| Title: | Data Driven Smooth Tests |
|---|---|
| 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. |
| Authors: | Przemyslaw Biecek [aut, cre], Teresa Ledwina [aut], Grzegorz Wylupek [aut] |
| Maintainer: | Przemyslaw Biecek <[email protected]> |
| License: | GPL-2 |
| Version: | 1.6.10 |
| Built: | 2024-09-15 06:04:59 UTC |
| Source: | https://github.com/pbiecek/ddst |
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).
ddst.againststochdom.test( x, y, k.N = 4, alpha = 0.05, t, nr = 1e+05, compute.cv = FALSE )ddst.againststochdom.test( x, y, k.N = 4, alpha = 0.05, t, nr = 1e+05, compute.cv = FALSE )
x |
a (non-empty) numeric vector of data |
y |
a (non-empty) numeric vector of data |
k.N |
an integer specifying a level of complexity of the grid considered, only for advanced users |
alpha |
a significance level |
t |
an alpha-dependent tunning parameter in the penalty in the model selection rule |
nr |
an integer specifying the number of runs for a p-value and a critical value computation if any |
compute.cv |
a logical value indicating whether to compute a critical value corresponding to the significance level alpha or not |
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
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)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)
Performs data driven smooth test for composite hypothesis of extreme value distribution.
Null density is given by
, .
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, ... )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, ... )
x |
a (non-empty) numeric vector of data values |
base |
a function which returns an orthonormal system, possible choice: |
d.n |
an integer specifying the maximum dimension considered, only for advanced users |
c |
a calibrating parameter in the penalty in the model selection rule |
nr |
an integer specifying the number of runs for a p-value and a critical value computation if any |
compute.p |
a logical value indicating whether to compute a p-value or not |
alpha |
a significance level |
compute.cv |
a logical value indicating whether to compute a critical value corresponding to the significance level alpha or not |
... |
further arguments |
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.
An object of class htest
statistic |
the value of the test statistic. |
parameter |
the number of choosen coordinates (k). |
method |
a character string indicating the parameters of performed test. |
data.name |
a character string giving the name(s) of the data. |
p.value |
the p-value for the test, computed only if |
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. 27, 251–261.
Janic-Wroblewska, A. (2004). Data-driven smooth test for extreme value distribution. 38, 413–426.
Janic, A. and Ledwina, T. (2008). Data-driven tests for a location-scale family revisited. .
Kallenberg, W.C.M., Ledwina, T. (1997). Data driven smooth tests for composite hypotheses: Comparison of powers. 59, 101–121.
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)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)
Performs data driven smooth test for composite hypothesis of exponentiality.
Null density is given by for z >= 0 and 0 otherwise.
Modelling alternatives similarly as in Kallenberg and Ledwina (1997 a,b).
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, ... )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, ... )
x |
a (non-empty) numeric vector of data values |
base |
a function which returns an orthonormal system, possible choice: |
d.n |
an integer specifying the maximum dimension considered, only for advanced users |
c |
a calibrating parameter in the penalty in the model selection rule |
nr |
an integer specifying the number of runs for a p-value and a critical value computation if any |
compute.p |
a logical value indicating whether to compute a p-value or not |
alpha |
a significance level |
compute.cv |
a logical value indicating whether to compute a critical value corresponding to the significance level alpha or not |
... |
further arguments |
An object of class htest
statistic |
the value of the test statistic. |
parameter |
the number of choosen coordinates (k). |
method |
a character string indicating the parameters of performed test. |
data.name |
a character string giving the name(s) of the data. |
p.value |
the p-value for the test, computed only if |
Kallenberg, W.C.M., Ledwina, T. (1997 a). Data driven smooth tests for composite hypotheses: Comparison of powers. 59, 101–121.
Kallenberg, W.C.M., Ledwina, T. (1997 b). Data driven smooth tests when the hypothesis is composite. 92, 1094–1104.
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$p.value plot(t) ## End(Not run)set.seed(7) # H0 is true z <- rexp(80,4) ## Not run: t <- ddst.exp.test (z, compute.p = TRUE, d.n = 10) t plot(t) # H0 is false z = rchisq(80,4) (t = ddst.exp.test (z, compute.p = TRUE, d.n = 10)) t$p.value plot(t) ## End(Not run)
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).
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 )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 )
x |
a (non-empty) numeric vector of data |
y |
a (non-empty) numeric vector of data |
K.N |
an integer specifying a level of complexity of the grid considered, only for advanced users |
alpha |
a significance level |
t |
an alpha-dependent tunning parameter in the penalty in the model selection rule |
nr |
an integer specifying the number of runs for a p-value and a critical value computation if any |
compute.p |
a logical value indicating whether to compute a p-value or not |
compute.cv |
a logical value indicating whether to compute a critical value corresponding to the significance level alpha or not |
Nonparametric tests for stochastic ordering. Ledwina and Wyłupek (2012) <doi:10.1007/s11749-011-0278-7>
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)$statistic }) quantile(samp, 0.95) plot(ecdf(samp)) ## End(Not run)set.seed(7) library("rmutil", warn.conflicts = FALSE) # 1. Pareto(1)/Pareto(1.5) # H0 is false x <- rpareto(50, 2, 2) y <- rpareto(50, 1.5, 1.5) t <- ddst.forstochdom.test(x, y, t = 2.2, K.N = 4) t plot(t) # 2. Laplace(0,1)/Laplace(1,25) # H0 is false x <- rlaplace(50, 0, 1) y <- rlaplace(50, 1, 25) t <- ddst.forstochdom.test(x, y, t = 2.2, K.N = 4) t plot(t) # 3. LN(0.85,0.6)/LN(1.2,0.2) # H0 is true x <- rlnorm(50, 0.85, 0.6) y <- rlnorm(50, 1.2, 0.2) t <- ddst.forstochdom.test(x, y, t = 2.2, K.N = 4) t plot(t) ## Not run: # Generate distribution of test statistic N <- 1000 samp <- replicate(N, { x <- runif(30) y <- runif(30) # statistics with Schwartz penalty ddst.forstochdom.test(x, y)$statistic }) quantile(samp, 0.95) plot(ecdf(samp)) ## End(Not run)
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).
ddst.ksample.test( x, d.N = 12, c = 2.3, nr = 1e+05, compute.p = TRUE, alpha = 0.05, compute.cv = TRUE )ddst.ksample.test( x, d.N = 12, c = 2.3, nr = 1e+05, compute.p = TRUE, alpha = 0.05, compute.cv = TRUE )
x |
a list of k (non-empty) numeric vectors of data |
d.N |
an integer specifying the maximum dimension considered, only for advanced users |
c |
a calibrating parameter in the penalty in the model selection rule |
nr |
an integer specifying the number of runs for a p-value and a critical value computation if any |
compute.p |
a logical value indicating whether to compute a p-value or not |
alpha |
a significance level |
compute.cv |
a logical value indicating whether to compute a critical value corresponding to the significance level alpha or not |
Data-driven k-sample tests. Wylupek (2010) https://www.jstor.org/stable/40586684?seq=1
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)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)
Performs data driven smooth test for composite hypothesis of normality
Null density is given by
for .
We model alternatives similarly as in Kallenberg and Ledwina (1997 a,b) using Legendre's polynomials or cosine basis.
ddst.normbounded.test( x, base = ddst.base.legendre, d.n = 10, c = 100, compute.p = TRUE, alpha = 0.05, compute.cv = TRUE, ... )ddst.normbounded.test( x, base = ddst.base.legendre, d.n = 10, c = 100, compute.p = TRUE, alpha = 0.05, compute.cv = TRUE, ... )
x |
a (non-empty) numeric vector of data values |
base |
a function which returns an orthonormal system, possible choice: |
d.n |
an integer specifying the maximum dimension considered, only for advanced users |
c |
a calibrating parameter in the penalty in the model selection rule |
compute.p |
a logical value indicating whether to compute a p-value or not |
alpha |
a significance level |
compute.cv |
a logical value indicating whether to compute a critical value corresponding to the significance level alpha or not |
... |
further arguments |
An object of class htest
statistic |
the value of the test statistic. |
parameter |
the number of choosen coordinates (k). |
method |
a character string indicating the parameters of performed test. |
data.name |
a character string giving the name(s) of the data. |
p.value |
the p-value for the test, computed only if |
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.
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)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)
Performs data driven smooth test for composite hypothesis of normality.
Null density is given by
for .
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)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)
x |
a (non-empty) numeric vector of data values |
d.n |
an integer specifying the maximum dimension considered, only for advanced users |
e.0 |
a (non-empty) numeric vector being the Monte Carlo estimate of the mean of the vector (C2; ... ;Cd.n) calculated using the function |
v.0 |
a (non-empty) numeric vector being the Monte Carlo estimate of the variance of the vector (C2; ... ;Cd.n) calculated using the function |
r.alpha |
a critical value of the alpha level R.n test |
s.n.alpha |
a penalty in the auxiliary model selection rule |
alpha |
a significance level |
nr |
an integer specifying the number of runs for a critical value computation |
compute.cv |
a logical value indicating whether to compute a critical value corresponding to the significance level alpha or not |
n |
sample size |
Ledwina, T., Wyłupek, G. (2015) Detection of non-Gaussianity by Ledwina and Wyłupek Journal of Statistical Computation and Simulation 17, 3480-3497.
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)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)
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).
ddst.twosample.test( x, y, d.N = 12, c = 2, B = 1e+05, compute.p = TRUE, alpha = 0.05, compute.cv = TRUE )ddst.twosample.test( x, y, d.N = 12, c = 2, B = 1e+05, compute.p = TRUE, alpha = 0.05, compute.cv = TRUE )
x |
a (non-empty) numeric vector of data |
y |
a (non-empty) numeric vector of data |
d.N |
an integer specifying the maximum dimension considered, only for advanced users |
c |
a calibrating parameter in the penalty in the model selection rule |
B |
an integer specifying the number of runs for a p-value and a critical value computation if any |
compute.p |
a logical value indicating whether to compute a p-value or not |
alpha |
a significance level |
compute.cv |
a logical value indicating whether to compute a critical value corresponding to the significance level alpha or not |
Data-driven k-sample tests. Wylupek (2010) https://www.jstor.org/stable/40586684?seq=1
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)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)
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).
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 )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 )
x |
a list of k (non-empty) numeric vectors of data |
p |
a peak of the umbrella pattern |
r.N |
a (p(p-1)=2+(k-p)(k-p+1)/2)-dimensional vector specifying the levels of complexity of the grids considered, only for advanced users |
alpha |
a significance level |
t.p |
an alpha-dependent (p(p-1)=2+(k-p)(k-p+1)/2)-dimensional vector of the tunning parameters in the penalties in the model selection rules T.o |
t.n |
an alpha-dependent (k-1)-dimensional vector of the tunning parameters in the penalties in the model selection rules T.tilde |
nr |
an integer specifying the number of runs for a p-value and a critical value computation if any |
compute.cv |
a logical value indicating whether to compute a critical value corresponding to the significance level alpha or not |
An automatic test for the umbrella alternatives. Wylupek (2016) https://onlinelibrary.wiley.com/doi/abs/10.1111/sjos.12231
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)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
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 )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 )
x |
a list of k (non-empty) numeric vectors of data |
r.N |
a (p(p-1)=2+(k-p)(k-p+1)/2)-dimensional vector specifying the levels of complexity of the grids considered, only for advanced users |
alpha |
a significance level |
t.p.aux |
an auxiliary alpha-dependent k x (k - 1) matrix of the tunning parameters in the penalties in the model selection rules To aux employed for estimation of the peak |
t.p |
an alpha-dependent (p(p-1)=2+(k-p)(k-p+1)/2)-dimensional vector of the tunning parameters in the penalties in the model selection rules T.o |
t.n |
an alpha-dependent (k-1)-dimensional vector of the tunning parameters in the penalties in the model selection rules T.tilde |
nr |
an integer specifying the number of runs for a p-value and a critical value computation if any |
compute.cv |
a logical value indicating whether to compute a critical value corresponding to the significance level alpha or not |
An automatic test for the umbrella alternatives. Wylupek (2016) https://onlinelibrary.wiley.com/doi/abs/10.1111/sjos.12231
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)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)
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).
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, ... )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, ... )
x |
a (non-empty) numeric vector of data |
base |
a function which returns an orthonormal system, possible choice: |
d.n |
an integer specifying the maximum dimension considered, only for advanced users |
c |
a calibrating parameter in the penalty in the model selection rule |
nr |
an integer specifying the number of runs for a p-value and a critical value computation if any |
compute.p |
a logical value indicating whether to compute a p-value or not |
alpha |
a significance level |
compute.cv |
a logical value indicating whether to compute a critical value corresponding to the significance level alpha or not |
... |
further arguments |
An object of class htest
statistic |
the value of the test statistic. |
parameter |
the number of choosen coordinates (k). |
method |
a character string indicating the parameters of performed test. |
data.name |
a character string giving the name(s) of the data. |
p.value |
the p-value for the test, computed only if |
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.
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$p.value plot(t) ## End(Not run)set.seed(7) # H0 is true z <- runif(80) ## Not run: t <- ddst.uniform.test(z, compute.p = TRUE, d.n = 10) t plot(t) # known fixed alternative z <- rnorm(80,10,16) t <- ddst.uniform.test(pnorm(z, 10, 16), compute.p = TRUE, d.n = 10) t plot(t) # H0 is false z <- rbeta(80,4,2) (t <- ddst.uniform.test(z, compute.p = TRUE, d.n = 10)) t$p.value plot(t) ## End(Not run)
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.
ddst.upwardtrend.test( x, r.N = rep(4, length(x) - 1), alpha = 0.05, t.p, t.n, nr = 1e+05, compute.cv = TRUE )ddst.upwardtrend.test( x, r.N = rep(4, length(x) - 1), alpha = 0.05, t.p, t.n, nr = 1e+05, compute.cv = TRUE )
x |
a list of k (non-empty) numeric vectors of data |
r.N |
a (k-1)-dimensional vector specifying the levels of complexity of the grids considered, only for advanced users |
alpha |
a significance level |
t.p |
an alpha-dependent (k-1)-dimensional vector of the tunning parameters in the penalties in the model selection rules T.o |
t.n |
an alpha-dependent (k-1)-dimensional vector of the tunning parameters in the penalties in the model selection rules T.tilde |
nr |
an integer specifying the number of runs for a p-value and a critical value computation if any |
compute.cv |
a logical value indicating whether to compute a critical value corresponding to the significance level alpha or not |
t |
an alpha-dependent tunning parameter in the penalty in the model selection rule |
An automatic test for the umbrella alternatives. Wylupek (2016) https://onlinelibrary.wiley.com/doi/abs/10.1111/sjos.12231
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)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)
Plots coordinates for selected test statistics
## S3 method for class 'ddst.test' plot(x, ...)## S3 method for class 'ddst.test' plot(x, ...)
x |
result from ddst test function |
... |
currently not used |
# H0 is true x <- runif(80) y <- runif(80) z <- runif(80) t <- ddst.ksample.test(list(x, y, z)) plot(t)# H0 is true x <- runif(80) y <- runif(80) z <- runif(80) t <- ddst.ksample.test(list(x, y, z)) plot(t)