma, df, log = FALSE)

rmvt(n, mu, Sigma, df)

Arguments

x

vector or matrix of quantiles

mu

vector or matrix of location parameters (mean if df > 1)

Sigma

positive definite scale matrix (proportional to the covariance matrix if df > 2)

df

degrees of freedom; must be positive

log

logical; if TRUE, densities p are returned as \log(p).

n

number of random values to return.

Details

This implementation of dmvt allows for automatic differentiation with RTMB.

Note: for df \le 1 the mean is undefined, and for df \le 2 the covariance is infinite. For df > 2, the covariance is df/(df-2) * Sigma.

Value

dmvt gives the density, rmvt generates random deviates.

Examples

# single mu
mu <- c(1,2,3)
Sigma <- diag(c(1,1,1))
df <- 5
x <- rmvt(2, mu, Sigma, df)
d <- dmvt(x, mu, Sigma, df)
# vectorised over mu
mu <- rbind(c(1,2,3), c(0, 0.5, 1))
x <- rmvt(2, mu, Sigma, df)
d <- dmvt(x, mu, Sigma, df)

Reparameterised negative binomial distribution

Description

Probability mass function, distribution function, quantile function, and random generation for the negative binomial distribution reparameterised in terms of mean and size.

Usage

dnbinom2(x, mu, size, log = FALSE)

pnbinom2(q, mu, size, lower.tail = TRUE, log.p = FALSE)

qnbinom2(p, mu, size, lower.tail = TRUE, log.p = FALSE)

rnbinom2(n, mu, size)

pnbinom(q, size, prob, lower.tail = TRUE, log.p = FALSE)

Arguments

x, q

vector of quantiles

mu

mean parameter, must be positive.

size

size parameter, must be positive.

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

p

vector of probabilities

n

number of random values to return.

prob

probability of success in each trial. 0 < prob <= 1.

Details

This implementation allows for automatic differentiation with RTMB.

pnbinom is an AD-compatible implementation of the standard parameterisation of the CDF, missing from RTMB.

Value

dnbinom2 gives the density, pnbinom2 gives the distribution function, qnbinom2 gives the quantile function, and rnbinom2 generates random deviates.

Examples

set.seed(123)
x <- rnbinom2(1, 1, 2)
d <- dnbinom2(x, 1, 2)
p <- pnbinom2(x, 1, 2)
q <- qnbinom2(p, 1, 2)

One-inflated beta distribution

Description

Density, distribution function, and random generation for the one-inflated beta distribution.

Usage

doibeta(x, shape1, shape2, oneprob = 0, log = FALSE)

poibeta(q, shape1, shape2, oneprob = 0, lower.tail = TRUE, log.p = FALSE)

roibeta(n, shape1, shape2, oneprob = 0)

Arguments

x, q

vector of quantiles

shape1, shape2

non-negative shape parameters of the beta distribution

oneprob

zero-inflation probability between 0 and 1.

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

Value

doibeta gives the density, poibeta gives the distribution function, and roibeta generates random deviates.

Examples

set.seed(123)
x <- roibeta(1, 2, 2, 0.5)
d <- doibeta(x, 2, 2, 0.5)
p <- poibeta(x, 2, 2, 0.5)

Reparameterised one-inflated beta distribution

Description

Density, distribution function, and random generation for the one-inflated beta distribution reparameterised in terms of mean and concentration.

Usage

doibeta2(x, mu, phi, oneprob = 0, log = FALSE)

poibeta2(q, mu, phi, oneprob = 0, lower.tail = TRUE, log.p = FALSE)

roibeta2(n, mu, phi, oneprob = 0)

Arguments

x, q

vector of quantiles

mu

mean parameter, must be in the interval from 0 to 1.

phi

concentration parameter, must be positive.

oneprob

zero-inflation probability between 0 and 1.

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

Value

doibeta2 gives the density, poibeta2 gives the distribution function, and roibeta2 generates random deviates.

Examples

set.seed(123)
x <- roibeta2(1, 0.6, 2, 0.5)
d <- doibeta2(x, 0.6, 2, 0.5)
p <- poibeta2(x, 0.6, 2, 0.5)

Pareto distribution

Description

Density, distribution function, quantile function, and random generation for the pareto distribution.

Usage

dpareto(x, mu = 1, log = FALSE)

ppareto(q, mu = 1, lower.tail = TRUE, log.p = FALSE)

qpareto(p, mu = 1, lower.tail = TRUE, log.p = FALSE)

rpareto(n, mu = 1)

Arguments

x, q

vector of quantiles

mu

location parameter, must be positive.

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE (default), probabilities are P[X \le x], otherwise P[X > x].

p

vector of probabilities

n

number of random values to return

Details

This implementation of dpareto and ppareto allows for automatic differentiation with RTMB while the other functions are imported from gamlss.dist package. See gamlss.dist::PARETO for more details.

Value

dpareto gives the density, ppareto gives the distribution function, qpareto gives the quantile function, and rpareto generates random deviates.

References

Examples

set.seed(123)
x <- rpareto(1, mu = 5)
d <- dpareto(x, mu = 5)
p <- ppareto(x, mu = 5)
q <- qpareto(p, mu = 5)

Power generalized Weibull distribution

Description

Survival, hazard, cumulative distribution, density, quantile and sampling function for the power generalized Weibull (PgW) distribution with parameters scale, shape and powershape.

Usage

spgweibull(x, scale = 1, shape = 1, powershape = 1, log = FALSE)

hpgweibull(x, scale = 1, shape = 1, powershape = 1, log = FALSE)

ppgweibull(x, scale = 1, shape = 1, powershape = 1,
           lower.tail = TRUE, log.p = FALSE)

dpgweibull(x, scale = 1, shape = 1, powershape = 1, log = FALSE)

qpgweibull(p, scale = 1, shape = 1, powershape = 1)

rpgweibull(n, scale = 1, shape = 1, powershape = 1)

Arguments

x

vector of quantiles

scale

positive scale parameter

shape

positive shape parameter

powershape

positive power shape parameter

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE (default), probabilities are P[X \le x], otherwise P[X > x].

p

vector of probabilities

n

number of observations

Details

The survival function of the PgW distribution is:

S(x) = \exp \left\{ 1 - \left[ 1 + \left(\frac{x}{\theta}\right)^{\nu}\right]^{\frac{1}{\gamma}} \right\}.

The hazard function is

\frac{\nu}{\gamma\theta^{\nu}}\cdot x^{\nu-1}\cdot \left[ 1 + \left(\frac{x}{\theta}\right)^{\nu}\right]^{\frac{1}{\gamma-1}}

The cumulative distribution function is then F(x) = 1 - S(x) and the density function is S(x)\cdot h(x).

If both shape parameters equal 1, the PgW distribution reduces to the exponential distribution (see dexp) with \texttt{rate} = 1/\texttt{scale} If the power shape parameter equals 1, the PgW distribution simplifies to the Weibull distribution (see dweibull) with the same parametrization.

Value

dpgweibull gives the density, ppgweibull gives the distribution function, qpgweibull gives the quantile function, and rpgweibull generates random deviates. spgweibull gives the survival function and hpgweibull gives the hazard function.

Examples

x <- rpgweibull(1, 2, 2, 3)
d <- dpgweibull(x, 2, 2, 3)
p <- ppgweibull(x, 2, 2, 3)
q <- qpgweibull(p, 2, 2, 3)
s <- spgweibull(x, 2, 2, 3)
h <- hpgweibull(x, 2, 2, 3)

Power Exponential distribution (PE and PE2)

Description

Density, distribution function, quantile function, and random generation for the Power Exponential distribution (two versions).

Usage

dpowerexp(x, mu = 0, sigma = 1, nu = 2, log = FALSE)

ppowerexp(q, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)

qpowerexp(p, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)

rpowerexp(n, mu = 0, sigma = 1, nu = 2)

dpowerexp2(x, mu = 0, sigma = 1, nu = 2, log = FALSE)

ppowerexp2(q, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)

qpowerexp2(p, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)

rpowerexp2(n, mu = 0, sigma = 1, nu = 2)

Arguments

x, q

vector of quantiles

mu

location parameter

sigma

scale parameter, must be positive

nu

shape parameter (real)

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p)

lower.tail

logical; if TRUE (default), probabilities are P[X \le x], otherwise P[X > x]

p

vector of probabilities

n

number of random values to return

Details

This implementation of the densities and distribution functions allow for automatic differentiation with RTMB while the other functions are imported from gamlss.dist package.

For powerexp, mu is the mean and sigma is the standard deviation while this does not hold for powerexp2.

See gamlss.dist::PE for more details.

Value

dpowerexp gives the density, ppowerexp gives the distribution function, qpowerexp gives the quantile function, and rpowerexp generates random deviates.

References

Examples

# PE
x <- rpowerexp(1, mu = 0, sigma = 1, nu = 2)
d <- dpowerexp(x, mu = 0, sigma = 1, nu = 2)
p <- ppowerexp(x, mu = 0, sigma = 1, nu = 2)
q <- qpowerexp(p, mu = 0, sigma = 1, nu = 2)

# PE2
x <- rpowerexp2(1, mu = 0, sigma = 1, nu = 2)
d <- dpowerexp2(x, mu = 0, sigma = 1, nu = 2)
p <- ppowerexp2(x, mu = 0, sigma = 1, nu = 2)
q <- qpowerexp2(p, mu = 0, sigma = 1, nu = 2)

Skew normal distribution

Description

Density, distribution function, quantile function, and random generation for the skew normal distribution.

Usage

dskewnorm(x, xi = 0, omega = 1, alpha = 0, log = FALSE)

pskewnorm(q, xi = 0, omega = 1, alpha = 0, ...)

qskewnorm(p, xi = 0, omega = 1, alpha = 0, ...)

rskewnorm(n, xi = 0, omega = 1, alpha = 0)

Arguments

x, q

vector of quantiles

xi

location parameter

omega

scale parameter, must be positive.

alpha

skewness parameter, +/- Inf is allowed.

log

logical; if TRUE, probabilities/ densities p are returned as \log(p).

...

additional parameters to be passed to the sn package functions for pskewnorm and qskewnorm.

p

vector of probabilities

n

number of random values to return

Details

This implementation of dskewnorm allows for automatic differentiation with RTMB while the other functions are imported from the sn package. See sn::dsn for more details.

Value

dskewnorm gives the density, pskewnorm gives the distribution function, qskewnorm gives the quantile function, and rskewnorm generates random deviates.

Examples

# alpha is skew parameter
x <- rskewnorm(1, alpha = 1)
d <- dskewnorm(x, alpha = 1)
p <- pskewnorm(x, alpha = 1)
q <- qskewnorm(p, alpha = 1)

Reparameterised skew normal distribution

Description

Density, distribution function, quantile function and random generation for the skew normal distribution reparameterised in terms of mean, standard deviation and skew magnitude

Usage

dskewnorm2(x, mean = 0, sd = 1, alpha = 0, log = FALSE)

pskewnorm2(q, mean = 0, sd = 1, alpha = 0, ...)

qskewnorm2(p, mean = 0, sd = 1, alpha = 0, ...)

rskewnorm2(n, mean = 0, sd = 1, alpha = 0)

Arguments

x, q

vector of quantiles

mean

mean parameter

sd

standard deviation, must be positive.

alpha

skewness parameter, +/- Inf is allowed.

log

logical; if TRUE, probabilities/ densities p are returned as \log(p).

...

additional parameters to be passed to the sn package functions for pskewnorm and qskewnorm.

p

vector of probabilities

n

number of random values to return

Details

This implementation of dskewnorm2 allows for automatic differentiation with RTMB while the other functions are imported from the sn package.

Value

dskewnorm2 gives the density, pskewnorm2 gives the distribution function, qskewnorm2 gives the quantile function, and rskewnorm2 generates random deviates.

Examples

# alpha is skew parameter
x <- rskewnorm2(1, alpha = 1)
d <- dskewnorm2(x, alpha = 1)
p <- pskewnorm2(x, alpha = 1)
q <- qskewnorm2(p, alpha = 1)

Skewed students t distribution

Description

Density, distribution function, quantile function, and random generation for the skew t distribution (type 2).

Usage

dskewt(x, mu = 0, sigma = 1, skew = 0, df = 100, log = FALSE)

pskewt(q, mu = 0, sigma = 1, skew = 0, df = 100,
       method = 0, lower.tail = TRUE, log.p = FALSE)

qskewt(p, mu = 0, sigma = 1, skew = 0, df = 100,
       tol = 1e-8, method = 0)

rskewt(n, mu = 0, sigma = 1, skew = 0, df = 100)

Arguments

x, q

vector of quantiles

mu

location parameter

sigma

scale parameter, must be positive.

skew

skewness parameter, can be positive or negative.

df

degrees of freedom, must be positive.

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

method

an integer value between 0 and 5 which selects the computing method; see ‘Details’ in the pst documentation below for the meaning of these values. If method=0 (default value), an automatic choice is made among the four actual computing methods, depending on the other arguments.

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

p

vector of probabilities

tol

a scalar value which regulates the accuracy of the result of qsn, measured on the probability scale.

n

number of random values to return.

Details

This corresponds to the skew t type 2 distribution in GAMLSS (ST2), see pp. 411-412 of Rigby et al. (2019) and the version implemented in the sn package. This implementation of dskewt allows for automatic differentiation with RTMB while the other functions are imported from the sn package. See sn::dst for more details.

Caution: In a numerial optimisation, the skew parameter should NEVER be initialised with exactly zero. This will cause the initial and all subsequent derivatives to be exactly zero and hence the parameter will remain at its initial value.

Value

dskewt gives the density, pskewt gives the distribution function, qskewt gives the quantile function, and rskewt generates random deviates.

Examples

x <- rskewt(1, 1, 2, 5, 2)
d <- dskewt(x, 1, 2, 5, 2)
p <- pskewt(x, 1, 2, 5, 2)
q <- qskewt(p, 1, 2, 5, 2)

Moment-parameterised skew t distribution

Description

Density, distribution function, quantile function, and random generation for the skew t distribution reparameterised so that mean and sd correspond to the true mean and standard deviation.

Usage

dskewt2(x, mean = 0, sd = 1, skew = 0, df = 100, log = FALSE)

pskewt2(q, mean = 0, sd = 1, skew = 0, df = 100,
        method = 0, lower.tail = TRUE, log.p = FALSE)

qskewt2(p, mean = 0, sd = 1, skew = 0, df = 100, tol = 1e-08, method = 0)

rskewt2(n, mean = 0, sd = 1, skew = 0, df = 100)

Arguments

x, q

vector of quantiles

mean

mean parameter

sd

standard deviation parameter, must be positive.

skew

skewness parameter, can be positive or negative.

df

degrees of freedom, must be positive.

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

method

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

p

vector of probabilities

tol

a scalar value which regulates the accuracy of the result of qsn, measured on the probability scale.

n

number of random values to return.

Details

This corresponds to the skew t type 2 distribution in GAMLSS (ST2), see pp. 411-412 of Rigby et al. (2019) and the version implemented in the sn package. However, it is reparameterised in terms of a standard deviation parameter sd rather than just a scale parameter sigma. Details of this reparameterisation are given below. This implementation of dskewt allows for automatic differentiation with RTMB while the other functions are imported from the sn package. See sn::dst for more details.

For given skew = \alpha and df = \nu, define

\delta = \alpha / \sqrt{1 + \alpha^2}, \qquad b_\nu = \sqrt{\nu / \pi}\, \Gamma((\nu-1)/2)/\Gamma(\nu/2),

then

E(X) = \mu + \sigma \delta b_\nu,\quad Var(X) = \sigma^2 \left( \frac{\nu}{\nu-2} - \delta^2 b_\nu^2 \right).

Value

dskewt2 gives the density, pskewt2 gives the distribution function, qskewt2 gives the quantile function, and rskewt2 generates random deviates.

Examples

x <- rskewt2(1, 1, 2, 5, 5)
d <- dskewt2(x, 1, 2, 5, 5)
p <- pskewt2(x, 1, 2, 5, 5)
q <- qskewt2(p, 1, 2, 5, 5)

Student t distribution with location and scale

Description

Density, distribution function, quantile function, and random generation for the t distribution with location and scale parameters.

Usage

dt2(x, mu, sigma, df, log = FALSE)

pt2(q, mu, sigma, df)

rt2(n, mu, sigma, df)

qt2(p, mu, sigma, df)

pt(q, df)

Arguments

x, q

vector of quantiles

mu

location parameter

sigma

scale parameter, must be positive.

df

degrees of freedom, must be positive.

log

logical; if TRUE, probabilities/ densities p are returned as \log(p).

n

number of random values to return.

p

vector of probabilities

Details

This implementation of dt2 allows for automatic differentiation with RTMB.

Value

dt2 gives the density, pt2 gives the distribution function, qt2 gives the quantile function, and rt2 generates random deviates.

Examples

x <- rt2(1, 1, 2, 5)
d <- dt2(x, 1, 2, 5)
p <- pt2(x, 1, 2, 5)
q <- qt2(p, 1, 2, 5)

Truncated normal distribution

Description

Density, distribution function, quantile function, and random generation for the truncated normal distribution.

Usage

dtruncnorm(x, mean = 0, sd = 1, min = -Inf, max = Inf, log = FALSE)

ptruncnorm(q, mean = 0, sd = 1, min = -Inf, max = Inf,
           lower.tail = TRUE, log.p = FALSE)

qtruncnorm(p, mean = 0, sd = 1, min = -Inf, max = Inf,
           lower.tail = TRUE, log.p = FALSE)

rtruncnorm(n, mean = 0, sd = 1, min = -Inf, max = Inf)

Arguments

x, q

vector of quantiles

mean

mean parameter, must be positive.

sd

standard deviation parameter, must be positive.

min, max

truncation bounds.

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

p

vector of probabilities

n

number of random values to return.

Details

This implementation of dtruncnorm allows for automatic differentiation with RTMB.

Value

dtruncnorm gives the density, ptruncnorm gives the distribution function, qtruncnorm gives the quantile function, and rtruncnorm generates random deviates.

Examples

x <- rtruncnorm(1, mean = 2, sd = 2, min = -1, max = 5)
d <- dtruncnorm(x, mean = 2, sd = 2, min = -1, max = 5)
p <- ptruncnorm(x, mean = 2, sd = 2, min = -1, max = 5)
q <- qtruncnorm(p, mean = 2, sd = 2, min = -1, max = 5)

Truncated t distribution

Description

Density, distribution function, quantile function, and random generation for the truncated t distribution.

Usage

dtrunct(x, df, min = -Inf, max = Inf, log = FALSE)

ptrunct(q, df, min = -Inf, max = Inf, lower.tail = TRUE, log.p = FALSE)

qtrunct(p, df, min = -Inf, max = Inf, lower.tail = TRUE, log.p = FALSE)

rtrunct(n, df, min = -Inf, max = Inf)

Arguments

x, q

vector of quantiles

df

degrees of freedom parameter, must be positive.

min, max

truncation bounds.

log, log.p

logical; if TRUE, probabilities/densities p are returned as \log(p).

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise P[X > x].

p

vector of probabilities

n

number of random values to return.

Details

This implementation of dtrunct allows for automatic differentiation with RTMB.

Value

dtrunct gives the density, ptrunct gives the distribution function, qtrunct gives the quantile function, and rtrunct generates random deviates.

Examples

x <- rtrunct(1, df = 5, min = -1, max = 5)
d <- dtrunct(x, df = 5, min = -1, max = 5)
p <- ptrunct(x, df = 5, min = -1, max = 5)
q <- qtrunct(p, df = 5, min = -1, max = 5)

Truncated t distribution with location and scale

Description

Density, distribution function, quantile function, and random generation for the truncated t distribution with location mu and scale sigma.

Usage

dtrunct2(x, df, mu = 0, sigma = 1, min = -Inf, max = Inf, log = FALSE)

ptrunct2(q, df, mu = 0, sigma = 1, min = -Inf, max = Inf,
         lower.tail = TRUE, log.p = FALSE)

qtrunct2(p, df, mu = 0, sigma = 1, min = -Inf, max = Inf,
         lower.tail = TRUE, log.p = FALSE)

rtrunct2(n, df, mu = 0, sigma = 1, min = -Inf, max = Inf)

Arguments

x, q

vector of quantiles

df

degrees of freedom parameter, must be positive.

mu

location parameter.

sigma

scale parameter, must be positive.

min, max

truncation bounds.

log, log.p

logical; if TRUE, probabilities/densities p are returned as \log(p).

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise P[X > x].

p

vector of probabilities

n

number of random values to return.

Details

This implementation of dtrunct2 allows for automatic differentiation with RTMB.

Value

dtrunct2 gives the density, ptrunct2 gives the distribution function, qtrunct2 gives the quantile function, and rtrunct2 generates random deviates.

Examples

x <- rtrunct2(1, df = 5, mu = 2, sigma = 3, min = -1, max = 5)
d <- dtrunct2(x, df = 5, mu = 2, sigma = 3, min = -1, max = 5)
p <- ptrunct2(x, df = 5, mu = 2, sigma = 3, min = -1, max = 5)
q <- qtrunct2(p, df = 5, mu = 2, sigma = 3, min = -1, max = 5)

von Mises distribution

Description

Density, distribution function, and random generation for the von Mises distribution.

Usage

dvm(x, mu = 0, kappa = 1, log = FALSE)

pvm(q, mu = 0, kappa = 1, from = NULL, tol = 1e-20)

rvm(n, mu = 0, kappa = 1, wrap = TRUE)

Arguments

x, q

vector of angles measured in radians at which to evaluate the density function.

mu

mean direction of the distribution measured in radians.

kappa

non-negative numeric value for the concentration parameter of the distribution.

log

logical; if TRUE, densities are returned on the log scale.

from

value from which the integration for CDF starts. If NULL, is set to mu - pi.

tol

the precision in evaluating the distribution function

n

number of random values to return.

wrap

logical; if TRUE, generated angles are wrapped to the interval from -pi to pi.

Details

This implementation of dvm allows for automatic differentiation with RTMB. rvm and pvm are simply wrappers of the corresponding functions from circular.

Value

dvm gives the density, pvm gives the distribution function, and rvm generates random deviates.

Examples

set.seed(1)
x <- rvm(10, 0, 1)
d <- dvm(x, 0, 1)
p <- pvm(x, 0, 1)

von Mises-Fisher distribution

Description

Density, distribution function, and random generation for the von Mises-Fisher distribution.

Usage

dvmf(x, mu, kappa, log = FALSE)

rvmf(n, mu, kappa)

Arguments

x

unit vector or matrix (with each row being a unit vector) of evaluation points

mu

unit mean vector

kappa

non-negative numeric value for the concentration parameter of the distribution.

log

logical; if TRUE, densities are returned on the log scale.

n

number of random values to return.

Details

This implementation of dvmf allows for automatic differentiation with RTMB. rvmf is a reparameterised import from movMF::rmovMF.

Value

dvmf gives the density and rvm generates random deviates.

Examples

set.seed(123)
# single parameter set
mu <- rep(1, 3) / sqrt(3)
kappa <- 4
x <- rvmf(1, mu, kappa)
d <- dvmf(x, mu, kappa)

# vectorised over parameters
mu <- matrix(mu, nrow = 1)
mu <- mu[rep(1,10), ]
kappa <- rep(kappa, 10)
x <- rvmf(10, mu, kappa)
d <- dvmf(x, mu, kappa)

Reparameterised von Mises-Fisher distribution

Description

Density, distribution function, and random generation for the von Mises-Fisher distribution.

Usage

dvmf2(x, theta, log = FALSE)

rvmf2(n, theta)

Arguments

x

unit vector or matrix (with each row being a unit vector) of evaluation points

theta

direction and concentration vector. The direction of theta determines the mean direction on the sphere. The norm of theta is the concentration parameter of the distribution.

log

logical; if TRUE, densities are returned on the log scale.

n

number of random values to return.

Details

In this parameterisation, \theta = \kappa \mu, where \mu is a unit vector and \kappa is the concentration parameter.

dvmf2 allows for automatic differentiation with RTMB. rvmf2 is imported from movMF::rmovMF.

Value

dvmf gives the density and rvm generates random deviates.

Examples

set.seed(123)
# single parameter set
theta <- c(1,2,3)
x <- rvmf2(1, theta)
d <- dvmf2(x, theta)

# vectorised over parameters
theta <- matrix(theta, nrow = 1)
theta <- theta[rep(1,10), ]
x <- rvmf2(10, theta)
d <- dvmf2(x, theta)

Wishart distribution

Description

Density and random generation for the wishart distribution

Usage

dwishart(x, nu, Sigma, log = FALSE)

rwishart(n, nu, Sigma)

Arguments

x

positive definite p x p matrix of evaluation points

nu

degrees of freedom, needs to be greater than p - 1

Sigma

scale matrix, needs to be positive definite and match the dimension of x.

log

logical; if TRUE, densities p are returned as \log(p).

n

number of random deviates to return

Value

dwishart gives the density, rwishart generates random deviates (matrix for n = 1, array for n > 1)

Examples

x <- rwishart(1, nu = 5, Sigma = diag(3))
d <- dwishart(x, nu = 5, Sigma = diag(3))

wrapped Cauchy distribution

Description

Density and random generation for the wrapped Cauchy distribution.

Usage

dwrpcauchy(x, mu = 0, rho, log = FALSE)

rwrpcauchy(n, mu = 0, rho, wrap = TRUE)

Arguments

x

vector of angles measured in radians at which to evaluate the density function.

mu

mean direction of the distribution measured in radians.

rho

concentration parameter of the distribution, must be in the interval from 0 to 1.

log

logical; if TRUE, densities are returned on the log scale.

n

number of random values to return.

wrap

logical; if TRUE, generated angles are wrapped to the interval from -pi to pi.

Details

This implementation of dwrpcauchy allows for automatic differentiation with RTMB. rwrpcauchy is simply a wrapper for rwrappedcauchyimported from circular.

Value

dwrpcauchy gives the density and rwrpcauchy generates random deviates.

Examples

set.seed(1)
x <- rwrpcauchy(10, 0, 0.5)
d <- dwrpcauchy(x, 0, 0.5)

Zero-inflated density constructer

Description

Constructs a zero-inflated density function from a given probability density function

Usage

zero_inflate(dist, discrete = NULL)

Arguments

dist

either a probability density function or a probability mass function

discrete

logical; if TRUE, the density for x = 0 will be zeroprob + (1-zeroprob) * dist(0, ...). Otherwise it will just be zeroprob. In standard cases, this will be determined automatically. For non-standard cases, set this to TRUE or FALSE depending on the type of dist. See details.

Details

The definition of zero-inflation is different for discrete and continuous distributions. For discrete distributions with p.m.f. f and zero-inflation probability p, we have

\Pr(X = 0) = p + (1 - p) \cdot f(0),

and

\Pr(X = x) = (1 - p) \cdot f(x), \quad x > 0.

For continuous distributions with p.d.f. f, we have

f_{\text{zinfl}}(x) = p \cdot \delta_0(x) + (1 - p) \cdot f(x),

where \delta_0 is the Dirac delta function at zero.

Value

zero-inflated density function with first argument x, second argument zeroprob, and additional arguments ... that will be passed to dist.

Examples

# Zero-inflated normal distribution
dzinorm <- zero_inflate(dnorm)
dzinorm(c(NA, 0, 2), 0.5, mean = 1, sd = 1)

# Zero-inflated Poisson distribution
zipois <- zero_inflate(dpois)
zipois(c(NA, 0, 1), 0.5, 1)

# Non-standard case: Zero-inflated reparametrised beta distribution
dzibeta2 <- zero_inflate(dbeta2, discrete = FALSE)

Zero-inflated beta distribution

Description

Density, distribution function, and random generation for the zero-inflated beta distribution.

Usage

dzibeta(x, shape1, shape2, zeroprob = 0, log = FALSE)

pzibeta(q, shape1, shape2, zeroprob = 0, lower.tail = TRUE, log.p = FALSE)

rzibeta(n, shape1, shape2, zeroprob = 0)

Arguments

x, q

vector of quantiles

shape1, shape2

non-negative shape parameters of the beta distribution

zeroprob

zero-inflation probability between 0 and 1.

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

Value

dzibeta gives the density, pzibeta gives the distribution function, and rzibeta generates random deviates.

Examples

set.seed(123)
x <- rzibeta(1, 2, 2, 0.5)
d <- dzibeta(x, 2, 2, 0.5)
p <- pzibeta(x, 2, 2, 0.5)

Reparameterised zero-inflated beta distribution

Description

Density, distribution function, and random generation for the zero-inflated beta distribution reparameterised in terms of mean and concentration.

Usage

dzibeta2(x, mu, phi, zeroprob = 0, log = FALSE)

pzibeta2(q, mu, phi, zeroprob = 0, lower.tail = TRUE, log.p = FALSE)

rzibeta2(n, mu, phi, zeroprob = 0)

Arguments

x, q

vector of quantiles

mu

mean parameter, must be in the interval from 0 to 1.

phi

concentration parameter, must be positive.

zeroprob

zero-inflation probability between 0 and 1.

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE (default), probabilities are P[X \le x], otherwise P[X > x].

n

number of random values to return.

p

vector of probabilities

Details

This implementation allows for automatic differentiation with RTMB.

Value

dzibeta2 gives the density, pzibeta2 gives the distribution function, and rzibeta2 generates random deviates.

Examples

set.seed(123)
x <- rzibeta2(1, 0.5, 1, 0.5)
d <- dzibeta2(x, 0.5, 1, 0.5)
p <- pzibeta2(x, 0.5, 1, 0.5)

Zero-inflated binomial distribution

Description

Probability mass function, distribution function, and random generation for the zero-inflated binomial distribution.

Usage

dzibinom(x, size, prob, zeroprob = 0, log = FALSE)

pzibinom(q, size, prob, zeroprob = 0, lower.tail = TRUE, log.p = FALSE)

rzibinom(n, size, prob, zeroprob = 0)

Arguments

x, q

vector of quantiles

size

number of trials (zero or more).

prob

probability of success on each trial.

zeroprob

zero-inflation probability between 0 and 1

log, log.p

logical; return log-density if TRUE

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

Value

dzibinom gives the probability mass function, pzibinom gives the distribution function, and rzibinom generates random deviates.

Examples

set.seed(123)
x <- rzibinom(1, size = 10, prob = 0.5, zeroprob = 0.5)
d <- dzibinom(x, size = 10, prob = 0.5, zeroprob = 0.5)
p <- pzibinom(x, size = 10, prob = 0.5, zeroprob = 0.5)

Zero-inflated gamma distribution

Description

Density, distribution function, and random generation for the zero-inflated gamma distribution.

Usage

dzigamma(x, shape, scale, zeroprob = 0, log = FALSE)

pzigamma(q, shape, scale, zeroprob = 0, lower.tail = TRUE, log.p = FALSE)

rzigamma(n, shape, scale, zeroprob = 0)

Arguments

x, q

vector of quantiles

shape

positive shape parameter

scale

positive scale parameter

zeroprob

zero-inflation probability between 0 and 1.

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

n

number of random values to return

Details

This implementation allows for automatic differentiation with RTMB.

Value

dzigamma gives the density, pzigamma gives the distribution function, and rzigamma generates random deviates.

Examples

x <- rzigamma(1, 1, 1, 0.5)
d <- dzigamma(x, 1, 1, 0.5)
p <- pzigamma(x, 1, 1, 0.5)

Zero-inflated and reparameterised gamma distribution

Description

Density, distribution function, and random generation for the zero-inflated gamma distribution reparameterised in terms of mean and standard deviation.

Usage

dzigamma2(x, mean = 1, sd = 1, zeroprob = 0, log = FALSE)

pzigamma2(q, mean = 1, sd = 1, zeroprob = 0)

rzigamma2(n, mean = 1, sd = 1, zeroprob = 0)

Arguments

x, q

vector of quantiles

mean

mean parameter, must be positive.

sd

standard deviation parameter, must be positive.

zeroprob

zero-inflation probability between 0 and 1.

log

logical; if TRUE, probabilities/ densities p are returned as \log(p).

n

number of random values to return

Details

This implementation allows for automatic differentiation with RTMB.

Value

dzigamma2 gives the density, pzigamma2 gives the distribution function, and rzigamma generates random deviates.

Examples

x <- rzigamma2(1, 2, 1, 0.5)
d <- dzigamma2(x, 2, 1, 0.5)
p <- pzigamma2(x, 2, 1, 0.5)

Zero-inflated inverse Gaussian distribution

Description

Density, distribution function, and random generation for the zero-inflated inverse Gaussian distribution.

Usage

dziinvgauss(x, mean = 1, shape = 1, zeroprob = 0, log = FALSE)

pziinvgauss(q, mean = 1, shape = 1, zeroprob = 0, lower.tail = TRUE, log.p = FALSE)

rziinvgauss(n, mean = 1, shape = 1, zeroprob = 0)

Arguments

x, q

vector of quantiles

mean

location parameter

shape

shape parameter, must be positive.

zeroprob

zero-probability, must be in [0, 1].

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

n

number of random values to return

Details

This implementation of zidinvgauss allows for automatic differentiation with RTMB.

Value

dziinvgauss gives the density, pziinvgauss gives the distribution function, and rziinvgauss generates random deviates.

Examples

x <- rziinvgauss(1, 1, 2, 0.5)
d <- dziinvgauss(x, 1, 2, 0.5)
p <- pziinvgauss(x, 1, 2, 0.5)

Zero-inflated log normal distribution

Description

Density, distribution function, and random generation for the zero-inflated log normal distribution.

Usage

dzilnorm(x, meanlog = 0, sdlog = 1, zeroprob = 0, log = FALSE)

pzilnorm(q, meanlog = 0, sdlog = 1, zeroprob = 0,
         lower.tail = TRUE, log.p = FALSE)

rzilnorm(n, meanlog = 0, sdlog = 1, zeroprob = 0)

plnorm(q, meanlog = 0, sdlog = 1, lower.tail = TRUE, log.p = FALSE)

Arguments

x, q

vector of quantiles

meanlog, sdlog

mean and standard deviation of the distribution on the log scale with default values of 0 and 1 respectively.

zeroprob

zero-inflation probability between 0 and 1.

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

n

number of random values to return

Details

This implementation allows for automatic differentiation with RTMB.

Value

dzilnorm gives the density, pzilnorm gives the distribution function, and rzilnorm generates random deviates.

Examples

x <- rzilnorm(1, 1, 1, 0.5)
d <- dzilnorm(x, 1, 1, 0.5)
p <- pzilnorm(x, 1, 1, 0.5)

Zero-inflated negative binomial distribution

Description

Probability mass function, distribution function, quantile function, and random generation for the zero-inflated negative binomial distribution.

Usage

dzinbinom(x, size, prob, zeroprob = 0, log = FALSE)

pzinbinom(q, size, prob, zeroprob = 0, lower.tail = TRUE, log.p = FALSE)

rzinbinom(n, size, prob, zeroprob = 0)

Arguments

x, q

vector of (non-negative integer) quantiles

size

size parameter, must be positive.

prob

mean parameter, must be positive.

zeroprob

zero-inflation probability between 0 and 1.

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

n

number of random values to return.

p

vector of probabilities

Details

This implementation allows for automatic differentiation with RTMB.

Value

dzinbinom gives the density, pzinbinom gives the distribution function, and rzinbinom generates random deviates.

Examples

set.seed(123)
x <- rzinbinom(1, size = 2, prob = 0.5, zeroprob = 0.5)
d <- dzinbinom(x, size = 2, prob = 0.5, zeroprob = 0.5)
p <- pzinbinom(x, size = 2, prob = 0.5, zeroprob = 0.5)

Zero-inflated and reparameterised negative binomial distribution

Description

Probability mass function, distribution function, quantile function and random generation for the zero-inflated negative binomial distribution reparameterised in terms of mean and size.

Usage

dzinbinom2(x, mu, size, zeroprob = 0, log = FALSE)

pzinbinom2(q, mu, size, zeroprob = 0, lower.tail = TRUE, log.p = FALSE)

rzinbinom2(n, mu, size, zeroprob = 0)

Arguments

x, q

vector of (non-negative integer) quantiles

mu

mean parameter, must be positive.

size

size parameter, must be positive.

zeroprob

zero-inflation probability between 0 and 1.

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

n

number of random values to return.

p

vector of probabilities

Details

This implementation allows for automatic differentiation with RTMB.

Value

dzinbinom2 gives the density, pzinbinom2 gives the distribution function, and rzinbinom2 generates random deviates.

Examples

set.seed(123)
x <- rzinbinom2(1, 2, 1, zeroprob = 0.5)
d <- dzinbinom2(x, 2, 1, zeroprob = 0.5)
p <- pzinbinom2(x, 2, 1, zeroprob = 0.5)

Zero-inflated Poisson distribution

Description

Probability mass function, distribution function, and random generation for the zero-inflated Poisson distribution.

Usage

dzipois(x, lambda, zeroprob = 0, log = FALSE)

pzipois(q, lambda, zeroprob = 0, lower.tail = TRUE, log.p = FALSE)

rzipois(n, lambda, zeroprob = 0)

Arguments

x, q

integer vector of counts

lambda

vector of (non-negative) means

zeroprob

zero-inflation probability between 0 and 1

log, log.p

logical; return log-density if TRUE

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

Value

dzipois gives the probability mass function, pzipois gives the distribution function, and rzipois generates random deviates.

Examples

set.seed(123)
x <- rzipois(1, 0.5, 1)
d <- dzipois(x, 0.5, 1)
p <- pzipois(x, 0.5, 1)

Zero- and one-inflated beta distribution

Description

Density, distribution function, and random generation for the zero-one-inflated beta distribution.

Usage

dzoibeta(x, shape1, shape2, zeroprob = 0, oneprob = 0, log = FALSE)

pzoibeta(q, shape1, shape2, zeroprob = 0, oneprob = 0,
         lower.tail = TRUE, log.p = FALSE)

rzoibeta(n, shape1, shape2, zeroprob = 0, oneprob = 0)

Arguments

x, q

vector of quantiles

shape1, shape2

non-negative shape parameters of the beta distribution

zeroprob

zero-inflation probability between 0 and 1.

oneprob

zero-inflation probability between 0 and 1.

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

Value

dzoibeta gives the density, pzoibeta gives the distribution function, and rzoibeta generates random deviates.

Examples

set.seed(123)
x <- rzoibeta(1, 2, 2, 0.2, 0.3)
d <- dzoibeta(x, 2, 2, 0.2, 0.3)
p <- pzoibeta(x, 2, 2, 0.2, 0.3)

Reparameterised zero- and one-inflated beta distribution

Description

Density, distribution function, and random generation for the zero-one-inflated beta distribution reparameterised in terms of mean and concentration.

Usage

dzoibeta2(x, mu, phi, zeroprob = 0, oneprob = 0, log = FALSE)

pzoibeta2(q, mu, phi, zeroprob = 0, oneprob = 0,
         lower.tail = TRUE, log.p = FALSE)

rzoibeta2(n, mu, phi, zeroprob = 0, oneprob = 0)

Arguments

x, q

vector of quantiles

mu

mean parameter, must be in the interval from 0 to 1.

phi

concentration parameter, must be positive.

zeroprob

zero-inflation probability between 0 and 1.

oneprob

zero-inflation probability between 0 and 1.

log, log.p

logical; if TRUE, probabilities/ densities p are returned as \log(p).

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

Value

dzoibeta2 gives the density, pzoibeta2 gives the distribution function, and rzoibeta2 generates random deviates.

Examples

set.seed(123)
x <- rzoibeta2(1, 0.6, 2, 0.2, 0.3)
d <- dzoibeta2(x, 0.6, 2, 0.2, 0.3)
p <- pzoibeta2(x, 0.6, 2, 0.2, 0.3)

Zero-truncated Binomial distribution

Description

Probability mass function, distribution function, and random generation for the zero-truncated Binomial distribution.

Usage

dztbinom(x, size, prob, log = FALSE)

pztbinom(q, size, prob, lower.tail = TRUE, log.p = FALSE)

rztbinom(n, size, prob)

Arguments

x, q

integer vector of counts

size

number of trials

prob

success probability in each trial

log, log.p

logical; return log-density if TRUE

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

By definition, this distribution only has support on the positive integers (1, ..., size). Any zero-truncated distribution is defined as

P(X=x | X>0) = P(X=x) / (1 - P(X=0)),

where P(X=x) is the probability mass function of the corresponding untruncated distribution.

Value

dztbinom gives the probability mass function, pztbinom gives the distribution function, and rztbinom generates random deviates.

Examples

set.seed(123)
x <- rztbinom(1, size = 10, prob = 0.3)
d <- dztbinom(x, size = 10, prob = 0.3)
p <- pztbinom(x, size = 10, prob = 0.3)

Zero-truncated Negative Binomial distribution

Description

Probability mass function, distribution function, and random generation for the zero-truncated Negative Binomial distribution.

Usage

dztnbinom(x, size, prob, log = FALSE)

pztnbinom(q, size, prob, lower.tail = TRUE, log.p = FALSE)

rztnbinom(n, size, prob)

Arguments

x, q

integer vector of counts

size

target for number of successful trials, or dispersion parameter (the shape parameter of the gamma mixing distribution). Must be strictly positive, need not be integer.

prob

probability of success in each trial. 0 < prob <= 1.

log, log.p

logical; return log-density if TRUE

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

By definition, this distribution only has support on the positive integers (1, 2, ...). Any zero-truncated distribution is defined as

P(X=x | X>0) = P(X=x) / (1 - P(X=0)),

where P(X=x) is the probability mass function of the corresponding untruncated distribution.

Value

dztnbinom gives the probability mass function, pztnbinom gives the distribution function, and rztnbinom generates random deviates.

Examples

set.seed(123)
x <- rztnbinom(1, size = 2, prob = 0.5)
d <- dztnbinom(x, size = 2, prob = 0.5)
p <- pztnbinom(x, size = 2, prob = 0.5)

Reparameterised zero-truncated negative binomial distribution

Description

Probability mass function, distribution function, quantile function, and random generation for the zero-truncated negative binomial distribution reparameterised in terms of mean and size.

Usage

dztnbinom2(x, mu, size, log = FALSE)

pztnbinom2(q, mu, size, lower.tail = TRUE, log.p = FALSE)

rztnbinom2(n, mu, size)

Arguments

x, q

integer vector of counts

mu

mean parameter, must be positive

size

size/dispersion parameter, must be positive

log, log.p

logical; return log-density if TRUE

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

By definition, this distribution only has support on the positive integers (1, 2, ...). Any zero-truncated distribution is defined as

P(X=x | X>0) = P(X=x) / (1 - P(X=0)),

where P(X=x) is the probability mass function of the corresponding untruncated distribution.

Value

dztnbinom2 gives the probability mass function, pztnbinom2 gives the distribution function, and rztnbinom2 generates random deviates.

Examples

set.seed(123)
x <- rztnbinom2(1, mu = 2, size = 1)
d <- dztnbinom2(x, mu = 2, size = 1)
p <- pztnbinom2(x, mu = 2, size = 1)

Zero-truncated Poisson distribution

Description

Probability mass function, distribution function, and random generation for the zero-truncated Poisson distribution.

Usage

dztpois(x, lambda, log = FALSE)

pztpois(q, lambda, lower.tail = TRUE, log.p = FALSE)

rztpois(n, lambda)

Arguments

x, q

integer vector of counts

lambda

vector of (non-negative) means

log, log.p

logical; return log-density if TRUE

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

By definition, this distribution only has support on the positive integers (1, 2, ...). Any zero-truncated distribution is defined as

P(X=x | X>0) = P(X=x) / (1 - P(X=0)),

where P(X=x) is the probability mass function of the corresponding untruncated distribution.

Value

dztpois gives the probability mass function, pztpois gives the distribution function, and rztpois generates random deviates.

Examples

set.seed(123)
x <- rztpois(1, 0.5)
d <- dztpois(x, 0.5)
p <- pztpois(x, 0.5)

Smooth approximation to the absolute value function

Description

Usage

Arguments

Details

Value

Examples

Box–Cox Cole and Green distribution (BCCG)

Description

Usage

Arguments

Details

Value

References

Examples

Box-Cox Power Exponential distribution (BCPE)

Description

Usage

Arguments

Details

Value

References

Examples

Box–Cox t distribution (BCT)

Description

Usage

Arguments

Details

Value

References

Examples

Reparameterised beta distribution

Description

Usage

Arguments

Details

Value

Examples

Beta-binomial distribution

Description

Usage

Arguments

Details

Value

Examples

Clayton copula constructors

Description

Usage

Arguments

Details

Value

See Also

Examples

Frank copula constructor

Description

Usage

Arguments

Details

Value

See Also

Examples

Gaussian copula constructor

Description

Usage

Arguments

Value

See Also

Examples

Multivariate Gaussian copula constructor parameterised by inverse correlation matrix

Description

Usage

Arguments

Details

Value

See Also

Examples

Gumbel copula constructors

Description

Usage

Arguments