ABEV1 {gofIG}R Documentation

The first Allison-Betsch-Ebner-Visagie test statistic

Description

This function computes the first test statistic of the goodness-of-fit tests for the inverse Gaussian family due to Allison et al. (2022). Two different estimation procedures are implemented, namely the method of moment and the maximum likelihood method.

Usage

ABEV1(data, a = 10, meth = "MME")

Arguments

data

a vector of positive numbers.

a

positive tuning parameter.

meth

method of estimation used. Possible values are 'MME' for moment estimation and 'MLE' for maximum likelihood estimation.

Details

The numerically stable test statistic for the first Allison-Betsch-Ebner-Visagie test is defined as:

ABEV1_{n,a} = \frac{1}{4n} \sum_{j,k=1}^{n} \left( \hat{\varphi}_n + \frac{3}{Y_{n,j}} - \frac{\hat{\varphi}_n}{Y_{n,j}^2} \right) \left( \hat{\varphi}_n + \frac{3}{Y_{n,k}} - \frac{\hat{\varphi}_n}{Y_{n,k}^2} \right) h_{1,a}(Y_{n,j}, Y_{n,k})

- 2 \left( \hat{\varphi}_n + \frac{3}{Y_{n,j}} - \frac{\hat{\varphi}_n}{Y_{n,j}^2} \right) h_{2,a}(Y_{n,j}, Y_{n,k})

- 2 \left( \hat{\varphi}_n + \frac{3}{Y_{n,k}} - \frac{\hat{\varphi}_n}{Y_{n,k}^2} \right) h_{2,a}(Y_{n,k}, Y_{n,j})

+ \frac{4}{a} e^{-a \max(Y_{n,j}, Y_{n,k})},

with \hat{\varphi}_n = \frac{\hat{\lambda}_n}{\hat{\mu}_n}, where \hat{\mu}_n,\hat{\lambda}_n are consistent estimators of \mu, \lambda, respectively, the parameters of the inverse Gaussian distribution. Furthermore Y_{n,j} = \frac{X_j}{\hat{\mu}_n}, j = 1,...,n, for (X_j)_{j = 1,...,n}, a sequence of independent observations of a positive random variable X. The functions h_{i,a}(s,t), i = 1,2, are defined in Allison et al. (2022), section 5.1. The null hypothesis is rejected for large values of the test statistic ABEV1_{n,a}.

Value

value of the test statistic.

References

Allison, J.S., Betsch, S., Ebner, B., Visagie, I.J.H. (2022) "On Testing the Adequacy of the Inverse Gaussian Distribution". LINK

Examples

ABEV1(rmutil::rinvgauss(20,2,1),a=10,meth='MLE')


[Package gofIG version 1.0 Index]