Title:	Distributed Online Goodness-of-Fit Tests for Distributed Datasets
Date:	2025-07-28
Version:	0.3
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:	7.3.2
Imports:	stats
LazyData:	true
Suggests:	testthat (≥ 3.0.0)
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2025-07-28 15:52:20 UTC; ASUS
Author:	Guangbao Guo [aut, cre], Di Chang [aut]
Maintainer:	Guangbao Guo <ggb11111111@163.com>
Depends:	R (≥ 3.5.0)
Repository:	CRAN
Date/Publication:	2025-07-29 04:50:02 UTC

Two-Sample Anderson-Darling Test (Bootstrap Version)

Description

Performs a two-sample Anderson-Darling (AD) goodness-of-fit test using bootstrap resampling to compare whether two samples come from the same distribution. This test is sensitive to differences in both location and shape between the two distributions.

Usage

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

Arguments

Details

The test computes the Anderson-Darling statistic using the pooled empirical distribution functions (ECDFs) of the two samples. A bootstrap procedure resamples the group labels to approximate the null distribution and compute a p-value. If 'p.value = 0', it is adjusted to '1 / (2 * nboots)' for stability.

Value

A list of class '"htest"' containing:

statistic

The observed Anderson-Darling test statistic.

p.value

The estimated bootstrap p-value.

alternative

The alternative hypothesis used.

method

A character string describing the test.

bootstraps

(Optional) A numeric vector of bootstrap statistics if 'keep.boots = TRUE'.

Examples

set.seed(123)
x <- rnorm(100, mean = 0, sd = 4)
y <- rnorm(100, mean = 2, sd = 4)
AD2gof(x, y)

Anderson-Darling Goodness-of-Fit Test for a Specified Distribution

Description

Performs the Anderson-Darling (AD) goodness-of-fit test for a given univariate distribution. The function computes the AD statistic and returns an approximate p-value based on adjusted formulas.

Usage

ADgof(
  x,
  dist = c("norm", "exp", "unif", "lnorm", "weibull", "gamma", "t", "chisq"),
  ...,
  eps = 1e-15
)

Arguments

Details

This implementation supports several common distributions. Parameters of the null distribution must be supplied via .... The p-value is calculated using the approximations suggested by Stephens (1986) and other refinements. For small samples or custom distributions, a bootstrap version may be preferred.

Value

A list of class "htest" with components:

statistic

The value of the Anderson-Darling test statistic.

p.value

The approximate p-value computed using adjustment formulas.

method

A description of the test performed.

data.name

The name of the input data.

Examples

set.seed(123)
x1 <- rnorm(500, mean = 5, sd = 2)
ADgof(x1, dist = "norm", mean = 5, sd = 2)

x2 <- rexp(400, rate = 1.5)
ADgof(x2, dist = "exp")
ADgof(x2, dist = "exp", rate = 1.5)

x3 <- runif(300, min = -2, max = 4)
ADgof(x3, dist = "unif", min = -2, max = 4)

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 Test (Bootstrap Version)

Description

Performs a nonparametric two-sample Cramér–von Mises test using a permutation-based bootstrap method to assess whether two samples come from the same distribution.

Usage

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

Arguments

Details

The test compares two empirical cumulative distribution functions (ECDFs). The bootstrap procedure permutes group labels to generate the null distribution. Tailored one-sided tests use one-sided squared differences of ECDFs.

Value

An object of class "htest" with elements:

statistic

Observed Cramér–von Mises test statistic.

p.value

Bootstrap-based p-value.

alternative

The alternative hypothesis used.

method

A description of the test.

bootstraps

(Optional) Vector of bootstrap test statistics if keep.boots = TRUE.

Examples

set.seed(123)
x <- rnorm(100, mean = 0, sd = 4)
y <- rnorm(100, mean = 2, sd = 4)
CVM2gof(x, y)

# One-sided test
CVM2gof(x, y, alternative = "greater")

# Store bootstrap replicates
res <- CVM2gof(x, y, keep.boots = TRUE)
hist(res$bootstraps, main = "Bootstrap Distribution", xlab = "Test Statistic")

One-Sample Cramér–von Mises Goodness-of-Fit Test

Description

Performs the one-sample Cramér–von Mises goodness-of-fit (GoF) test to assess whether a sample comes from a specified distribution using asymptotic p-value approximations.

Usage

CVMgof2(
  x,
  dist = c("norm", "exp", "unif", "lnorm", "weibull", "gamma", "t", "chisq"),
  ...,
  eps = 1e-15
)

Arguments

Details

The test uses the Cramér–von Mises statistic to assess how well the empirical distribution function (EDF) of the sample agrees with the cumulative distribution function (CDF) of the specified theoretical distribution. The p-value is computed using approximation formulas derived from the asymptotic distribution of the test statistic.

Value

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

statistic

The computed Cramér–von Mises test statistic.

p.value

The asymptotic p-value.

method

A description of the test and distribution.

data.name

The name of the data vector.

Examples

set.seed(123)
x1 <- rnorm(500, mean = 0, sd = 1)
CVMgof2(x1, dist = "norm", mean = 0, sd = 1)

x2 <- rexp(500, rate = 2)
CVMgof2(x2, dist = "exp", rate = 2)

x3 <- runif(200, min = -1, max = 3)
CVMgof2(x3, dist = "unif", min = -1, max = 3)

Two-Sample Kolmogorov–Smirnov Test with Bootstrap

Description

Performs a two-sample Kolmogorov–Smirnov (KS) test using a bootstrap method to assess whether two independent samples come from the same distribution.

Usage

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

Arguments

Details

This implementation performs a nonparametric KS test for equality of distributions by resampling under the null hypothesis. It supports one-sided and two-sided alternatives.

If keep.boots = TRUE, the function returns all bootstrap statistics, which can be used for further analysis (e.g., plotting).

If the p-value is zero due to no bootstrap statistic exceeding the observed value, it is adjusted to 1 / (2 * nboots) to avoid a zero p-value.

Value

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

statistic

The observed KS statistic.

p.value

The p-value based on the bootstrap distribution.

alternative

The alternative hypothesis.

method

Description of the test used.

Examples

set.seed(123)
x <- rnorm(100, mean = 0, sd = 4)
y <- rnorm(100, mean = 2, sd = 4)
KS2gof(x, y)

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

Description

Performs the one-sample Kolmogorov-Smirnov test for a specified theoretical distribution.

Usage

KSgof2(
  x,
  dist = c("norm", "exp", "unif", "lnorm", "weibull", "gamma", "t", "chisq"),
  ...,
  eps = 1e-15
)

Arguments

Details

The test compares the empirical distribution function of x with the cumulative distribution function of a specified theoretical distribution 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(1000, mean = 5, sd = 2)
KSgof2(x, dist = "norm", mean = 5, sd = 2)

y <- rexp(500, rate = 0.5)
KSgof2(y, dist = "exp", rate = 0.5)

u <- runif(300, min = 0, max = 10)
KSgof2(u, dist = "unif", min = 0, max = 10)

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)

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 dataset of the Portuguese "Vinho Verde" wine

Description

A white wine tasting preference data used in the study of Cortez, Cerdeira, Almeida, Matos, and Reis 2009. This white wine contains 4898 white vinho verde wine samples and 12 variables including the tasting preference score of white wine and its physicochemical characteristics.

Usage

data(WhiteWine)

Format

A data frame with 4898 rows, quality score, and 11 variables of physicochemical properties of wines.

• quality Tasting preference is a rating score provided by a minimum of three sensory with ordinal values from 0 (very bad) to 10 (excellent). The final sensory score is the median of these evaluations.

• fixed.acidity The fixed acidity is the physicochemical property in unit (g(tartaric acid)/dm^3).

• volatile.acidity The volatile acidity is in unit g(acetic acid)/dm^3.

• citric.acid The citric acidity is in unit g/dm^3.

• residual.sugar The residual sugar is in unit g/dm^3.

• chlorides The chlorides is in unit g(sodium chloride)/dm^3.

• free.sulfur.dioxide The free sulfur dioxide is in unit mg/dm^3.

• total.sulfur.dioxide The total sulfur dioxide is in unit mg/dm^3.

• density The density is in unit g/cm^3.

• pH The wine's pH value.

• sulphates The sulphates is in unit g(potassium sulphates)/dm^3.

• alcohol The alcohol is in unit \

References

Cortez, P., Cerdeira, A., Almeida, F., Matos, T., and Reis, J. (2009), “Modeling wine preferences by data mining from physicochemical properties,” Decision Support Systems, 47, 547–553. doi:10.1016/j.dss.2009.05.016

Examples

head(WhiteWine)

Perform the Cramer-von Mises Goodness-of-Fit Test for Normality

Description

Perform the Cramer-von Mises Goodness-of-Fit Test for Normality

Usage

cvmgof(x)

Arguments

Value

Examples

# Example usage:
set.seed(123)
x <- rnorm(100)  # Generate a sample from a normal distribution
result <- cvmgof(x)
print(result)

# Example with non-normal data:
y <- rexp(100)  # Generate a sample from an exponential distribution
result <- cvmgof(y)
print(result)

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)

Perform the Lilliefors (Kolmogorov-Smirnov) Goodness-of-Fit Test for Normality

Description

Perform the Lilliefors (Kolmogorov-Smirnov) Goodness-of-Fit Test for Normality

Usage

ksgof(x)

Arguments

Value

Examples

# Example usage:
set.seed(123)
x <- rnorm(100)  # Generate a sample from a normal distribution
result <- ksgof(x)
print(result)

# Example with non-normal data:
y <- rexp(100)  # Generate a sample from an exponential distribution
result <- ksgof(y)
print(result)

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)

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

Description

This function performs a simple Cramer-von Mises goodness-of-fit test to assess whether a given sample comes from a uniform distribution. The test statistic and p-value are calculated based on the sorted sample data.

Usage

simpleCvMgof(X)

Arguments

Value

Examples

# Example usage:
set.seed(123)
X <- runif(100)  # Generate a sample from a uniform distribution
result <- simpleCvMgof(X)
print(result)

# Example with non-uniform data:
Y <- rnorm(100)  # Generate a sample from a normal distribution
result <- simpleCvMgof(Y)
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"