## P-value of Anderson-Darling statistic

### Description

Compute the P-value of the given Anderson-Darling statistic A^2 using imhof function in CompQuadForm.

### Usage

AD.uniform.pvalue(a, neig = 100, verbose = FALSE)

AD.normal.pvalue(a, neig = 100, verbose = FALSE)

AD.gamma.pvalue(a, shape, neig = 100, verbose = FALSE)

AD.logistic.pvalue(a, neig = 100, verbose = FALSE)

AD.laplace.pvalue(a, neig = 100, verbose = FALSE)

AD.weibull.pvalue(a, neig = 100, verbose = FALSE)

AD.extremevalue.pvalue(a, neig = 100, verbose = FALSE)

AD.exp.pvalue(a, neig = 100, verbose = FALSE)


### Arguments

 a Anderson-Darling statistic A^2 with a given distribution. neig Number of eigenvalues used for imhof. verbose Logical; if TRUE, print warning messages. shape The shape parameter of Gamma distribution.

### Details

Parameters must be estimated by maximum likelihood (ML) in order for the P-values computed here to be asymptotically valid. They are computed using the fact that when parameters are estimated by maximum likelihood and the null hypothesis is true, the asymptotic distribution of the GOF statistic is the distribution of an infinite weighted sum of weighted chi-square random variables on 1 degree of freedom. The weights are eigenvalues of an integral equation. They depend on the distribution being tested, the statistic being used, and in some cases on the actual parameter values. These weights are then computed approximately by discretization of the integral equation; when that equation depends on one or more parameter values we use the MLE in the equation.

Some notes on the specific distributions: For the Normal, Logistic, Laplace, Extreme Value, Weibull and Exponential distributions, the limiting distributions do not depend on the parameters. For the Gamma distribution, the shape parameter affects the limiting distribution. The tests remain asymptotically valid when the MLE is used to approximate the limit distribution.

The Exponential distribution is a special case of the Weibull and Gamma families arising when the shape is known to be 1. Knowing a parameter and therefore not estimating it affects the distribution of the test statistic and the functions provided for the Exponential distribution allow for this.

If a data set X_1,...,X_n follows the Weibull distribution then Y_1 = log(X_1), ... ,Y_n = log(X_n) follows the Extreme Value distribution and vice versa. The two procedures give identical test statistics and P-values, in principal.

Some of the models have more than one common parametrization. For the Exponential, Gamma, and Weibull distributions, some writers use a rate parameter and some use the scale parameter which is the inverse of the rate. Our code uses the scale parameter.

For the Laplace distribution, some writers use the density f(x)= exp(-|x-\mu|/\beta)/(2\beta) in which \beta is a scale parameter. Others use the standard deviation \sigma = \beta/\sqrt{2}. Our code uses the scale parameter.

For the Uniform distribution, we offer code for the two parameter Uniform distribution on the range \theta_1 to \theta_2. These are estimated by the sample minimum and sample maximum. The probability integral transforms of the remaining n-2 points are then tested for uniformity on the range 0 to 1. This procedure is justified because the these probability integral transforms have exactly this distribution if the original data had a uniform distribution over any interval.

It is not unusual to test the hypothesis that a sample follows the standard uniform distribution on [0,1]. In this case the parameters should not be estimated. Instead use AD(z) or CvM(z) or Watson(z) to compute the statistic values and then get P-values from AD.uniform.pvalue(a) or CvM.uniform.pvalue(w) or Watson.uniform.pvalue(u) whichever is wanted.

### Value

P-value of the Anderson-Darling statistic.

AD for calculating Anderson-Darling statistic; CvM.pvalue for calculating P-value of CramÃ©r-von Mises statistic; Watson.pvalue for calculating P-value of Watson statistic.

### Examples

x0=runif(n=100,min=-1,max=1)

x1=rnorm(n=100,mean=0,sd=1)

x2=rgamma(n=100,shape=1,scale=1)

x3=rlogis(n=100,location=0,scale=1)

x4=rmutil::rlaplace(n=100,m=0,s=1)