Package {Dogoftest}

Title:	Distributed Online Goodness-of-Fit Tests for Distributed Datasets
Version:	0.4.2
Description:	Distributed Online Goodness-of-Fit Test can process the distributed datasets. The philosophy of the package is described in Guo G.(2024) <doi:10.1016/j.apm.2024.115709>.
License:	MIT + file LICENSE
Encoding:	UTF-8
RoxygenNote:	7.3.3
Imports:	stats
LazyData:	true
Suggests:	testthat (≥ 3.0.0)
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2026-10-12 11:15:09 UTC; admin
Author:	Guangbao Guo [aut, cre], Di Chang [aut]
Maintainer:	Guangbao Guo <ggb11111111@163.com>
Depends:	R (≥ 3.5.0)
Repository:	CRAN
Date/Publication:	2026-10-12 11:50:02 UTC

Two-Sample Anderson-Darling Permutation Test

Description

Permutation two-sample Anderson-Darling test. AD statistic puts heavier weights on tail regions of distribution.

Usage

AD2gof(
  x,
  y,
  alternative = c("two.sided", "less", "greater"),
  nboots = 2000,
  keep.perms = FALSE
)

Arguments

Value

An htest list object:

statistic

Observed Anderson-Darling statistic.

p.value

Permutation p-value.

alternative

Specified alternative.

method

Test description.

data.name

Sample names.

permutations

Permutation statistics vector (if keep.perms=TRUE).

Examples

set.seed(123)
x <- rnorm(100)
y <- rnorm(100, mean=0.2)
out <- AD2gof(x, y, nboots = 200)
out$p.value

Anderson-Darling Test for Normality

Description

Performs the Anderson-Darling goodness-of-fit test for normality, where the mean and standard deviation of the normal distribution are estimated from the data. The p-value is computed using asymptotic approximation formulas adjusted for parameter estimation.

Usage

ADgof(x, eps = 1e-15)

Arguments

Details

This function tests the null hypothesis that the data come from a normal distribution, with unknown parameters replaced by their maximum likelihood estimates (sample mean and sample standard deviation).

The Anderson-Darling statistic is computed as:

A = -n - \frac{1}{n} \sum_{i=1}^n (2i - 1) \left[ \log F(x_{(i)}) + \log(1 - F(x_{(n+1-i)})) \right],

where F is the CDF of the fitted normal distribution. A small-sample adjustment A^* = A \cdot (1 + 0.75/n + 2.25/n^2) is applied before computing the p-value, though the reported statistic is the unadjusted A.

The p-value is obtained via piecewise exponential-polynomial approximations derived for the case where parameters are estimated from the data (composite null hypothesis). These approximations are based on empirical studies (e.g., Stephens, 1986) and are valid for moderate to large sample sizes.

Note: Because the parameters are estimated from the data, this is not the standard Anderson-Darling test for a fully specified distribution. P-values are therefore approximate.

Value

An object of class "htest" with the following components:

statistic

The Anderson-Darling test statistic (AD).

p.value

The approximate p-value based on adjusted asymptotic formulas.

method

A character string describing the test performed.

data.name

The name of the input data vector.

parameter

Named vector containing the estimated mean and sd.

Examples

set.seed(100)
x <- rnorm(60, mean = 0, sd = 5)
ADgof(x)

Data set

Description

psi21k, psi26k, and psi31k are from Birnbaum and Saunders (1969). The fatigue lifetimes of aluminum specimens exposed to a maximum stress of 21,000 psi, 26,000 psi, 31,000 psi, respectively.

bearings is from McCool (1974). The fatigue lifetimes (in hours) of ten bearings.

fatigue is from Brown and Miller (1978). The fatigue lifetimes of cylindrical specimens subjected to combined torsional and axial loads over constant-amplitude cycles until failure.

repair is from Hsieh (1990). This is a maintenance data set on active repair times (in hours) for an airborne communications transceiver.

Usage

data(BSdata)

References

Birnbaum, Z. W. and Saunders, S. C. (1969). A new family of life distributions. J. Appl. Probab. 6(2): 637-652.

McCool, J. I. (1974). Inferential techniques for Weibull populations. Aerospace Research Laboratories Report ARL T R74-0180, Wright-Patterson Air Force Base, Dayton, OH.

Rieck, J. R. and Nedelman, J. (1991). A Log-Linear Model for the Birnbaum-Saunders Distribution. Technometrics. 33, 51-60.

Brown, M. W. and Miller, K. J. (1978). Biaxial Fatigue Data. Report CEMR1/78. University of Sheffield, Dept. of Mechanical Engineering.

Hsieh, H. K. (1990). Estimating the Critical Time of Inverse Gaussian Hazard Rate. IEEE Transactions on Reliability, 39(10): 342-345.

Examples

# Attach data sets
data(BSdata)

Two-Sample Cramér-von Mises Permutation Test

Description

Two-sample Cramér-von Mises goodness-of-fit test using permutation null distribution. Returns htest object.

Usage

CVM2gof(
  x,
  y,
  alternative = c("two.sided", "less", "greater"),
  nboots = 2000,
  keep.boots = FALSE
)

Arguments

Value

An htest list:

statistic

Observed CvM statistic.

p.value

Permutation p-value.

alternative

Alternative hypothesis.

method

Test name.

data.name

Sample names.

bootstraps

Permutation statistics vector, only for keep.boots=TRUE.

Examples

set.seed(123)
x <- rnorm(100)
y <- rnorm(100, mean=0.2)
out <- CVM2gof(x, y, nboots = 200)
out$p.value

One-Sample Cramér–von Mises Test for Normality

Description

Performs the one-sample Cramér–von Mises goodness-of-fit test for normality, where the mean and standard deviation of the normal distribution are estimated from the data. The p-value is computed using asymptotic approximations.

Usage

CVMgof(x, eps = 1e-15)

Arguments

Details

This function tests the null hypothesis that the sample comes from a normal distribution, with unknown parameters estimated by the sample mean and standard deviation.

The Cramér–von Mises statistic is computed as:

W = \frac{1}{12n} + \sum_{i=1}^n \left( F(x_{(i)}) - \frac{2i - 1}{2n} \right)^2,

where F is the CDF of the fitted normal distribution. A small-sample correction (1 + 0.5/n) \cdot W is applied before computing the p-value.

The p-value is obtained via piecewise exponential-polynomial approximations based on the asymptotic distribution under the composite null hypothesis (i.e., when parameters are estimated from the data). These approximations are accurate for moderate to large sample sizes but remain approximate.

Note: Because the parameters are estimated from the data, this is not the classical Cramér–von Mises test (which assumes a fully specified distribution).

Value

An object of class "htest" with the following components:

statistic

The corrected Cramér–von Mises test statistic (W).

p.value

The approximated asymptotic p-value.

method

A character string describing the test.

data.name

The name of the input data vector.

parameter

Named vector containing the estimated mean and sd.

Examples

set.seed(123)
x <- rnorm(50)
CVMgof(x)

DAD This function implements a resampling-based AD goodness-of-fit test. It draws K bootstrap subsamples, performs AD goodness-of-fit test on each subsample, returns the average AD statistic and the minimum p-value across all K subsamples.

Description

DAD This function implements a resampling-based AD goodness-of-fit test. It draws K bootstrap subsamples, performs AD goodness-of-fit test on each subsample, returns the average AD statistic and the minimum p-value across all K subsamples.

Usage

DAD(X, K)

Arguments

Value

A list containing two elements:

stat

scalar, average of the K AD test statistics.

pval

scalar, minimum p-value from the K subsample AD tests.

See Also

ADgof

Examples

## Not run: 
set.seed(123)
X <- rnorm(200)
DAD(X, K = 5)

## End(Not run)

Distributed Two-Sample Anderson-Darling Test via Fisher Combination

Description

Implements a distributed two-sample Anderson-Darling (AD) test: randomly draw subsets from samples into K partitions, run local bootstrap AD test on each subset pair, then combine partition p-values using Fisher's method.

Usage

DAD2(x, y, K, nboots = 50)

Arguments

Value

A list containing:

stat

Mean of local AD statistics across all K partitions.

pval

Combined p-value obtained from Fisher's combination method.

method

Character string describing the test method.

partition_pvals

Numeric vector of p-values from each partition.

partition_stats

Numeric vector of AD statistics from each partition.

Examples

set.seed(123)
x <- rnorm(200)
y <- rnorm(200, mean = 0.3)
out <- DAD2(x, y, K = 3, nboots = 30)
out$pval

DCVM This function implements a resampling-based CVM goodness-of-fit test. It draws K bootstrap subsamples, performs CVM goodness-of-fit test on each subsample, returns the average CVM statistic and the minimum p-value across all K subsamples.

Description

DCVM This function implements a resampling-based CVM goodness-of-fit test. It draws K bootstrap subsamples, performs CVM goodness-of-fit test on each subsample, returns the average CVM statistic and the minimum p-value across all K subsamples.

Usage

DCVM(X, K)

Arguments

Value

A list containing two elements:

stat

scalar, average of the K CVM test statistics.

pval

scalar, minimum p-value from the K subsample CVM tests.

See Also

CVMgof

Examples

## Not run: 
set.seed(123)
X <- rnorm(200)
DCVM(X, K = 5)

## End(Not run)

Distributed Two-Sample Cramér-von Mises Test via Fisher Combination

Description

Implements a distributed two-sample Cramér-von Mises (CvM) test: randomly split samples into K subsets, run local bootstrap CvM test on each partition, then combine the partition p-values using Fisher's method.

Usage

DCVM2(x, y, K, nboots = 50)

Arguments

Value

A list with components:

stat

Mean of the local CvM statistics across all K partitions.

pval

Combined p-value from Fisher's method.

method

Character string describing the test method.

partition_pvals

Numeric vector of p-values from each partition.

partition_stats

Numeric vector of CvM statistics from each partition.

Examples

set.seed(123)
x <- rnorm(200)
y <- rnorm(200, mean = 0.3)
out <- DCVM2(x, y, K = 3, nboots = 30)
out$pval

DKS This function implements a resampling-based KS test. It draws K bootstrap subsamples, performs KS goodness-of-fit test on each subsample, returns the average KS statistic and the minimum p-value across all K subsamples.

Description

DKS This function implements a resampling-based KS test. It draws K bootstrap subsamples, performs KS goodness-of-fit test on each subsample, returns the average KS statistic and the minimum p-value across all K subsamples.

Usage

DKS(X, K)

Arguments

Value

A list containing two elements:

stat

scalar, average of the K KS test statistics.

pval

scalar, minimum p-value from the K subsample KS tests.

See Also

KSgof

Examples

## Not run: 
set.seed(123)
X <- rnorm(200)
DKS(X, K = 5)

## End(Not run)

Distributed Two-Sample KS Test via Fisher Combination

Description

Implements a distributed two-sample KS test: randomly split samples into K subsets, run bootstrap KS test on each partition, then combine partition p-values using Fisher's method.

Usage

DKS2(x, y, K, nboots = 50)

Arguments

Value

A list containing:

stat

Mean of KS statistics across all K partitions.

pval

Combined p-value from Fisher's method.

method

Character string describing test method.

partition_pvals

Vector of p-values from each partition.

partition_stats

Vector of KS statistics from each partition.

Examples

set.seed(123)
x <- rnorm(200)
y <- rnorm(200, mean = 0.3)
out <- DKS2(x, y, K = 3, nboots = 30)
out$pval

DPECVM Performs the divided PECVM goodness-of-fit test. The original sample is split into K bootstrap subsamples, PECVM test is applied on each subsample. The final test statistic is the average of subsample statistics, and the p-value is taken as the minimum of subsample p-values.

Description

DPECVM Performs the divided PECVM goodness-of-fit test. The original sample is split into K bootstrap subsamples, PECVM test is applied on each subsample. The final test statistic is the average of subsample statistics, and the p-value is taken as the minimum of subsample p-values.

Usage

DPECVM(X, K, F0, rF0)

Arguments

Value

A list containing two elements:

stat

Numeric, averaged PECVM test statistic across K subsamples.

pval

Numeric, minimum p-value from K subsample PECVM tests.

Examples

## Not run: 
# Null: standard normal
set.seed(123)
X <- rnorm(200)
F0 <- pnorm
rF0 <- rnorm
out <- DPECVM(X, K = 5, F0 = F0, rF0 = rF0)
out$stat
out$pval

## End(Not run)

Distributed Power-Enhanced Two-Sample Cramer-von Mises Test

Description

Implements a distributed power-enhanced two-sample Cramer-von Mises (PE-CvM) test. Randomly draw subsets from samples into K partitions, compute local power-enhanced CvM test for each partition via PE2CVM, then combine partition p-values using Fisher's combination method.

Usage

DPECVM2(
  x,
  y,
  K,
  lambda = 0.2,
  threshold = 0.99,
  B = 100,
  nboots = 50,
  na.rm = FALSE
)

Arguments

Value

A list of class htest compatible structure containing:

statistic

Named numeric vector with mean values across partitions: Enhanced, Original.CvM, PE.Component.

p.value

Combined p-value obtained by Fisher's combination of K partition p-values.

method

Character string describing the test name.

data.name

Character string naming the input samples x and y.

parameters

Named vector storing hyperparameters: K, lambda, threshold, B.

partition_pvals

Numeric vector of p-values from each partition.

partition_stats

Data frame containing three types of statistics for each partition.

Examples

set.seed(123)
x <- rnorm(300)
y <- rnorm(300, mean = 0.4)
out <- DPECVM2(x, y, K = 3, lambda = 0.2, threshold = 0.99, B = 30)
out$p.value
out$statistic

Two-Sample Kolmogorov-Smirnov Test with Bootstrap/Permutation

Description

Compute two-sample Kolmogorov-Smirnov test using permutation bootstrap to calculate p-value. Returns an object of class htest.

Usage

KS2gof(
  x,
  y,
  alternative = c("two.sided", "less", "greater"),
  nboots = 5000,
  keep.boots = FALSE
)

Arguments

Value

An object of class htest, a list containing:

statistic

Observed KS statistic.

p.value

Permutation p-value.

alternative

Specified alternative hypothesis.

method

Name of test method.

bootstraps

Vector of bootstrap statistics, only present if keep.boots = TRUE.

Examples

set.seed(123)
x <- rnorm(100)
y <- rnorm(100, mean=0.2)
out <- KS2gof(x, y, alternative = "two.sided", nboots = 200)
out$p.value

One-sample Kolmogorov-Smirnov goodness-of-fit test

Description

Performs the one-sample Kolmogorov-Smirnov test against the normal distribution with parameters estimated from the data.

Usage

KSgof(x, eps = 1e-15)

Arguments

Details

The test compares the empirical distribution function of x with the cumulative distribution function of a normal distribution (mean and standard deviation estimated from x) using the Kolmogorov-Smirnov statistic. For large sample sizes, a p-value approximation based on the asymptotic distribution is used.

A correction is applied when sample size exceeds 100, adjusting the test statistic to approximate a fixed sample size. For very small or very large statistics, piecewise polynomial approximations are used to compute the p-value.

Value

An object of class "htest" containing the test statistic, p-value, method description, and data name.

Examples

set.seed(123)
x <- rnorm(100)
KSgof(x)

Two-Sample Kuiper Test with Bootstrap

Description

Performs a two-sample Kuiper test using bootstrap resampling to test whether two independent samples come from the same distribution.

Usage

Kuiper2gof(
  x,
  y,
  alternative = c("two.sided", "less", "greater"),
  nboots = 2000,
  keep.boots = FALSE
)

Arguments

Details

The Kuiper test is a nonparametric test similar to the Kolmogorov–Smirnov test, but sensitive to discrepancies in both location and shape between two distributions. This implementation uses bootstrap resampling to estimate the p-value.

The two.sided test uses the sum of maximum positive and negative ECDF differences. The greater and less options use one-sided variations.

If the observed test statistic exceeds all bootstrap values, the p-value is set to 1 / (2 * nboots) to avoid zero.

Value

An object of class "htest" containing:

statistic

The observed Kuiper statistic.

p.value

The p-value computed from the bootstrap distribution.

alternative

The specified alternative hypothesis.

method

A character string describing the test.

bootstraps

(If requested) A numeric vector of bootstrap statistics.

Examples

set.seed(123)
x <- rnorm(100, 0, 4)
y <- rnorm(100, 2, 4)
Kuiper2gof(x, y)

Power-Enhanced Two-Sample Anderson-Darling Test (PE2AD)

Description

Performs a power-enhanced variant of the two-sample Anderson-Darling (AD) test. The proposed statistic augments the classical two-sample AD statistic by adding a penalty term constructed from extreme weighted deviations between empirical distribution functions. The proposed enhancement improves statistical power for detecting distributional differences concentrated in the tails. While the standard AD test already places higher weight on tail discrepancies, this method explicitly amplifies the contribution of extreme local deviations.

Usage

PE2AD(x, y, lambda = 1, threshold = 1, B = NULL)

Arguments

Details

The classical two-sample Anderson-Darling statistic is defined as

A^2 = \frac{n_1 n_2}{(n_1 + n_2)^2} \sum_{i=1}^{N} \frac{\left[F_{n_1}(z_i) - G_{n_2}(z_i)\right]^2}{H_N(z_i)\left[1 - H_N(z_i)\right]},

where

• z_i: pooled and sorted combined observations from the two samples,

• F_{n_1}, G_{n_2}: empirical cumulative distribution functions for sample x and sample y,

• H_N: pooled empirical CDF under the null hypothesis of identical distributions.

The denominator H_N(1-H_N) assigns larger weights to tail deviations, which makes AD test more sensitive to tail differences than Kolmogorov-Smirnov or Cramer-von Mises tests.

The PE2AD statistic is constructed in four steps:

1. Compute pointwise weighted squared deviations (summands of the A^2 statistic).

2. Standardize these marginal contributions: s_i = (m_i - \bar{m}) / \text{sd}(m).

3. Retain only points satisfying s_i > \texttt{threshold}.

4. Compute the PE component as the mean of squared standardized selected terms, which ensures scale compatibility with the original AD statistic.

The final enhanced test statistic:

T = A^2 + \lambda \times \text{PE\_comp}.

Under the null hypothesis that the two samples come from identical distributions, permutation inference randomly reassigns group labels while fixing the pooled ordered grid z. The permutation p-value is estimated by the fraction of permutation replicates whose simulated statistic is greater than or equal to the observed T.

Numerical stabilization: pooled CDF H_N is truncated to [\epsilon, 1 - \epsilon] where \epsilon equals .Machine$double.eps to avoid division by zero near the extreme tails.

Value

An object of S3 class "htest", a list containing the following elements:

statistic

Named numeric vector with three entries: Enhanced (final PE2AD test statistic), Original.AD (classical two-sample Anderson-Darling statistic), PE.Component (power-enhancement penalty term).

p.value

Permutation-based p-value when B > 0; otherwise NA.

method

Character string with the full test name.

data.name

Character string naming the two input samples.

parameters

Named numeric vector storing user-supplied lambda, threshold, and B.

Note

Permutation p-value computation can be computationally expensive for large B. The permutation scheme fixes the pooled ordered grid for efficiency. When all observations are identical across the pooled sample, marginal contributions become zero and the PE component returns zero.

References

Anderson, T. W., & Darling, D. A. (1952). Asymptotic theory of certain "goodness of fit" criteria based on stochastic processes. Annals of Mathematical Statistics.

Examples

set.seed(2025)
x <- rnorm(60, mean = 0, sd = 1)      # light-tailed Gaussian sample
y <- rt(60, df = 3)                   # heavy-tailed t-distribution sample

# PE2AD with permutation p-value
PE2AD(x, y, lambda = 0.4, threshold = 1.0, B = 2000)

# Quick calculation without permutation p-value
PE2AD(x, y)

Power-Enhanced Two-Sample Cramer-von Mises Test (PE2CVM)

Description

Performs a power-enhanced variant of the two-sample Cramer-von Mises (CvM) test. The enhanced statistic combines the classical two-sample CvM statistic with a power-enhancement component constructed from large squared deviations between empirical distribution functions. This modification improves detection power for subtle distributional differences compared with the standard CvM test.

Usage

PE2CVM(x, y, lambda = 0.4, threshold = 0.9, B = NULL, na.rm = FALSE)

Arguments

Details

The classical two-sample Cramer-von Mises statistic is defined as

W^2 = \frac{n_1 n_2}{(n_1 + n_2)^2}\sum_{i=1}^{N}\left[F_{n_1}(z_i)-G_{n_2}(z_i)\right]^2,

where

• z_i: pooled sorted observations from the two samples,

• F_{n_1}, G_{n_2}: empirical cumulative distribution functions for sample x and sample y,

• n_1,n_2: sample sizes, N=n_1+n_2.

The pointwise squared deviation (F_{n_1}(z_i)-G_{n_2}(z_i))^2 is treated as the local energy term. The energy values are standardized using median and MAD for robustness against outliers. Points whose standardized energy exceeds the quantile specified by threshold are selected as high-energy locations.

The raw power-enhancement term is the mean of squared standardized values among selected high-energy points, and then mapped via a sigmoid-like transformation:

\text{PE}_{\text{scaled}} = \frac{2}{1+\exp(-\text{PE}_{\text{raw}})}-1.

The final enhanced test statistic is

T = W^2 + \lambda \times \text{PE}_{\text{scaled}}.

Under the null hypothesis of identical distributions, permutation inference randomly reassigns group labels while keeping the pooled ordered grid z fixed. The permutation p-value is estimated as the proportion of permutation replicates with simulated statistic greater than or equal to the observed T.

Value

An object of S3 class "htest", a list containing:

statistic

Named numeric vector: Enhanced (final combined test statistic), Original.CVM (classical two-sample Cramer-von Mises statistic), PE.Component (scaled power-enhancement component).

p.value

Permutation-based p-value; NA when permutation is not performed.

method

Character string with the full test name.

data.name

Character string naming the two input samples.

parameters

Named numeric vector storing hyperparameters lambda, threshold, and B.

Note

Permutation p-value computation is computationally costly when B is large. The quantile threshold threshold must lie between 0 and 1. If all local energy terms are identical, MAD equals zero and is replaced by 1 to avoid division by zero.

References

Cramer, H. (1928). On the composition of elementary errors. Skandinavisk Aktuarietidskrift. von Mises, R. (1928). Wahrscheinlichkeitsrechnung und ihre Anwendung in der Statistik und theoretischen Physik.

Examples

set.seed(123)
x <- rnorm(100)
y <- rnorm(100, mean = 0.2)
out <- PE2CVM(x, y, lambda = 0.4, threshold = 0.9, B = 200)
out$p.value
out$statistic

# Quick run without permutation p-value
PE2CVM(x, y, lambda = 0.4, threshold = 0.9)

Power-Enhanced Two-Sample Kolmogorov–Smirnov Test (PE2KS)

Description

Performs a power-enhanced version of the two-sample Kolmogorov–Smirnov (KS) test that augments the classical KS statistic with a penalty term based on unusually large deviations across the entire empirical distribution function (EDF). This enhancement improves sensitivity to differences in shape or tail behavior that may not produce a large maximum deviation.

Usage

PE2KS(x, y, lambda = 1, threshold = 1, B = NULL)

Arguments

Details

The classical two-sample KS test uses the statistic

D_{\text{KS}} = \sup_t |F_n(t) - G_m(t)|,

which is sensitive primarily to the largest pointwise difference between empirical CDFs.

The PE2KS test enhances this by adding a term that captures **systematic or tail deviations**:

1. Compute the absolute deviation vector d_i = |F_n(z_i) - G_m(z_i)| over all pooled ordered points z_i.

2. Standardize these deviations: s_i = (d_i - \bar{d}) / \text{sd}(d).

3. Select points where s_i > \texttt{threshold}.

4. Define the power-enhancement (PE) component as the **mean squared standardized deviation** over selected points (or 0 if none are selected).

The final test statistic is:

T = D_{\text{KS}} + \lambda \times \text{PE\_comp}.

A permutation procedure (if B > 0) randomly reassigns group labels to approximate the null distribution of T, yielding a p-value as the proportion of permuted statistics exceeding the observed value.

Missing values are not explicitly handled; users should ensure inputs are clean.

Value

An object of class "htest" containing:

statistic

A named vector with three components: Enhanced (the final test statistic), Original.KS (classical KS statistic), and PE.Component (the power-enhancement term).

p.value

Permutation-based p-value if B > 0; otherwise NA.

method

A character string describing the test.

data.name

Names of the input data vectors.

parameters

Named vector of user-specified parameters (lambda, threshold, B).

Note

- The enhancement term is scale-normalized (via standardization and averaging) to be comparable in magnitude to D_{\text{KS}} (which lies in [0,1]). - Setting lambda = 0 recovers the standard KS test. - Very high lambda may reduce specificity; recommended values are in [0.1, 2].

Examples

set.seed(42)
x <- rnorm(60, mean = 0, sd = 1)
y <- rnorm(60, mean = 0.25, sd = 1)

# Basic usage (no p-value)
PE2KS(x, y)

# With permutation p-value
PE2KS(x, y, lambda = 0.5, threshold = 1.0, B = 2000)

Power-Enhanced Cramer-von Mises Goodness-of-Fit Test

Description

Performs the Power-Enhanced Cramer-von Mises (PECVM) goodness-of-fit test, which improves testing power by standardizing a Cramer-von Mises-type statistic with an estimate of its standard deviation under the null hypothesis. The null distribution must be fully specified via a cumulative distribution function (CDF).

Performs the Power-Enhanced Cramér–von Mises (PECVM) goodness-of-fit test, which improves testing power by standardizing a Cramér–von Mises-type statistic with an estimate of its standard deviation under the null hypothesis. The null distribution must be fully specified via a cumulative distribution function (CDF).

Usage

PECVM(x, F0, alpha = 0.05, B = 500, boot = TRUE)

PECVM(x, F0, alpha = 0.05, B = 500, boot = TRUE)

Arguments

Details

The base statistic is defined as:

T_n = n \cdot \frac{1}{n} \sum_{i=1}^n \left( \hat{F}_n(x_i) - F_0(x_i) \right)^2 = \sum_{i=1}^n \left( \hat{F}_n(x_i) - F_0(x_i) \right)^2,

where \hat{F}_n is the empirical CDF and F_0 is the user-specified null CDF.

The PECVM test enhances power by standardizing T_n with \hat{\sigma}_{02}, the bootstrap-estimated standard deviation of T_n under the null. The standardized statistic is T_n^{PE} = T_n / \hat{\sigma}_{02}.

Bootstrap inference proceeds as follows:

1. Estimate \hat{\sigma}_{02} using 500 resamples from the original data.

2. Generate B bootstrap replicates of T^* / \hat{\sigma}_{02} (using the same \hat{\sigma}_{02}).

3. Compute the p-value as the proportion of bootstrap statistics exceeding the observed T_n^{PE}.

This test assumes a simple null hypothesis: the CDF F0 must be completely known and must not depend on parameters estimated from x. Using a data-dependent F0 (e.g., plugging in sample mean) invalidates the test.

The base statistic is defined as:

T_n = n \cdot \frac{1}{n} \sum_{i=1}^n \left( \hat{F}_n(x_i) - F_0(x_i) \right)^2 = \sum_{i=1}^n \left( \hat{F}_n(x_i) - F_0(x_i) \right)^2,

where \hat{F}_n is the empirical CDF and F_0 is the user-specified null CDF.

The PECVM test enhances power by standardizing T_n with \hat{\sigma}_{02}, the bootstrap-estimated standard deviation of T_n under the null. The standardized statistic is T_n^{PE} = T_n / \hat{\sigma}_{02}.

Bootstrap inference proceeds as follows:

1. Estimate \hat{\sigma}_{02} using 500 resamples from the original data.

2. Generate B bootstrap replicates of T^* / \hat{\sigma}_{02} (using the same \hat{\sigma}_{02}).

3. Compute the p-value as the proportion of bootstrap statistics exceeding the observed T_n^{PE}.

This test assumes a **simple null hypothesis**: the CDF F0 must be completely known and must not depend on parameters estimated from x. Using a data-dependent F0 (e.g., plugging in sample mean) invalidates the test.

Value

A list with class "htest" containing:

statistic

The power-enhanced Cramer-von Mises statistic T_n^{PE} = T_n / \hat{\sigma}_{02}.

p.value

Bootstrap p-value (or NA if boot = FALSE).

critical.value

(1 - alpha) quantile of the bootstrap distribution (or NA).

sigma

Estimated standard deviation \hat{\sigma}_{02} of the base CVM-type statistic.

n

Sample size.

method

Description of the test performed.

data.name

Name of the input data vector.

A list with class "htest" containing:

statistic

The power-enhanced Cramér–von Mises statistic T_n^{PE} = T_n / \hat{\sigma}_{02}.

p.value

Bootstrap p-value (or NA if boot = FALSE).

critical.value

(1 - alpha) quantile of the bootstrap distribution (or NA).

sigma

Estimated standard deviation \hat{\sigma}_{02} of the base CVM-type statistic.

n

Sample size.

method

Description of the test performed.

data.name

Name of the input data vector.

Examples

set.seed(123)
x <- rnorm(100)               # Data generated under N(0,1)
F0 <- function(x) pnorm(x)    # Null hypothesis: standard normal CDF
res <- PECVM(x, F0 = F0, B = 999)
print(res)

set.seed(123)
x <- rnorm(100)               # Data generated under N(0,1)
F0 <- function(x) pnorm(x)    # Null hypothesis: standard normal CDF
res <- PECVM(x, F0 = F0, B = 999)
print(res)

Power-Enhanced Kolmogorov–Smirnov Goodness-of-Fit Test

Description

Performs the Power-Enhanced Kolmogorov–Smirnov (PEKS) goodness-of-fit test, which improves the power of the classical KS test by standardizing the test statistic with an estimate of its standard deviation under the null hypothesis. The null distribution must be fully specified via a cumulative distribution function (CDF).

Usage

PEKS(x, F0, alpha = 0.05, B = 500, boot = TRUE)

Arguments

Details

The PEKS test modifies the classical Kolmogorov–Smirnov statistic D_n by dividing it by an estimate of its standard deviation \hat{\sigma}_{01}, obtained via nonparametric bootstrap from the original sample. This standardization enhances the test's power, particularly against local alternatives.

The bootstrap procedure involves:

1. Computing the original KS statistic D_n.

2. Estimating \hat{\sigma}_{01} using 500 bootstrap resamples.

3. Generating B bootstrap replicates of the standardized statistic D^*/\hat{\sigma}_{01}.

The p-value is the proportion of bootstrap statistics exceeding the observed D_n^{PE}. Note that the null hypothesis must be **simple** (i.e., the CDF F0 is fully known); this test does not support composite hypotheses where parameters are estimated from the data.

Value

A list with class "htest" containing:

statistic

The power-enhanced KS statistic D_n^{PE} = D_n / \hat{\sigma}_{01}.

p.value

Bootstrap p-value (or NA if boot = FALSE).

critical.value

(1 - alpha) quantile of the bootstrap distribution (or NA).

sigma

Estimated standard deviation \hat{\sigma}_{01} of the KS statistic.

n

Sample size.

method

Description of the test performed.

data.name

Name of the input data vector.

Examples

set.seed(123)
x <- rnorm(100)               # Data from N(0,1)
F0 <- function(x) pnorm(x)    # Null: standard normal CDF
res <- PEKS(x, F0 = F0, B = 999)
print(res)

Snow Dataset

Description

Snowfall dataset

Format

vector of values

Details

This file contains observations of the annual snowfall amounts in Buffalo, New York. 63 as observed from 1910/11 to 1972/73 as listed in The autoregressive method: a method of approximating and estimating positive functions. Carmichael, Jean-Pierre. DTIC Document. 1976

Watson goodness-of-fit test Performs the Watson test for goodness-of-fit to a specified distribution.

Description

Watson goodness-of-fit test Performs the Watson test for goodness-of-fit to a specified distribution.

Usage

Wgof(x, dist = c("norm", "exp", "unif", "lnorm", "gamma"), ..., eps = 1e-15)

Arguments

Details

The Watson test is a modification of the Cramér–von Mises test, adjusting for mean deviations. It measures the squared distance between the empirical distribution function of the data and the specified theoretical cumulative distribution function, with a correction for location.

Value

An object of class "htest" containing the test statistic, p-value, method description, data name, and any distribution parameters used.

Examples

set.seed(123)
x_norm <- rnorm(1000, mean = 5, sd = 2)
Wgof(x_norm, dist = "norm", mean = 5, sd = 2)

x_exp <- rexp(500, rate = 0.5)
Wgof(x_exp, dist = "exp", rate = 0.5)

x_unif <- runif(300, min = 0, max = 10)
Wgof(x_unif, dist = "unif", min = 0, max = 10)

x_lnorm <- rlnorm(200, meanlog = 0, sdlog = 1)
Wgof(x_lnorm, dist = "lnorm", meanlog = 0, sdlog = 1)

x_gamma <- rgamma(400, shape = 1, rate = 1)
Wgof(x_gamma, dist = "gamma", shape = 1, rate = 1)

White Wine Quality Data

Description

Physicochemical measurements and sensory quality ratings for white wine samples of the Portuguese Vinho Verde wine. Quality was evaluated by wine experts and is recorded as an ordered factor from 3 (poor) to 9 (excellent).

Usage

WhiteWine

Format

A data frame with 4898 rows and 12 variables:

quality

Sensory quality score (ordered factor), levels 3 to 9; median of at least 3 expert evaluations.

fixed.acidity

Fixed acidity, g(tartaric acid)/dm^3.

volatile.acidity

Volatile acidity, g(acetic acid)/dm^3.

citric.acid

Citric acid, g/dm^3.

residual.sugar

Residual sugar, g/dm^3.

chlorides

Chlorides, g(sodium chloride)/dm^3.

free.sulfur.dioxide

Free sulfur dioxide, mg/dm^3.

total.sulfur.dioxide

Total sulfur dioxide, mg/dm^3.

density

Density, g/cm^3.

pH

pH value.

sulphates

Sulphates, g(potassium sulphate)/dm^3.

alcohol

Alcohol content, percent by volume.

Source

UCI Machine Learning Repository, https://archive.ics.uci.edu/ml/datasets/wine+quality P. Cortez, A. Cerdeira, F. Almeida, T. Matos and J. Reis (2009). Modeling wine preferences by data mining from physicochemical properties. Decision Support Systems, 47(4):547-553.

Zoometric measurements of goats

Description

Zoometric measurements of 27 week old creole goats collected by Dorantes-Coronado (2013).

Usage

data(goats)

Format

A data frame with 52 rows and 7 columns containing measurements (in kilograms and centimeters) on the following variables.

body.weight

body.length

trunk.length

withers.height

thoracic.perimeter

hip.length

ear.length

Source

Dorantes-Coronado (2013).

References

Dorantes-Coronado, E.J. (2013). Estudio preliminar para el establecimiento de un programa de mejoramiento genetico de cabras en el Estado de Mexico. Ph.D. Thesis. Colegio de Postgraduados, Mexico.

Examples

data(goats)
plot(goats)

Calculate the Quantile of the Cramer-von Mises Goodness-of-Fit Statistic

Description

This function calculates the quantile of the Cramer-von Mises goodness-of-fit statistic using the 'uniroot' function to find the root of the given function.

Usage

qCvMgof(X, p)

Arguments

Value

Examples

# Example usage:
set.seed(123)
X <- rnorm(100)  # Generate a sample from a normal distribution
p <- 0.95        # Desired quantile probability
result <- qCvMgof(X, p)
print(result)

Compute False Discovery Proportion (FDP) via Monte Carlo simulation for DKS, DCVM, DAD and DPECVM tests.

Description

Compute False Discovery Proportion (FDP) via Monte Carlo simulation for DKS, DCVM, DAD and DPECVM tests.

Usage

run_FDP_sim(rgen, n_values, M, K, B, F0, rF0, alpha = 0.05, seed = 123)

Arguments

Value

A data frame with columns:

n

sample size

DKS

average FDP of DKS test

DCVM

average FDP of DCVM test

DAD

average FDP of DAD test

DPECVM

average FDP of DPECVM test

Examples

## Not run: 
F0  <- function(x) pnorm(x,0,1)
rF0 <- function(n) rnorm(n,0,1)
rgen_t6 <- function(n) rt(n, df=6)
res <- run_FDP_sim(rgen = rgen_t6,
                   n_values = c(200,300),
                   M = 10, K = 2, B = 50,
                   F0 = F0, rF0 = rF0, seed=123)
print(res)

## End(Not run)

Run four goodness-of-fit tests for normality

Description

Perform Anderson-Darling, Cramer-von Mises, Lilliefors (KS), and PECVM tests on a single sample. Return 0/1 indicators indicating whether the null hypothesis is rejected at level alpha.

Usage

run_custom_tests(X, F0 = pnorm, rF0 = rnorm, B_boot = 500, alpha = 0.05)

Arguments

Value

A named numeric vector of 0/1 indicators for AD, CVM, KS and PECVM.

See Also

ADgof, CVMgof, KSgof, PECVM

Examples

## Not run: 
set.seed(123)
x <- rt(50, df = 5)
run_custom_tests(x)

## End(Not run)

Perform a Simple Cramer-von Mises Goodness-of-Fit Test

Description

This function performs a simple Cramer-von Mises goodness-of-fit test. It assumes the input X are probability integral transformed values under the null distribution (uniform on [0,1]). The test statistic and an asymptotic approximate p-value are computed.

Usage

simpleCvMgof(X)

Arguments

Value

A list with class "htest" containing:

statistic

The value of the Cramer-von Mises test statistic.

p.value

Asymptotic approximate p-value for the test.

method

Description of the test performed.

data.name

Name of input data vector.

Examples

# Example usage: null uniform data
set.seed(123)
X <- runif(100)
result <- simpleCvMgof(X)
print(result)

Compressive strength of maize seeds

Description

Compressive strength and strain of maize seeds.

Usage

data("strength")

Format

A data frame with 90 observations on the following 2 variables.

strain

a numeric vector giving the relative change in length under compression stress in millimeters.

cstrength

a numeric vector giving the compressive strength in Newtons.

Details

These data correspond to maize seeds with floury endosperm and 8% of moisture.

Source

Mancera-Rico, A. (2014).

References

Mancera-Rico, A. (2014). Contenido de humedad y tipo de endospermo en la resistencia a compresion en semillas de maiz. Ph.D. Thesis. Colegio de Postgraduados, Mexico.

Examples

data(strength)
plot(strength)     # plot of "strain" versus "cstrength"