Package 'gofgamma'

statistic of the Anderson-Darling goodness-of-fit test for the gamma family

Description

This function computes the goodness-of-fit test statistic for the gamma family in the spirit of Anderson and Darling. Note that this tests the composite hypothesis of fit to the family of gamma distributions, i.e. a bootstrap procedure is implemented to perform the test, see crit.values.

Usage

AD(data, k_estimator)

Arguments

data	a vector of positive numbers. NOTE: data has to be the rescaled data, i.e. devided by the estimated scale parameter!
k_estimator	value of the estimated shape parameter.

Details

The Anderson-Darling test is computed as described in Henze et. al. (2012). Values of k_estimator are found by gamma_est.

Value

value of the test statistic

References

Henze, N., Meintanis, S.G., Ebner, B. (2012) "Goodness-of-fit tests for the Gamma distribution based on the empirical Laplace transform". Communications in Statistics - Theory and Methods, 41(9): 1543-1556. DOI

Examples

X=stats::rgamma(20,3,6)
AD(X,k_estimator=gamma_est(X)[1])

---

statistic of the Betsch-Ebner test

Description

This function computes the statistic of the goodness-of-fit test for the gamma family due to Betsch and Ebner (2019).

Usage

BE(data, k_estimator, a)

Arguments

data	a vector of positive numbers. NOTE: data has to be the rescaled data, i.e. devided by the estimated scale parameter!
k_estimator	value of the estimated shape parameter.
a	positive tuning parameter.

Details

The test is of weighted $L^2$ type and uses a characterization of the distribution function of the gamma distribution. Values of k_estimator are found by gamma_est.

Value

value of the test statistic

References

Betsch, S., Ebner, B. (2019) "A new characterization of the Gamma distribution and associated goodness of fit tests", Metrika, 82(7):779-806. DOI

Examples

X=stats::rgamma(20,3,6)
BE(X,k_estimator=gamma_est(X)[1],a=2)

---

statistic of the Cramer-von Mises goodness-of-fit test for the gamma family

Description

This function computes the goodness-of-fit test statistic for the gamma family in the spirit of Cramer and von Mises. Note that this tests the composite hypothesis of fit to the family of gamma distributions, i.e. a bootstrap procedure is implemented to perform the test, see crit.values.

Usage

CM(data, k_estimator)

Arguments

data	a vector of positive numbers. NOTE: data has to be the rescaled data, i.e. devided by the estimated scale parameter!
k_estimator	value of the estimated shape parameter.

Details

The Cramér-von Mises test is computed as described in Henze et. al. (2012). Values of k_estimator are found by gamma_est.

Value

value of the test statistic

References

Henze, N., Meintanis, S.G., Ebner, B. (2012) "Goodness-of-fit tests for the Gamma distribution based on the empirical Laplace transform". Communications in Statistics - Theory and Methods, 41(9): 1543-1556. DOI

Examples

X=stats::rgamma(20,3,6)
CM(X,k_estimator=gamma_est(X)[1])

---

bootstrap critical value of statistic

Description

bootstrap critical value of statistic

Usage

crit.values(
  samplesize,
  statistic,
  tuning = NULL,
  k_estimator,
  boot.param = 500,
  alpha = 0.05
)

Arguments

samplesize	number of observations in the sample
statistic	test statistic to be used
tuning	tuning parameter used for the test statistic (NULL stands for no tuning parameter needed)
k_estimator	value of the estimated shape parameter
boot.param	number of bootstrap iterations
alpha	significance level of the test

Value

returns the critical value for the goodness-of-fit test using the statistic.

Examples

crit.values(samplesize=20,statistic=HME1,tuning=1,k_estimator=2,boot.param=100,alpha=0.05)

---

Maximum-likelihood estimation of parameters for the gamma distribution

Description

The maximum-likelihood estimators for the shape and scale parameters of a gamma distribution are computed due to the method of Bhattacharya (2001).

Usage

gamma_est(data)

Arguments

data	vector of positive valued observations

Value

returns a bivariate vector containing (shape,scale) estimated parameter vector.

References

Bhattacharya, B. (2001) "Testing equality of scale parameters against restricted alternatives for $m \ge 3$ gamma distributions with unknown common shape parameter". Journal of Statistical Computation and Simulations, 69(4):353-368, DOI

Examples

gamma_est(stats::rgamma(100,shape=3,scale=6))

---

statistic of the first Henze-Meintanis-Ebner goodness-of-fit test for the gamma family

Description

This function computes the goodness-of-fit test statistic for the gamma family due to the first test in Henze, Meintanis and Ebner (2012).

Usage

HME1(data, k_estimator, a = 1)

Arguments

data	a vector of positive numbers. NOTE: data has to be the rescaled data, i.e. devided by the estimated scale parameter!
k_estimator	value of the estimated shape parameter.
a	positive tuning parameter.

Details

The test statistic is of weighted $L^2$ type and uses a characterization of the distribution function of the gamma distribution.

Value

value of the test statistic

References

Henze, N., Meintanis, S.G., Ebner, B. (2012) "Goodness-of-fit tests for the Gamma distribution based on the empirical Laplace transform". Communications in Statistics - Theory and Methods, 41(9): 1543-1556. DOI

Examples

X=stats::rgamma(20,3,6)
HME1(X,k_estimator=gamma_est(X)[1],a=1)

---

statistic of the second Henze-Meintanis-Ebner goodness-of-fit test for the gamma family

Description

This function computes the goodness-of-fit test statistic for the gamma family due to the second test in Henze, Meintanis and Ebner (2012).

Usage

HME2(data, k_estimator, a = 4)

Arguments

data	a vector of positive numbers. NOTE: data has to be the rescaled data, i.e. devided by the estimated scale parameter!
k_estimator	value of the estimated shape parameter.
a	positive tuning parameter.

Details

The test statistic is of weighted $L^2$ type and uses a characterization of the distribution function of the gamma distribution.

Value

value of the test statistic

References

Henze, N., Meintanis, S.G., Ebner, B. (2012) "Goodness-of-fit tests for the Gamma distribution based on the empirical Laplace transform". Communications in Statistics - Theory and Methods, 41(9): 1543-1556. DOI

Examples

X=stats::rgamma(20,3,6)
HME2(X,k_estimator=gamma_est(X)[1],a=1)

---

statistic of the Kolmogorov-Smirnov goodness-of-fit test for the gamma family

Description

This function computes the goodness-of-fit test statistic for the gamma family in the spirit of Kolmogorov and Smirnov. Note that this tests the composite hypothesis of fit to the family of gamma distributions, i.e. a bootstrap procedure is implemented to perform the test, see crit.values.

Usage

KS(data, k_estimator)

Arguments

data	a vector of positive numbers. NOTE: data has to be the rescaled data, i.e. devided by the estimated scale parameter!
k_estimator	value of the estimated shape parameter.

Details

The Kolmogorov-Smirnov test is computed as described in Henze et. al. (2012). Values of k_estimator are found by gamma_est.

Value

value of the test statistic

References

Henze, N., Meintanis, S.G., Ebner, B. (2012) "Goodness-of-fit tests for the Gamma distribution based on the empirical Laplace transform". Communications in Statistics - Theory and Methods, 41(9): 1543-1556. DOI

Examples

X=stats::rgamma(20,3,6)
KS(X,k_estimator=gamma_est(X)[1])

---

Print method for tests of Gamma distribution

Description

Printing objects of class "gofgamma".

Usage

## S3 method for class 'gofgamma'
print(x, ...)

Arguments

x	object of class "gofgamma".
...	further arguments to be passed to or from methods.

Details

A gofgamma object is a named list of numbers and character string, supplemented with test (the name of the teststatistic). test is displayed as a title. The remaining elements are given in an aligned "name = value" format.

Value

the argument x, invisibly, as for all print methods.

Examples

print(test.BE(rgamma(20,1)))

---

The Anderson-Darling goodness-of-fit test for the gamma family

Description

This function computes the goodness-of-fit test for the gamma family in the spirit of Anderson and Darling. Note that this tests the composite hypothesis of fit to the family of gamma distributions, i.e. a bootstrap procedure is implemented to perform the test.

Usage

test.AD(data, boot = 500, alpha = 0.05)

Arguments

data	a vector of positive numbers.
boot	number of bootstrap iterations used to obtain critical value.
alpha	level of significance of the test.

Details

The Anderson-Darling test is computed as described in Henze et. al. (2012). Critical values are obtained by a parametric bootstrap procedure, see crit.values.

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$T.value

the value of the test statistic.

$cv

the approximated critical value.

$par.est

number of points used in approximation.

$Decision

the comparison of the critical value and the value of the test statistic.

$sig.level

level of significance chosen.

$boot.run

number of bootstrap iterations.

References

Henze, N., Meintanis, S.G., Ebner, B. (2012) "Goodness-of-fit tests for the Gamma distribution based on the empirical Laplace transform". Communications in Statistics - Theory and Methods, 41(9): 1543-1556. DOI

Examples

test.AD(stats::rgamma(20,3,6),boot=100)

---

The Betsch-Ebner goodness-of-fit test for the gamma family

Description

This function computes the goodness-of-fit test for the gamma family due to Betsch and Ebner (2019).

Usage

test.BE(data, a = 1, boot = 500, alpha = 0.05)

Arguments

data	a vector of positive numbers.
a	positive tuning parameter.
boot	number of bootstrap iterations used to obtain critical value.
alpha	level of significance of the test.

Details

The test is of weighted $L^2$ type and uses a characterization of the distribution function of the gamma distribution. Critical values are obtained by a parametric bootstrap procedure, see crit.values.

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$T.value

the value of the test statistic.

$cv

the approximated critical value.

$par.est

number of points used in approximation.

$Decision

the comparison of the critical value and the value of the test statistic.

$sig.level

level of significance chosen.

$boot.run

number of bootstrap iterations.

References

Betsch, S., Ebner, B. (2019) "A new characterization of the Gamma distribution and associated goodness of fit tests", Metrika, 82(7):779-806. DOI

Examples

test.BE(stats::rgamma(20,3,6),boot=100)

---

The Cramer-von Mises goodness-of-fit test for the gamma family

Description

This function computes the goodness-of-fit test for the gamma family in the spirit of Cramer and von Mises. Note that this tests the composite hypothesis of fit to the family of gamma distributions, i.e. a bootstrap procedure is implemented to perform the test.

Usage

test.CM(data, boot = 500, alpha = 0.05)

Arguments

data	a vector of positive numbers.
boot	number of bootstrap iterations used to obtain critical value.
alpha	level of significance of the test.

Details

The Cramér-von Mises test is computed as described in Henze et. al. (2012). Critical values are obtained by a parametric bootstrap procedure, see crit.values.

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$T.value

the value of the test statistic.

$cv

the approximated critical value.

$par.est

number of points used in approximation.

$Decision

the comparison of the critical value and the value of the test statistic.

$sig.level

level of significance chosen.

$boot.run

number of bootstrap iterations.

References

Henze, N., Meintanis, S.G., Ebner, B. (2012) "Goodness-of-fit tests for the Gamma distribution based on the empirical Laplace transform". Communications in Statistics - Theory and Methods, 41(9): 1543-1556. DOI

Examples

test.CM(stats::rgamma(20,3,6),boot=100)

---

The first Henze-Meintanis-Ebner goodness-of-fit test for the gamma family

Description

This function computes the first goodness-of-fit test for the gamma family due to Henze, Meintanis and Ebner (2012).

Usage

test.HME1(data, a = 1, boot = 500, alpha = 0.05)

Arguments

data	a vector of positive numbers.
a	positive tuning parameter.
boot	number of bootstrap iterations used to obtain critical value.
alpha	level of significance of the test.

Details

The test is of weighted $L^2$ type and uses a characterization of the distribution function of the gamma distribution. Critical values are obtained by a parametric bootstrap procedure, see crit.values.

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$T.value

the value of the test statistic.

$cv

the approximated critical value.

$par.est

number of points used in approximation.

$Decision

the comparison of the critical value and the value of the test statistic.

$sig.level

level of significance chosen.

$boot.run

number of bootstrap iterations.

References

Henze, N., Meintanis, S.G., Ebner, B. (2012) "Goodness-of-fit tests for the Gamma distribution based on the empirical Laplace transform". Communications in Statistics - Theory and Methods, 41(9): 1543-1556. DOI

Examples

test.HME1(stats::rgamma(20,3,6),boot=100)

---

The second Henze-Meintanis-Ebner goodness-of-fit test for the gamma family

Description

This function computes the second goodness-of-fit test for the gamma family due to Henze, Meintanis and Ebner (2012).

Usage

test.HME2(data, a = 4, boot = 500, alpha = 0.05)

Arguments

data	a vector of positive numbers.
a	positive tuning parameter.
boot	number of bootstrap iterations used to obtain critical value.
alpha	level of significance of the test.

Details

The test is of weighted $L^2$ type and uses a characterization of the distribution function of the gamma distribution. Critical values are obtained by a parametric bootstrap procedure, see crit.values.

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$T.value

the value of the test statistic.

$cv

the approximated critical value.

$par.est

number of points used in approximation.

$Decision

the comparison of the critical value and the value of the test statistic.

$sig.level

level of significance chosen.

$boot.run

number of bootstrap iterations.

References

Henze, N., Meintanis, S.G., Ebner, B. (2012) "Goodness-of-fit tests for the Gamma distribution based on the empirical Laplace transform". Communications in Statistics - Theory and Methods, 41(9): 1543-1556. DOI

Examples

test.HME2(stats::rgamma(20,3,6),boot=100)

---

The Kolmogorov-Smirnov goodness-of-fit test for the gamma family

Description

This function computes the goodness-of-fit test for the gamma family in the spirit of Kolmogorov and Smirnov. Note that this tests the composite hypothesis of fit to the family of gamma distributions, i.e. a bootstrap procedure is implemented to perform the test.

Usage

test.KS(data, boot = 500, alpha = 0.05)

Arguments

data	a vector of positive numbers.
boot	number of bootstrap iterations used to obtain critical value.
alpha	level of significance of the test.

Details

The Kolmogorov Smirnov test is computed as described in Henze et. al. (2012). Critical values are obtained by a parametric bootstrap procedure, see crit.values.

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$T.value

the value of the test statistic.

$cv

the approximated critical value.

$par.est

number of points used in approximation.

$Decision

the comparison of the critical value and the value of the test statistic.

$sig.level

level of significance chosen.

$boot.run

number of bootstrap iterations.

References

Henze, N., Meintanis, S.G., Ebner, B. (2012) "Goodness-of-fit tests for the Gamma distribution based on the empirical Laplace transform". Communications in Statistics - Theory and Methods, 41(9): 1543-1556. DOI

Examples

test.KS(stats::rgamma(20,3,6),boot=100)

---

The Watson goodness-of-fit test for the gamma family

Description

This function computes the goodness-of-fit test for the gamma family in the spirit of Watson. Note that this tests the composite hypothesis of fit to the family of gamma distributions, i.e. a bootstrap procedure is implemented to perform the test.

Usage

test.WA(data, boot = 500, alpha = 0.05)

Arguments

data	a vector of positive numbers.
boot	number of bootstrap iterations used to obtain critical value.
alpha	level of significance of the test.

Details

The Watson test is computed as described in Henze et. al. (2012). Critical values are obtained by a parametric bootstrap procedure, see crit.values.

Value

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$T.value

the value of the test statistic.

$cv

the approximated critical value.

$par.est

number of points used in approximation.

$Decision

the comparison of the critical value and the value of the test statistic.

$sig.level

level of significance chosen.

$boot.run

number of bootstrap iterations.

References

Henze, N., Meintanis, S.G., Ebner, B. (2012) "Goodness-of-fit tests for the Gamma distribution based on the empirical Laplace transform". Communications in Statistics - Theory and Methods, 41(9): 1543-1556. DOI

Examples

test.WA(stats::rgamma(20,3,6),boot=100)

---

statistic of the Watson goodness-of-fit test for the gamma family

Description

This function computes the goodness-of-fit test statistic for the gamma family in the spirit of Watson. Note that this tests the composite hypothesis of fit to the family of gamma distributions, i.e. a bootstrap procedure is implemented to perform the test, see crit.values.

Usage

WA(data, k_estimator)

Arguments

data	a vector of positive numbers. NOTE: data has to be the rescaled data, i.e. devided by the estimated scale parameter!
k_estimator	value of the estimated shape parameter.

Details

The Watson test is computed as described in Henze et. al. (2012). Values of k_estimator are found by gamma_est.

Value

value of the test statistic

References

Henze, N., Meintanis, S.G., Ebner, B. (2012) "Goodness-of-fit tests for the Gamma distribution based on the empirical Laplace transform". Communications in Statistics - Theory and Methods, 41(9): 1543-1556. DOI

Examples

X=stats::rgamma(20,3,6)
WA(X,k_estimator=gamma_est(X)[1])

---