New weighted $L^2$-type tests for the inverse Gaussian distribution

Abstract

We propose a new class of goodness-of-fit tests for the inverse Gaussian distribution. The proposed tests are weighted L 2 -type tests depending on a tuning parameter. We develop the asymptotic theory under the null hypothesis and under a broad class of alternative distributions. These results are used to show that the parametric bootstrap procedure, which we employ to implement the test, is asymptotically valid and that the whole test procedure is consistent. A comparative simulation study for finite sample sizes shows that the new procedure is competitive to classical and recent tests, outperforming these other methods almost uniformly over a large set of alternative distributions. The use of the newly proposed test is illustrated with two observed data sets.

Stein's Method Meets Statistics: A Review of Some Recent Developments

“…In a similar vein, Betsch and Ebner (2021) and Betsch et al (2020b) provide new characterizations of continuous and discrete parametric families of distributions through the density approach for novel tests for univariate normality (Betsch and Ebner, 2020), the gamma family (Betsch and Ebner, 2019) and the inverse Gaussian law (Allison et al, 2019). Note that other test statistics of type (37) based on Stein operators are implicitly proposed in tests for parametric families, although originally motivated by characterizing (partial) differential equations for integral transforms; see, for instance, Baringhaus and Henze (1991) for a test of exponentiality, Baringhaus and Henze (1992) for a test of Poissonity, and Henze et al (2012) for a test of the gamma law.…”