An ‘apples to apples’ comparison of various tests for exponentiality

2019

Ann Inst Stat Math

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...

Section: Tests For the Gamma Distributionmentioning

confidence: 99%

Fixed point characterizations of continuous univariate probability distributions and their applications

2019

Ann Inst Stat Math

“…A test based on G (1) n or G (2) n rejects H 0 for large values of the statistic. A suggestion for the choice of the weight function and equivalent expressions for G (1) n and G (2) n that are suitable for computations can be found in section 6.…”

Section: The Proposed Test Statisticsmentioning

confidence: 99%

“…[42]). Stating that X has a standard normal distribution if, and only if, E f ′ (X) − Xf (X) = 0 (2) holds for each absolutely continuous function f for which the expectation exists, it appears reasonable to regard the left hand side of (2), for a suitable function f , as an estimate of Eh(X) − Eh(N ) since both terms ought to be small whenever the distribution of X is close to standard normal. In practice, solving the differential equation…”

mentioning

confidence: 99%

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

2019

TEST

We propose two families of tests for the classical goodness-of-fit problem to univariate normality. The new procedures are based on L 2 -distances of the empirical zero-bias transformation to the normal distribution or the empirical distribution of the data, respectively. Weak convergence results are derived under the null hypothesis, under fixed alternatives as well as under contiguous alternatives. Empirical critical values are provided and a comparative finite-sample power study shows the competitiveness to classical procedures. IntroductionTesting normality is commonly known as the most used and discussed goodness-of-fit technique, motivated by the model assumption of normality in classical models. To be specific, let X, X 1 , X 2 , . . . be real valued independent and identically distributed (iid.) random variables.The problem of interest is to test the hypothesisagainst general alternatives. So far, a great variety of goodness-of-fit tests have been proposed and research is of ongoing interest, as witnessed by the recent papers [8, 21, 45] and comparative simulation studies like [35, 46]. Classical procedures in goodness-of-fit methodology as the Kolmogorov-Smirnov and the Cramér-von Mises test approach the testing problem by American Mathematical Society 2010 subject classifications. Primary 62G10 Secondary 60F05

“…The resulting test is consistent against general alternatives (cf. [6]) and has already been included in the extensive simulation study [3] delivering a remarkable performance in terms of the tests power. We also want to emphasize that our Theorem 1.2 is a vast generalization of the characterization (8).…”

mentioning

confidence: 99%

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

2019

Metrika

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.