Abstract
We propose a new class of goodness-of-fit tests for the inverse Gaussian distribution based on a characterization of the cumulative distribution function (CDF). The new tests are of weighted L2-type depending on a tuning parameter. We develop the asymptotic theory under the null hypothesis and under a broad class of alternative distributions. These results guarantee 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.
Highlights
The inverse Gaussian distribution was first heuristically observed by [1], and derived by [2] as the distribution of the first passage time of Brownian motion with drift, see [3] for a historical summary
Since the characterization is directly related to the theory of Stein characterizations, we refer to the corresponding characterization of the generalized inverse Gaussian distribution, as in Theorem 3.2 of [20], and to the connection with the Stein operator for the special case (p, a, b) = (−1/2, λ/μ2, λ), using the authors’ notation
Our novel testing procedure is motivated by Theorem 1: We estimate both sides of (3) by their empirical counterparts, calculate a weighted L2-distance of the difference. These L2-type statistics are widely used in goodness-of-fit testing; see [25]
Summary
The inverse Gaussian distribution ( known as the Wald distribution) was first heuristically observed by [1], and derived by [2] as the distribution of the first passage time of Brownian motion with drift, see [3] for a historical summary. In [10] a connection to the so-called random walk distribution is used, and [11] proposes exact tests based on the empirical distribution function of transformations characterizing the inverse Gaussian law, which are corrected in [12]. The authors of [17] tackle the testing problem for the generalized inverse Gaussian family exploiting the ULAN property in connection to Le Cam theory. Since the characterization is directly related to the theory of Stein characterizations (for details on Stein operators, see [24]), we refer to the corresponding characterization of the generalized inverse Gaussian distribution, as in Theorem 3.2 of [20], and to the connection with the Stein operator for the special case (p, a, b) = (−1/2, λ/μ2, λ), using the authors’ notation. Our novel testing procedure is motivated by Theorem 1: We estimate both sides of (3) by their empirical counterparts, calculate a weighted L2-distance of the difference These L2-type statistics are widely used in goodness-of-fit testing; see [25].
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