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

Betsch, Steffen; Ebner, Bruno

doi:10.1007/s00184-019-00708-7

Cited by 26 publications

(29 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Note that this result has been proven explicitly, and with a similar line of proof as our general results above, by Betsch and Ebner (2019a).…”

Section: Univariate Distributions With Semi-bounded Supportsupporting

confidence: 76%

“…Remark 3.2. Note that some contributions to the scientific literature [like Ley and Swan (2011), Ley and Swan (2013b), Betsch and Ebner (2019a)] claim that the function…”

Section: The Density Approach Identitymentioning

confidence: 99%

“…As witnessed by Baringhaus and Henze (2000), Betsch and Ebner (2019a), and Betsch and Ebner and their empirical pendants, or on a differential equation that characterizes the Laplace transformation, see Henze and Klar (2002) and Henze et al (2012). All of these share the unpleasant feature that in order to establish the theoretic basis for the test statistics, one has to have explicit knowledge about these transformations for the distribution in consideration.…”

Section: Applications To Goodness-of-fit Testingmentioning

confidence: 99%

“…Betsch and Ebner (2019a), the authors establish the result of Corollary 5.6 for the special case of the Gamma distribution and examine the corresponding goodness-of-fit statistic. Denote byp ϑ (x) = λ −k Γ(k) x k−1 e −x/λ , x > 0, where ϑ = (k, λ) ∈ (0, ∞) 2 = Θ,the density function of the Gamma distribution with shape parameter k and scale parameter λ.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Fixed point characterizations of continuous univariate probability distributions and their applications

Betsch

Ebner

2019

Ann Inst Stat Math

Self Cite

View full text Add to dashboard 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...

show abstract

“…Note that this result has been proven explicitly, and with a similar line of proof as our general results above, by Betsch and Ebner (2019a).…”

Section: Univariate Distributions With Semi-bounded Supportsupporting

confidence: 76%

“…Remark 3.2. Note that some contributions to the scientific literature [like Ley and Swan (2011), Ley and Swan (2013b), Betsch and Ebner (2019a)] claim that the function…”

Section: The Density Approach Identitymentioning

confidence: 99%

Section: Applications To Goodness-of-fit Testingmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Fixed point characterizations of continuous univariate probability distributions and their applications

Betsch

Ebner

2019

Ann Inst Stat Math

Self Cite

View full text Add to dashboard Cite

show abstract

“…Chen, Goldstein & Shao (2011). The aim of this work is to investigate these characterizations, which where already used to construct goodness-of-fit tests (see Betsch & Ebner, 2019b, more closely in the context of parameter estimation. An advantage of the resulting estimators lies in the way the density function of the underlying model appears in the characterization, and thus also in the estimation method.…”

Section: Introductionmentioning

confidence: 99%

MinimumL^q‐distance estimators for non‐normalized parametric models

2020

Self Cite

View full text Add to dashboard Cite

We propose and investigate a new estimation method for the parameters of models consisting of smooth density functions on the positive half axis. The procedure is based on a recently introduced characterization result for the respective probability distributions, and is to be classified as a minimum distance estimator, incorporating as a distance function the L q -norm. Throughout, we deal rigorously with issues of existence and measurability of these implicitly defined estimators. Moreover, we provide consistency results in a common asymptotic setting, and compare our new method with classical estimators for the exponential, the Rayleigh and the Burr Type XII distribution in Monte Carlo simulation studies. We also assess the performance of different estimators for non-normalized models in the context of an exponential-polynomial family.

show abstract

Characteristic function and Laplace transform‐based tests for exponentiality in the presence of random right censoring

Bothma

Allison

Cockeran

et al. 2021

Stat

View full text Add to dashboard Cite

In this paper, the composite hypothesis that lifetimes follow an exponential distribution is tested based on observed randomly right censored data. Testing this hypothesis is complicated by the presence of this censoring, due to the fact that not all lifetimes are observed. To account for this complication, we propose modifications to tests based on the empirical characteristic function and Laplace transform. In the full sample case, these empirical functions can be expressed as integrals with respect to the empirical distribution function of the lifetimes. We propose replacing this estimate of the distribution function by the Kaplan–Meier estimate. The resulting test statistics can be expressed in easily calculable forms in terms of summations of functionals of the observed data. Additionally, a general framework for goodness‐of‐fit testing, in the presence of random right censoring, is outlined. A Monte Carlo study is performed, the results of which indicate that the newly modified tests generally outperform the existing tests. A practical application, concerning initial remission times of leukaemia patients, is discussed along with some concluding remarks and avenues for future research.

show abstract

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

Cited by 26 publications

References 24 publications

Fixed point characterizations of continuous univariate probability distributions and their applications

Fixed point characterizations of continuous univariate probability distributions and their applications

MinimumL^q‐distance estimators for non‐normalized parametric models

Characteristic function and Laplace transform‐based tests for exponentiality in the presence of random right censoring

Contact Info

Product

Resources

About

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

Cited by 26 publications

References 24 publications

Fixed point characterizations of continuous univariate probability distributions and their applications

Fixed point characterizations of continuous univariate probability distributions and their applications

MinimumLq‐distance estimators for non‐normalized parametric models

Characteristic function and Laplace transform‐based tests for exponentiality in the presence of random right censoring

Contact Info

Product

Resources

About

MinimumL^q‐distance estimators for non‐normalized parametric models