Goodness-of-Fit Tests for the Gamma Distribution Based on the Empirical Laplace Transform

We propose a class of goodness-of-fit tests for the gamma distribution that utilizes the empirical Laplace transform. The consistency of the tests as well as their asymptotic distribution under the null hypothesis are investigated. As the decay of the weight function tends to infinity, the test statistics approach limit values related to the first non zero component of Neyman's smooth test for the gamma law. The new tests are compared with other omnibus tests for the gamma distribution.     

Jacobi and Laguerre polynomial approximations for the distributions of statistics useful in testing for outliers in exponential and gamma samples

Recently, Sanjel and Balakrishnan [A Laguerre Polynomial Approximation for a goodness-of-fit test for exponential distribution based on progressively censored data, J. Stat. Comput. Simul. 78 (2008), pp. 503–513] proposed the use of Laguerre orthogonal polynomials for a goodness-of-fit test for the exponential distribution based on progressively censored data. In this paper, we use Jacobi and Laguerre orthogonal polynomials in order to obtain density approximants for some test statistics useful in testing for outliers in gamma and exponential samples. We first obtain the exact moments of the statistics and then the density approximants, based on these moments, are expressed in terms of Jacobi and Laguerre polynomials. A comparative study is carried out of the critical values obtained by using the proposed methods to the corresponding results given by Barnett and Lewis [Outliers in Statistical Data, 3rd ed., John Wiley & Sons, New York, 1993]. This reveals that the proposed techniques provide very accurate approximations to the distributions. Finally, we present some numerical examples to illustrate the proposed approximations. Monte Carlo simulations suggest that the proposed approximate densities are very accurate.     

Exponentiality test based on Renyi distance between equilibrium distributions

ABSTRACT

On the basis of Csiszar's φ-divergence discrimination information, we propose a measure of discrepancy between equilibriums associated with two distributions. Proving that a distribution can be characterized by associated equilibrium distribution, a Renyi distance of the equilibrium distributions is constructed that made us to propose an EDF-based goodness-of-fit test for exponential distribution. For comparing the performance of the proposed test, some well-known EDF-based tests and some entropy-based tests are considered. Based on the simulation results, the proposed test has better powers than those of competing entropy-based tests for the alternatives with decreasing hazard rate function. The use of the proposed test is evaluated in an illustrative example.     

Inference procedures for the variance gamma model and applications

Goodness-of-fit tests for the family of the four-parameter normal–variance gamma distribution are constructed. The tests are based on a weighted integral incorporating the empirical characteristic function of suitably standardized data. Non-standard algorithms are employed for the computation of the maximum-likelihood estimators of the parameters involved in the test statistic, while Monte Carlo results are used in order to compare the new test with some classical goodness-of-fit methods. A real-data application is also included.     

A gamma goodness-of-fit test based on characteristic independence of the mean and coefficient of variation

Characterization theorems in probability and statistics are widely appreciated for their role in clarifying the structure of the families of probability distributions. Less well known is the role characterization theorems have as a natural, logical and effective starting point for constructing goodness-of-fit tests. The characteristic independence of the mean and variance and of the mean and the third central moment of a normal sample were used, respectively, by Lin and Mudholkar [1980. A simple test for normality against asymmetric alternatives. Biometrika 67, 455–461] and by Mudholkar et al. [2002a. Independence characterizations and testing normality against skewness-kurtosis alternatives. J. Statist. Plann. Inference 104, 485–501] for developing tests of normality. The characteristic independence of the maximum likelihood estimates of the population parameters was similarly used by Mudholkar et al. [2002b. Independence characterization and inverse Gaussian goodness-of-fit. Sankhya A 63, 362–374] to develop a test of the composite inverse Gaussian hypothesis. The gamma models are extensively used for applied research in the areas of econometrics, engineering and biomedical sciences; but there are few goodness-of-fit tests available to test if the data indeed come from a gamma population. In this paper we employ Hwang and Hu's [1999. On a characterization of the gamma distribution: the independence of the sample mean and the sample coefficient of variation. Ann. Inst. Statist. Math. 51, 749–753] characterization of the gamma population in terms of the independence of sample mean and coefficient of variation for developing such a test. The asymptotic null distribution of the proposed test statistic is obtained and empirically refined for use with samples of moderate size.     

A review of model selection procedures relevant to the weibull distribution

A number of goodness-of-fit and model selection procedures related to the Weibull distribution are reviewed. These procedures include probability plotting, correlation type goodness-of-fit tests, and chi-square goodness-of-fit tests. Also the Kolmogorow-Smirniv, Kuiper, and Cramer-Von Mises test statistics for completely specified hypothesis based on censored data are reviewed, and these test statistics based on complete samples for the unspecified parameters case are considered. Goodness-of-fit tests based on sample spacings, and a goodness-of-fit test for the Weibull process, is also discussed.

Model selection procedures for selecting between a Weibull and gamma model, a Weibull and lognormal model, and for selecting from among all three models are considered. Also tests of exponential versus Weibull and Weibull versus generalized gamma are mentioned.     

Goodness-of-Fit Tests for the Skew-Normal Distribution When the Parameters Are Estimated from the Data

In this article, tests are developed which can be used to investigate the goodness-of-fit of the skew-normal distribution in the context most relevant to the data analyst, namely that in which the parameter values are unknown and are estimated from the data. We consider five test statistics chosen from the broad Cramér–von Mises and Kolmogorov–Smirnov families, based on measures of disparity between the distribution function of a fitted skew-normal population and the empirical distribution function. The sampling distributions of the proposed test statistics are approximated using Monte Carlo techniques and summarized in easy to use tabular form. We also present results obtained from simulation studies designed to explore the true size of the tests and their power against various asymmetric alternative distributions.     

Goodness-of-fit tests for the Gompertz distribution

Abstract

While the Gompertz distribution is often fitted to lifespan data, testing whether the fit satisfies theoretical criteria is being neglected. Here four goodness-of-fit measures – the Anderson–Darling statistic, the correlation coefficient test, a statistic using moments, and a nested test against the generalized extreme value distributions – are discussed. Along with an application to laboratory rat data, critical values calculated by the empirical distribution of the test statistics are also presented.     

An extension of the gamma distribution

ABSTRACT

The gamma distribution has been widely used in many research areas such as engineering and survival analysis. We present an extension of this distribution, called the Kummer beta gamma distribution, having greater flexibility to model scenarios involving skewed data. We derive analytical expressions for some mathematical quantities. The estimation of parameters is approached by the maximum likelihood method and Bayesian analysis. The likelihood ratio and formal goodness-of-fit tests are used to compare the presented distribution with some of its sub-models and non nested models. A real data set is used to illustrate the importance of the distribution.     

Kolmogorov-Smirnov Type Test-Statistics For The Gamma,Erlang-2 And The Inverse Gaussian Distributions When The Parameters Are Unknown.

Critical values are presented for the Kolmogorov-Smirnov type test statistics for the following three cases: (i) the gamma distribution when both the scale and the shape parameters are not known, (ii) the scale parameter of the gamma distribution is not known and (iii) the inverse Gaussian distribution when both the parameters are unknown. This study was motivated by the necessity to fit the gamma, the Erlang-2 and the inverse Gaussian distributions to the interpurchase times of individuals for coffee in marketing research.     

Powers of Discrete Goodness-of-Fit Test Statistics for a Uniform Null Against a Selection of Alternative Distributions

The comparative powers of six discrete goodness-of-fit test statistics for a uniform null distribution against a variety of fully specified alternative distributions are discussed. The results suggest that the test statistics based on the empirical distribution function for ordinal data (Kolmogorov–Smirnov, Cramér–von Mises, and Anderson–Darling) are generally more powerful for trend alternative distributions. The test statistics for nominal (Pearson's chi-square and the nominal Kolmogorov–Smirnov) and circular data (Watson's test statistic) are shown to be generally more powerful for the investigated triangular (∨), flat (or platykurtic type), sharp (or leptokurtic type), and bimodal alternative distributions.     

Estimation of the parameters of the Gompertz distribution under the first failure-censored sampling plan

This article presents the goodness-of-fit tests for the Laplace distribution based on its maximum entropy characterization result. The critical values of the test statistics estimated by Monte Carlo simulations are tabulated for various window and sample sizes. The test statistics use an entropy estimator depending on the window size; so, the choice of the optimal window size is an important problem. The window sizes for yielding the maximum power of the tests are given for selected sample sizes. Power studies are performed to compare the proposed tests with goodness-of-fit tests based on the empirical distribution function. Simulation results report that entropy-based tests have consistently higher power than EDF tests against almost all alternatives considered.     

Marshall–Olkin gamma–Weibull distribution with applications

Abstract

A Marshall–Olkin variant of the Provost type gamma–Weibull probability distribution is being introduced in this paper. Some of its statistical functions and numerical characteristics among others characteristics function, moment generalizing function, central moments of real order are derived in the computational series expansion form and various illustrative special cases are discussed. This density function is utilized to model two real data sets. The new distribution provides a better fit than related distributions as measured by the Anderson–Darling and Cramér–von Mises statistics. The proposed distribution could find applications for instance in the physical and biological sciences, hydrology, medicine, meteorology, engineering, etc.     

Goodness-of-fit testing by transforming to normality: comparison between classical and characteristic function-based methods

Chen and Balakrishnan [Chen, G. and Balakrishnan, N., 1995, A general purpose approximate goodness-of-fit test. Journal of Quality Technology, 27, 154–161] proposed an approximate method of goodness-of-fit testing that avoids the use of extensive tables. This procedure first transforms the data to normality, and subsequently applies the classical tests for normality based on the empirical distribution function, and critical points thereof. In this paper, we investigate the potential of this method in comparison to a corresponding goodness-of-fit test which instead of the empirical distribution function, utilizes the empirical characteristic function. Both methods are in full generality as they may be applied to arbitrary laws with continuous distribution function, provided that an efficient method of estimation exists for the parameters of the hypothesized distribution.     

Graphical procedures and goodness-of-fit tests for the compound poisson-exponential distribution

ABSTRACT

The compound Poisson-exponential distribution is a basic model in risk analysis and stochastic hydrology. Graphical procedures for assessing this distribution are proposed which utilize the residuals from a regression involving the moment generating function. Plots furnished with a 95% simultaneous confidence band are constructed. The band and critical points of the equivalent goodness-of-fit test are found by utilizing asymptotic results and fitted regressions involving the supremum of the standardized residuals, the sample size, and the estimated Poisson mean. Simulation results indicate that the tests have good level stability and appreciable power against competing compound Poisson distributions of a mixed type.     

Customizing Generalizations of the Kolmogorov-Smirnov Goodness-of-fit Test

This paper presents a method of customizing goodness-of-fit tests that transforms the empirical distribution function in such a way as to create tests for certain alternatives. Using the @ , g transform described in Blom(1958), one can create non-parametric tests for an assortment of alternative distributions. As examples, three new ( f , g )-corrected Kolmogorov-Smirnov tests for goodness-of-fit are discussed. One of these tests is powerful for testing whether or not the data come from an alternative that is heavier in the tails. Another test identifies whether or not the data come from an alternative which is heavier in the middle of the distribution. The last test identifies if the data come from an alternative in which the first or third quartile is far from the corresponding quartile of the hypothesized distribution. The behavior of the three new tests is investigated through a power study.     

Cumulant Plots for Assessing the Gamma Distribution

This article introduces graphical procedures for assessing the fit of the gamma distribution. The procedures are based on a standardized version of the cumulant generating function. Plots with bands of 95% simultaneous confidence level are developed by utilizing asymptotic and finite-sample results. The plots have linear scales and do not rely on the use of tables or values of special functions. Further, it is found through simulation, that the goodness-of-fit test implied by these plots compares favorably with respect to power to other known tests for the gamma distribution in samples drawn from lognormal and inverse Gaussian distributions.     

Entropy estimation and goodness-of-fit tests for the inverse Gaussian and Laplace distributions using paired ranked set sampling

ABSTRACT

In this paper, Vasicek [A test for normality based on sample entropy. J R Stat Soc Ser B. 1976;38:54–59] entropy estimator is modified using paired ranked set sampling (PRSS) method. Also, two goodness-of-fit tests using PRSS are suggested for the inverse Gaussian and Laplace distributions. The new suggested entropy estimator and goodness-of-fit tests using PRSS are compared with their counterparts using simple random sampling (SRS) via Monte Carlo simulations. The critical values of the suggested tests are obtained, and the powers of the tests based on several alternatives hypotheses using SRS and PRSS are calculated. It turns out that the proposed PRSS entropy estimator is more efficient than the SRS counterpart in terms of root mean square error. Also, the proposed PRSS goodness-of-fit tests have higher powers than their counterparts using SRS for all alternative considered in this study.     

Testing for spherical symmetry via the empirical characteristic function

Kolmogorov–Smirnov-type and Cramér–von Mises-type goodness-of-fit tests are proposed for the null hypothesis that the distribution of a random vector X is spherically symmetric. The test statistics utilize the fact that X has a spherical symmetric distribution if, and only if, the characteristic function of X is constant over surfaces of spheres centred at the origin. Both tests come in convenient forms that are straightforwardly applicable with the computer. The asymptotic null distribution of the test statistics as well as the consistency of the tests is investigated under general conditions. Since both the finite sample and the asymptotic null distribution depend on the unknown distribution of the Euclidean norm of X, a conditional Monte Carlo procedure is used to actually carry out the tests. Results on the behaviour of the test in finite-samples are included along with a real-data example.     

An Estimator of Shannon Entropy of Beta-Generated Distributions and a Goodness-of-Fit Test

The aim of this article is twofold: on the one hand to introduce and study some of the statistical properties of an estimator for the Shannon entropy and on the other hand to develop a goodness-of-fit test for beta-generated distributions and the distribution of order statistics. Beta-generated distributions are a broad class of univariate distributions which has received great attention during the last 15 years, as it obeys nice properties and it extends the distribution of order statistics. The proposed estimator of Shannon entropy of beta-generated distributions is motivated by the respective Vasicek’s estimator, as the latter one is tailored to the class of the beta-generated distributions and the distribution of order statistics. The estimator of Shannon entropy is defined and its consistency is studied. It is, moreover, exploited to build a goodness-of-fit test for the beta-generated distribution and the distribution of order statistics. Simulations are performed to examine the small- and moderate-sample properties of the proposed estimator and to compare the power of the proposed test with the power of competitors under a variety of alternatives.