A Goodness of Fit Test for Normality Based on the Empirical Moment Generating Function

Zghoul, Ahmad A.

doi:10.1080/03610918.2010.490318

Cited by 12 publications

(11 citation statements)

References 15 publications

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“…where γ > 2 is a smoothing parameter. Based on the finite-sample performance reported in [36], the author recommended setting γ equal to either 3 or 15. The numerical results presented below include the powers obtained using both of these values for the smoothing parameter; the resulting tests are denoted by Z 3 and Z 15 respectively.…”

Section: Monte Carlo Resultsmentioning

confidence: 99%

Testing for normality in any dimension based on a partial differential equation involving the moment generating function

Henze

Visagie

2019

Ann Inst Stat Math

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We use a system of first-order partial differential equations that characterize the moment generating function of the d-variate standard normal distribution to construct a class of affine invariant tests for normality in any dimension. We derive the limit null distribution of the resulting test statistics, and we prove consistency of the tests against general alternatives. In the case d > 1, a certain limit of these tests is connected with two measures of multivariate skewness. The new tests show strong power performance when compared to well-known competitors, especially against heavy-tailed distributions, and they are illustrated by means of a real data set.

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Section: Monte Carlo Resultsmentioning

confidence: 99%

Testing for normality in any dimension based on a partial differential equation involving the moment generating function

Henze

Visagie

2019

Ann Inst Stat Math

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“…An advantage lies in the range of their applicability. A substantial proportion of known procedures relies on a comparison between theoretical moment generating functions, seeCabaña and Quiroz (2005),, andZghoul (2010), or characteristic functions, as employed byBaringhaus and Henze (1988),Epps andPulley (1983), andJiménez-Gamero et al (2009),…”

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confidence: 99%

Fixed point characterizations of continuous univariate probability distributions and their applications

Betsch

Ebner

2019

Ann Inst Stat Math

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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...

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“…Wong and Sim (2000) proposed a test which is also based on an integral of the squared modulus of the difference between the empirical characteristic function and the normal characteristic function. Recently, Zghoul (2010) proposed a test based on the empirical moment generating function. This test is found to be more powerful than other tests when testing against stable distributions.…”

Section: Introductionmentioning

confidence: 99%

Normality Test Based on a Truncated Mean Characterization

Zghoul¹,

Awad²

2010

Communications in Statistics - Simulation and Computation

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This article generalizes a characterization based on a truncated mean to include higher truncated moments, and introduces a new normality goodness-of-fit test based on the truncated mean. The test is a weighted integral of the squared distance between the empirical truncated mean and its expectation. A closed form for the test statistic is derived. Assuming known parameters, the mean and the variance of the test are derived under the normality assumption. Moreover, a limiting distributionfor the proposed test as well as an approximation are obtained. Also, based on Monte Carlo simulations, the power of the test is evaluated against stable, symmetric, and skewed classes of distributions. The test proves compatibility with prominent tests and shows higher power for a wide range of alternatives.

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A Goodness of Fit Test for Normality Based on the Empirical Moment Generating Function

Cited by 12 publications

References 15 publications

Testing for normality in any dimension based on a partial differential equation involving the moment generating function

Testing for normality in any dimension based on a partial differential equation involving the moment generating function

Fixed point characterizations of continuous univariate probability distributions and their applications

Normality Test Based on a Truncated Mean Characterization

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