Package {ddst}

Title:	Data Driven Smooth Tests
Version:	1.6.11
Description:	Smooth tests are data driven (alternative hypothesis is dynamically selected based on data). In this package you will find two groups of smooth of test: goodness-of-fit tests and nonparametric tests for comparing distributions. Among goodness-of-fit tests there are tests for exponent, Gaussian, Gumbel and uniform distribution. Among nonparametric tests there are tests for stochastic dominance, k-sample test, test with umbrella alternatives and test for change-point problems.
License:	GPL-2
Encoding:	UTF-8
Depends:	R (≥ 2.7), polynom
Imports:	ggplot2, orthopolynom, evd
Suggests:	rmutil, knitr
URL:	https://pbiecek.github.io/ddst/
BugReports:	https://github.com/pbiecek/ddst/issues
VignetteBuilder:	knitr
RoxygenNote:	7.3.3
NeedsCompilation:	no
Packaged:	2026-09-03 20:06:00 UTC; pbiecek
Author:	Przemyslaw Biecek [aut, cre], Teresa Ledwina [aut], Grzegorz Wylupek [aut]
Maintainer:	Przemyslaw Biecek <przemyslaw.biecek@gmail.com>
Repository:	CRAN
Date/Publication:	2026-09-12 05:10:02 UTC

Data Driven Smooth Test Against Stochastic Dominance

Description

Performs data driven smooth non-parametric two-sample test against one-sided alternatives (stochastic dominance). Suppose that we have random samples from two distributions F and G. The null hypothesis is that F(x) < G(x) for some x while the alternative is that at F(x) >= G(x) for all x with strict inequality for at least one x. Detailed description of the test statistic is provided in Ledwina and Wylupek (2012).

Usage

ddst.againststochdom.test(
  x,
  y,
  k.N = 4,
  alpha = 0.05,
  t,
  nr = 1e+05,
  compute.cv = FALSE
)

Arguments

References

Two-sample test against one-sided alternatives. Ledwina and Wylupek (2012). https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1467-9469.2011.00787.x

Examples

set.seed(7)
# H0 is true
x <- runif(80)
y <- runif(80)
t <- ddst.againststochdom.test(x, y, alpha = 0.05, t = 2.2, k.N = 4)
t
plot(t)

# H0 is false
# known fixed alternative
x <- runif(80)
y <- rbeta(80,4,2)
t <- ddst.againststochdom.test(x, y, alpha = 0.05, t = 2.2, k.N = 4)
t
plot(t)

Data Driven Smooth Test for Extreme Value Distribution

Description

Performs data driven smooth test for composite hypothesis of extreme value distribution. Null density is given by f(z;\gamma)=1/\gamma_2 \exp((z-\gamma_1)/\gamma_2- \exp((z-\gamma_1)/\gamma_2)), z \in R.

Usage

ddst.evd.test(
  x,
  base = ddst.base.legendre,
  d.n = 10,
  c = 100,
  nr = 1e+05,
  compute.p = TRUE,
  alpha = 0.05,
  compute.cv = TRUE,
  ...
)

Arguments

Details

We model alternatives similarly as in Kallenberg and Ledwina (1997) and Janic-Wroblewska (2004) using Legendre's polynomials or cosines. For more details see: http://www.biecek.pl/R/ddst/description.pdf.

Value

An object of class htest

References

Hosking, J.R.M., Wallis, J.R., Wood, E.F. (1985). Estimation of the generalized extreme-value distribution by the method of probability-weighted moments. Technometrics 27, 251–261.

Janic-Wroblewska, A. (2004). Data-driven smooth test for extreme value distribution. Statistics 38, 413–426.

Janic, A. and Ledwina, T. (2008). Data-driven tests for a location-scale family revisited. J. Statist. Theory. Pract. Special issue on Modern Goodness of Fit Methods. accepted..

Kallenberg, W.C.M., Ledwina, T. (1997). Data driven smooth tests for composite hypotheses: Comparison of powers. J. Statist. Comput. Simul. 59, 101–121.

Examples

library(evd)
set.seed(7)

# for given vector of 19 numbers
z <- c(13.41, 6.04, 1.26, 3.67, -4.54, 2.92, 0.44, 12.93, 6.77, 10.09,
      4.10, 4.04, -1.97, 2.17, -5.38, -7.30, 4.75, 5.63, 8.84)
## Not run: 
t <- ddst.evd.test(z, compute.p = TRUE, d.n = 10)
t
plot(t)

# H0 is true
x <- -qgumbel(runif(100),-1,1)
t <- ddst.evd.test (x, compute.p = TRUE, d.n = 10)
t
plot(t)

# H0 is false
x <- rexp(80,4)
t <- ddst.evd.test (x, compute.p = TRUE, d.n = 10)
t
plot(t)

## End(Not run)

Data Driven Smooth Test for Exponentiality

Description

Performs data driven smooth test for composite hypothesis of exponentiality. Null density is given by f(z;gamma) = exp(-z/gamma) for z >= 0 and 0 otherwise. Modelling alternatives similarly as in Kallenberg and Ledwina (1997 a,b).

Usage

ddst.exp.test(
  x,
  base = ddst.base.legendre,
  d.n = 10,
  c = 100,
  nr = 1e+05,
  compute.p = TRUE,
  alpha = 0.05,
  compute.cv = TRUE,
  ...
)

Arguments

Value

An object of class htest

References

Kallenberg, W.C.M., Ledwina, T. (1997 a). Data driven smooth tests for composite hypotheses: Comparison of powers. J. Statist. Comput. Simul. 59, 101–121.

Kallenberg, W.C.M., Ledwina, T. (1997 b). Data driven smooth tests when the hypothesis is composite. J. Amer. Statist. Assoc. 92, 1094–1104.

Examples

set.seed(7)
# H0 is true
z <- rexp(80,4)
## Not run: 
t <- ddst.exp.test (z, compute.p = TRUE, d.n = 10)
t
plot(t)

# H0 is false
z = rchisq(80,4)
(t = ddst.exp.test (z, compute.p = TRUE, d.n = 10))
t$p.value
plot(t)

## End(Not run)

Data Driven Smooth Test for Stochastic Dominance in Two Samples

Description

Performs the data driven smooth test for detection of the stochastic ordering, as described in detail in Ledwina and Wyłupek (2012). Suppose that we have random samples from two distributions F and G. The null hypothesis is that F(x) >= G(x) for all x while the alternative is that at F(x) < G(x) for some x. Detailed description of the test statistic is provided in Ledwina and Wylupek (2012).

Usage

ddst.forstochdom.test(
  x,
  y,
  K.N = floor(log(length(x) + length(y), 2)) - 1,
  alpha = 0.05,
  t,
  nr = 1e+05,
  compute.p = TRUE,
  compute.cv = TRUE
)

Arguments

References

Nonparametric tests for stochastic ordering. Ledwina and Wyłupek (2012) <doi:10.1007/s11749-011-0278-7>

Examples

set.seed(7)
library("rmutil", warn.conflicts = FALSE)
# 1. Pareto(1)/Pareto(1.5)
# H0 is false
x <- rpareto(50, 2, 2)
y <- rpareto(50, 1.5, 1.5)
t <- ddst.forstochdom.test(x, y, t = 2.2, K.N = 4)
t
plot(t)

# 2. Laplace(0,1)/Laplace(1,25)
# H0 is false
x <- rlaplace(50, 0, 1)
y <- rlaplace(50, 1, 25)
t <- ddst.forstochdom.test(x, y, t = 2.2, K.N = 4)
t
plot(t)

# 3. LN(0.85,0.6)/LN(1.2,0.2)
# H0 is true
x <- rlnorm(50, 0.85, 0.6)
y <- rlnorm(50, 1.2, 0.2)
t <- ddst.forstochdom.test(x, y, t = 2.2, K.N = 4)
t
plot(t)

## Not run: 
# Generate distribution of test statistic
N <- 1000
samp <- replicate(N, {
   x <- runif(30)
   y <- runif(30)
   # statistics with Schwartz penalty
   ddst.forstochdom.test(x, y)$statistic
})
quantile(samp, 0.95)
plot(ecdf(samp))

## End(Not run)

Data Driven Smooth Test for k-Sample Problem

Description

Performs data driven smooth test for the classical k-sample problem. Suppose that we have random samples from k distributions F_i where i = 1, ..., k. The null hypothesis is that F_1 = ... = F_k while the alternative is that at least two distributions are different. Detailed description of the test statistic is provided in Wylupek (2010).

Usage

ddst.ksample.test(
  x,
  d.N = 12,
  c = 2.3,
  nr = 1e+05,
  compute.p = TRUE,
  alpha = 0.05,
  compute.cv = TRUE
)

Arguments

References

Data-driven k-sample tests. Wylupek (2010) https://www.jstor.org/stable/40586684?seq=1

Examples

set.seed(7)
# H0 is false
x <- runif(80)
y <- rexp(80, 1)
z <- runif(80)
t <- ddst.ksample.test(list(x, y, z))
t
plot(t)

# H0 is true
x <- runif(80)
y <- runif(80)
z <- runif(80)
t <- ddst.ksample.test(list(x, y, z))
t
plot(t)

Data Driven Smooth Test for Normality; Bounded Basis Functions

Description

Performs data driven smooth test for composite hypothesis of normality Null density is given by f(z;\gamma)=1/(\sqrt{2 \pi}\gamma_2) \exp(-(z-\gamma_1)^2/(2 \gamma_2^2)) for z \in R. We model alternatives similarly as in Kallenberg and Ledwina (1997 a,b) using Legendre's polynomials or cosine basis.

Usage

ddst.normbounded.test(
  x,
  base = ddst.base.legendre,
  d.n = 10,
  c = 100,
  compute.p = TRUE,
  alpha = 0.05,
  compute.cv = TRUE,
  ...
)

Arguments

Value

An object of class htest

References

Chen, L., Shapiro, S.S. (1995). An alternative test for normality based on normalized spacings. J. Statist. Comput. Simulation 53, 269–288.

Inglot, T., Ledwina, T. (2006). Towards data driven selection of a penalty function for data driven Neyman tests. Linear Algebra and its Appl. 417, 579–590.

Janic, A. and Ledwina, T. (2008). Data-driven tests for a location-scale family revisited. J. Statist. Theory. Pract. Special issue on Modern Goodness of Fit Methods..

Kallenberg, W.C.M., Ledwina, T. (1997 a). Data driven smooth tests for composite hypotheses: Comparison of powers. J. Statist. Comput. Simul. 59, 101–121.

Kallenberg, W.C.M., Ledwina, T. (1997 b). Data driven smooth tests when the hypothesis is composite. J. Amer. Statist. Assoc. 92, 1094–1104.

Examples

set.seed(7)
# H0 is true
z <- rnorm(100)
# let's look on first 10 coordinates
d.n <- 10
t <- ddst.normbounded.test(z, compute.p = TRUE, d.n = d.n)
t
plot(t)

# H0 is false
z <- rexp(100, 1)
t <- ddst.normbounded.test(z, compute.p = TRUE, d.n = d.n)
t
plot(t)

# for Tephra data
z <- c(-1.748789, -1.75753, -1.740102, -1.740102, -1.731467, -1.765523,
      -1.761521, -1.72522, -1.80371, -1.745624, -1.872957, -1.729121,
      -1.81529, -1.888637, -1.887761, -1.881645, -1.91518, -1.849769,
      -1.755141, -1.665687, -1.764721, -1.736171, -1.736956, -1.737742,
      -1.687537, -1.804534, -1.790593, -1.808661, -1.784081, -1.729903,
      -1.711263, -1.748789, -1.772755, -1.72756, -1.71358, -1.821116,
      -1.839588, -1.839588, -1.830321, -1.807835, -1.747206, -1.788147,
      -1.759923, -1.786519, -1.726779, -1.738528, -1.754345, -1.781646,
      -1.641949, -1.755936, -1.775175, -1.736956, -1.705103, -1.743255,
      -1.82613, -1.826967, -1.780025, -1.684504, -1.751168)
t <- ddst.normbounded.test(z, compute.p = TRUE, d.n = d.n)
t
plot(t)

Data Driven Smooth Test for Normality; Unbounded Basis Functions

Description

Performs data driven smooth test for composite hypothesis of normality. Null density is given by f(z;\gamma)=1/(\sqrt{2 \pi}\gamma_2) \exp(-(z-\gamma_1)^2/(2 \gamma_2^2)) for z \in R.

Usage

ddst.normubounded.test(
  x,
  d.n = 20,
  e.0,
  v.0,
  r.alpha,
  s.n.alpha,
  alpha = 0.05,
  nr = 10000,
  compute.cv = TRUE
)

ddst.normunbounded.bias(n = 100, d.n = 20, nr = 10000)

Arguments

References

Ledwina, T., Wyłupek, G. (2015) Detection of non-Gaussianity by Ledwina and Wyłupek Journal of Statistical Computation and Simulation 17, 3480-3497.

Examples

set.seed(7)
# H0 is true
z <- rnorm(100)
# let's look on first 20 coordinates
d.n <- 20

## Not run: 
# calculate finite sample corrections
# see 6.2. Composite null hypothesis H in the appendix materials
e.v <- ddst.normunbounded.bias(n = length(z))
e.v

# simulated 1-alpha qunatiles, s(n, alpha) and  s.o(n, alpha)
# see Table 1 in the JSCS article
s <- 4.4
r.alpha <- 2.708

t <- ddst.normubounded.test(z, d.n, e.v$e.0, e.v$v.0, r.alpha, s)
t
plot(t)

# for Tephra data
z <- c(-1.748789, -1.75753, -1.740102, -1.740102, -1.731467, -1.765523,
       -1.761521, -1.72522, -1.80371, -1.745624, -1.872957, -1.729121,
       -1.81529, -1.888637, -1.887761, -1.881645, -1.91518, -1.849769,
       -1.755141, -1.665687, -1.764721, -1.736171, -1.736956, -1.737742,
       -1.687537, -1.804534, -1.790593, -1.808661, -1.784081, -1.729903,
       -1.711263, -1.748789, -1.772755, -1.72756, -1.71358, -1.821116,
       -1.839588, -1.839588, -1.830321, -1.807835, -1.747206, -1.788147,
       -1.759923, -1.786519, -1.726779, -1.738528, -1.754345, -1.781646,
       -1.641949, -1.755936, -1.775175, -1.736956, -1.705103, -1.743255,
       -1.82613, -1.826967, -1.780025, -1.684504, -1.751168)

# calculate finite sample corrections
e.v <- ddst.normunbounded.bias(n = length(z))
e.v

s <- 3.3
so <- 3.6
r.alpha <- 2.142

t <- ddst.normubounded.test(z, d.n, e.v$e.0, e.v$v.0, r.alpha, s)
t
plot(t)

## End(Not run)

Data Driven Smooth Test for Two-Sample Problem

Description

Performs data driven smooth test for the classical two-sample problem. It is a special case of data driven test for k-samples. Detailed description of the test statistic is provided in Wylupek (2010).

Usage

ddst.twosample.test(
  x,
  y,
  d.N = 12,
  c = 2,
  B = 1e+05,
  compute.p = TRUE,
  alpha = 0.05,
  compute.cv = TRUE
)

Arguments

References

Data-driven k-sample tests. Wylupek (2010) https://www.jstor.org/stable/40586684?seq=1

Examples

set.seed(7)
# H0 is true
x <- runif(80)
y <- runif(80)
t <- ddst.twosample.test(x, y)
t
plot(t)

# H0 is false
x <- runif(80)
y <- rexp(80, 1)
t <- ddst.twosample.test(x, y)
t
plot(t)

Data Driven Smooth Test for Umbrella Alternatives; Known Peak

Description

Performs data driven smooth test for so-called umbrella alternatives in k-sample problem. Suppose that we have random samples from k distributions F_i where i = 1, ..., k. The null hypothesis is that there is no umbrella pattern, i.e. F_1 >= ... >= F_p <= ... <= F_k and F_i != F_j for some i and j. The alternative is that there is an umbrella pattern i.e. F_1 >= ... >= F_p <= ... <= F_k and F_i != F_j for some i and j. Detailed description of the test statistic is provided in Wylupek (2016).

Usage

ddst.umbrellaknownp.test(
  x,
  p,
  r.N = rep(4, length(x) - 1),
  alpha = 0.05,
  t.p,
  t.n,
  nr = 1e+05,
  compute.cv = TRUE
)

Arguments

References

An automatic test for the umbrella alternatives. Wylupek (2016) https://onlinelibrary.wiley.com/doi/abs/10.1111/sjos.12231

Examples

set.seed(7)
# H0 is true
x = runif(80)
y = runif(80) + 0.2
z = runif(80)
t <- ddst.umbrellaknownp.test(list(x, y, z), p = 2, t.p = 2.2, t.n = 2.2)
t
plot(t)

# known fixed alternative
x1 = rnorm(80)
x2 = rnorm(80) + 2
x3 = rnorm(80) + 4
x4 = rnorm(80) + 3
x5 = rnorm(80) + 2
x6 = rnorm(80) + 1
x7 = rnorm(80)
t <- ddst.umbrellaknownp.test(list(x1, x2, x3, x4, x5, x6, x7), p = 3, t.p = 2.2, t.n = 2.2)
t
plot(t)

t <- ddst.umbrellaknownp.test(list(x1, x2, x3, x4, x5, x6, x7), p = 5, t.p = 2.2, t.n = 2.2)
t
plot(t)

Data Driven Smooth Test for Umbrella Alternatives; Unknown Peak

Description

Data Driven Smooth Test for Umbrella Alternatives; Unknown Peak

Usage

ddst.umbrellaunknownp.test(
  x,
  r.N = rep(4, length(x) - 1),
  alpha = 0.05,
  t.p.aux,
  t.p,
  t.n,
  nr = 1e+05,
  compute.cv = TRUE
)

Arguments

References

An automatic test for the umbrella alternatives. Wylupek (2016) https://onlinelibrary.wiley.com/doi/abs/10.1111/sjos.12231

Examples

set.seed(7)
# H0 is true
x = runif(80)
y = runif(80) + 0.2
z = runif(80)
t <- ddst.umbrellaknownp.test(list(x, y, z), p = 2, t.p = 2.2, t.n = 2.2)
t
plot(t)

# known fixed alternative
x1 = rnorm(80)
x2 = rnorm(80) + 2
x3 = rnorm(80) + 4
x4 = rnorm(80) + 3
x5 = rnorm(80) + 2
x6 = rnorm(80) + 1
x7 = rnorm(80)
t <- ddst.umbrellaknownp.test(list(x1, x2, x3, x4, x5, x6, x7), p = 3, t.p = 2.2, t.n = 2.2)
t
plot(t)

t <- ddst.umbrellaknownp.test(list(x1, x2, x3, x4, x5, x6, x7), p = 5, t.p = 2.2, t.n = 2.2)
t
plot(t)

Data Driven Smooth Test for Uniformity

Description

Performs data driven smooth tests for simple hypothesis of uniformity on [0,1]. Embeding null model into the original exponential family introduced by Neyman (1937).

Usage

ddst.uniform.test(
  x,
  base = ddst.base.legendre,
  d.n = 10,
  c = 2.4,
  nr = 1e+05,
  compute.p = TRUE,
  alpha = 0.05,
  compute.cv = TRUE,
  ...
)

Arguments

Value

An object of class htest

References

Inglot, T., Ledwina, T. (2006). Towards data driven selection of a penalty function for data driven Neyman tests. Linear Algebra and its Appl. 417, 579–590.

Ledwina, T. (1994). Data driven version of Neyman's smooth test of fit. J. Amer. Statist. Assoc. 89 1000-1005.

Neyman, J. (1937). ‘Smooth test’ for goodness of fit. Skand. Aktuarietidskr. 20, 149-199.

Examples

set.seed(7)
# H0 is true
z <- runif(80)
## Not run: 
t <- ddst.uniform.test(z, compute.p = TRUE, d.n = 10)
t
plot(t)

# known fixed alternative
z <- rnorm(80,10,16)
t <- ddst.uniform.test(pnorm(z, 10, 16), compute.p = TRUE, d.n = 10)
t
plot(t)

# H0 is false
z <- rbeta(80,4,2)
(t <- ddst.uniform.test(z, compute.p = TRUE, d.n = 10))
t$p.value
plot(t)

## End(Not run)

Data Driven Smooth Test for Upward Trend Alternatives

Description

Performs data driven smooth test for upward trend in k-sample problem. Suppose that we have random samples from k distributions F_i where i = 1, ..., k. The null hypothesis is that there is lack of trend, i.e. F_1 >= ... >= F_k and F_i != F_j for some i and j. The alternative is that there is a trend i.e. F_1 >= ... >= F_k and F_i != F_j for some i and j. This test is implemented as a special case of an umbrella test.

Usage

ddst.upwardtrend.test(
  x,
  r.N = rep(4, length(x) - 1),
  alpha = 0.05,
  t.p,
  t.n,
  nr = 1e+05,
  compute.cv = TRUE
)

Arguments

References

An automatic test for the umbrella alternatives. Wylupek (2016) https://onlinelibrary.wiley.com/doi/abs/10.1111/sjos.12231

Examples

set.seed(7)
# H0 is true
x = runif(80)
y = runif(80) + 0.2
z = runif(80) + 0.4
t <- ddst.upwardtrend.test(list(x, y, z), t.p = 2.2, t.n = 2.2)
t
plot(t)

# H0 is false
# known fixed alternative
x1 = rnorm(80)
x2 = rnorm(80) + 2
x3 = rnorm(80) + 4
x4 = rnorm(80) + 3
t <- ddst.upwardtrend.test(list(x1, x2, x3, x4), t.p = 2.2, t.n = 2.2)
t
plot(t)

Plot Function fo Data Driven Tests

Description

Plots coordinates for selected test statistics

Usage

## S3 method for class 'ddst.test'
plot(x, ...)

Arguments

Examples

# H0 is true
x <- runif(80)
y <- runif(80)
z <- runif(80)
t <- ddst.ksample.test(list(x, y, z))
plot(t)