Testing normality via a distributional fixed point property in the Stein characterization

2019

Ann Inst Stat Math

Self Cite

By extrapolating the explicit formula of the zero-bias distribution occurring in the context of Stein's method, we construct characterization identities for a large class of absolutely continuous univariate distributions. Instead of trying to derive characterizing distributional transformations that inherit certain structures for the use in further theoretic endeavours, we focus on explicit representations given through a formula for the density-or distribution function. The results we establish with this ambition feature immediate applications in the area of goodness-of-fit testing. We draw up a blueprint for the construction of tests of fit that include procedures for many distributions for which little (if any) practicable tests are known.To illustrate this last point, we construct a test for the Burr Type XII distribution for which, to our knowledge, not a single test is known aside from the classical universal procedures.MSC 2010 subject classifications. Primary 62E10 Secondary 60E10, 62G10Over the last decades, Stein's method for distributional approximation has become a viable tool for proving limit theorems and establishing convergence rates. At it's heart lies the well-known Stein characterization which states that a real-valued random variable Z has a standard normal distribution if, and only if,holds for all functions f of a sufficiently large class of test functions. To exploit this characterization for testing the hypothesisof normality, where P X is the distribution of a real-valued random variable X, against general alternatives, Betsch and Ebner (2019b) used that (1.1) can be untied from the class of test functions with the help of the so-called zero-bias transformation introduced by Goldstein and Reinert (1997). To be specific, a real-valued random variable X * is said to have the X-zero-bias distri-bution ifholds for any of the respective test functions f . If EX = 0 and Var(X) = 1, the X-zero-bias distribution exists and is unique, and it has distribution function(1.3)By (1.1), the standard Gaussian distribution is the unique fixed point of the transformation P X → P X * . Thus, the distribution of X is standard normal if, and only if,where F X denotes the distribution function of X. In the spirit of characterization-based goodnessof-fit tests, an idea introduced by Linnik (1962), this fixed point property directly admits a new class of testing procedures as follows. Letting T X n be an empirical version of T X and F n the empirical distribution function, both based on the standardized sample, Betsch and Ebner (2019b) proposed a test for (1.2) based on the statisticwhere w is an appropriate weight function, which, in view of (1.4), rejects the normality hypothesis for large values of G n . As these tests have several desirable properties such as consistency against general alternatives, and since they show a very promising performance in simulations, we devote this work to the question to what extent the fixed point property and the class of goodness-of-fit procedures may be generalized to othe...

“…The goodness-of-fit tests for normality proposed by Betsch and Ebner (2019b) are also included in our framework (cf. Example 4.4).…”

Section: Tests For Normalitymentioning

confidence: 99%

Section: Tests For Normalitymentioning

confidence: 99%

Fixed point characterizations of continuous univariate probability distributions and their applications

Betsch

2019

Ann Inst Stat Math

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“…From this result we conclude that a test based on G n is able to detect these alternatives. To our knowledge, this setting has not yet been examined in the context of goodness-of-fit tests for Gamma distributions, however, the standard reasoning for this type of contiguous alternatives (see, for instance, [21] or [9]) works well.…”

Section: The Behaviour Under Contiguous Alternativesmentioning

confidence: 99%

“…Remark. Note that [9] used the similar zero bias transformation (introduced by [15]) for testing normality. By analogy with this transformation, the proof of Theorem 1.2 also shows that if X is chosen such that T X is a distribution function, there exists a random variable…”

mentioning

confidence: 99%

A new characterization of the Gamma distribution and associated goodness-of-fit tests

Betsch

2019

Metrika

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We propose a class of weighted L 2 -type tests of fit to the Gamma distribution. Our novel procedure is based on a fixed point property of a new transformation connected to a Steinian characterization of the family of Gamma distributions. We derive the weak limits of the statistic under the null hypothesis and under contiguous alternatives. Further, we establish the global consistency of the tests and apply a parametric bootstrap technique in a Monte Carlo simulation study to show the competitiveness to existing procedures.

“…Note that these alternatives can also be found in the simulation studies presented in Betsch and Ebner (2020), Dörr et al (2020), Romão et al (2010). We chose these alternatives in order to ease the comparison with many other existing tests.…”

Section: Simulationsmentioning

confidence: 99%

A new test of multivariate normality by a double estimation in a characterizing PDE

Dörr

Henze

2020

Metrika

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This paper deals with testing for nondegenerate normality of a d-variate random vector X based on a random sample X 1 ,. .. , X n of X. The rationale of the test is that the characteristic function ψ(t) = exp(− t 2 /2) of the standard normal distribution in R d is the only solution of the partial differential equation Δ f (t) = (t 2 −d) f (t), t ∈ R d , subject to the condition f (0) = 1, where Δ denotes the Laplace operator. In contrast to a recent approach that bases a test for multivariate normality on the difference Δψ n (t)− (t 2 − d)ψ(t), where ψ n (t) is the empirical characteristic function of suitably scaled residuals of X 1 ,. .. , X n , we consider a weighted L 2-statistic that employs Δψ n (t) − (t 2 − d)ψ n (t). We derive asymptotic properties of the test under the null hypothesis and alternatives. The test is affine invariant and consistent against general alternatives, and it exhibits high power when compared with prominent competitors. The main difference between the procedures are theoretically driven by different covariance kernels of the Gaussian limiting processes, which has considerable effect on robustness with respect to the choice of the tuning parameter in the weight function.