Gamma Spacings as Finite Gamma Mixtures

IJSES, 2021

Over the last few decades, mixture distributions are used in creating population from two or more distributions. Mixture distributions are a good application in the applications of medical science, biology, engineering, finance and economics. Gaussian mixture models have broad utility, including their usage for model-based clustering framework. Recently, there are indications to use of non-Gaussian mixture distributions to skewed and asymmetric data. We propose a mixture model of inverse power Gamma shape distributions (MIPGSD) to analyze positive data. Basic structural properties such raw and central moments and hazard rate function are obtained. Different estimation methods are studied to estimate the proposed model parameters. Simulation studies is done to present the performance and behavior of the different estimates of the proposed model parameters. A real data set is provided to compare the reliability of the new model with other models.

Journal of Statistical Planning and Inference, 2000

Let X 1; X2; : : : be a sequence of independent random variables with a common continuous distribution function F and let X1;n6 · · · 6Xn;n be order statistics based on random variables X1; : : : ; Xn; n = 1; 2; : : : : The dependence structure of order statistics X k; n and X l; n+m is discussed. Conditional distributions of generalized spacings X k; n − X l; n+m are given under the condition that X l; n+m is ÿxed. For exponential distributions independence of vectors (T1; : : : ; Tn) and (X1;n+m; X2;n+m; : : : ; X l; n+m ), where T k = max(0; X k; n − X l; n+m ), is proved for any n; m and 16l6m + n. The latter fact is useful in ÿnding joint moments of exponential order statistics X k; n and X l; n+m . A characterization of distributions by properties of regression functions '(x) = E(X1;n − X1;n+m|X1;n+m = x) is obtained.

Statistical Methods in Medical Research, 2021

In this paper, we concentrate on the statistical properties of Gamma-X family of distributions. A special case of this family is the Gamma-Weibull distribution. Therefore, the statistical properties of Gamma-Weibull distribution as a sub-model of Gamma-X family are discussed such as moments, variance, skewness, kurtosis and Rényi entropy. Also, the parameters of the Gamma-Weibull distribution are estimated by the method of maximum likelihood. Some sub-models of the Gamma-X are investigated, including the cumulative distribution, probability density, survival and hazard functions. The Monte Carlo simulation study is conducted to assess the performances of these estimators. Finally, the adequacy of Gamma-Weibull distribution in data modeling is verified by the two clinical real data sets. Mathematics Subject Classification: 62E99; 62E15

Journal of Statistical Computation and Simulation, 2015

It is the aim of this note to point out that the double gamma difference distribution recently introduced by Augustyniak and Doray (2012) is wellknown in financial econometrics: it is the symmetric variance gamma family of distributions. We trace back to the various origins of this distribution. In addition, we consider in some detail the difference of two independent gamma distributed random variables with different shape parameters.

Annals of Operations Research, 2012

Generalizations of the results of an earlier paper of the second author, related to the problem of fitting a multivarite gamma distribution to empirical data, are discussed in the paper. The multivariate gamma distribution under consideration is the one that was introduced in the paper of Prékopa and Szántai (1975), some earlier results on the fitting problem were given in the paper of Szántai (1984). In the present paper it is proved that the necessary conditions given earlier are not sufficient and some further new, mostly computational results are provided, too. Using the more efficient computation tools we are able now to give the sufficient conditions for dimensions 5 and 6 as well. For higher dimensions we have only necessary conditions and the invention of a suitable necessary and sufficient condition remains an open problem when n is greater than 6. The miscellaneousness of the necessary and sufficient conditions obtained in our new project for n = 6 indicates that finding necessary and sufficient conditions in general should be a very hard problem.

Statistics, Optimization & Information Computing, 2020

In this article a new bivariate distribution, whose both the marginals are finite mixture of gamma distribution has been defined. Several of its properties such moments, correlation coefficients, measure of skewness, moment generating function, Renyi and Shannon entropies have been derived. Simulation study have been conducted to evaluate the performance of maximum likelihood method.

Annals of the Institute of Statistical Mathematics, 2014

We determine the joint limiting distribution of adjacent spacings around a central, intermediate, or an extreme order statistic X k:n of a random sample of size n from a continuous distribution F. For central and intermediate cases, normalized spacings in the left and right neighborhoods are asymptotically i.i.d. exponential random variables. The associated independent Poisson arrival processes are independent of X k:n. For an extreme X k:n , the asymptotic independence property of spacings fails for F in the domain of attraction of Fréchet and Weibull (α = 1) distributions. This work also provides additional insight into the limiting distribution for the number of observations around X k:n for all three cases.

A bimodal extension of the generalized gamma distribution is proposed by using a mixing approach. Some distributional properties of the new distribution are investigated. The maximum likelihood (ML) estimators for the parameters of the new distribution are obtained. Real data examples are given to show the strength of the new distribution for modeling data.

2007

The purpose of this paper is to investigate conditions on the underlying distribution functions and the parameters, on which the generalized order statistics are based on, to establish the usual stochastic and the likelihood ratio orderings of general p-spacings by conditioning on the right tail of another lower indexed generalized order statistics. Some potential applications are also given.

The International Journal of Contemporary …, 2009

Order statistics from an exponentiated gamma (EG) distribution are considered. Exact expression for the single and double moments of order statistics from EG distribution are derived. Based on the moments of order statistics, the best linear unbiased estimators (BLUE's) for the location and scale parameters of EG distribution under Type-II censoring are obtained. The variances and covariances of these estimators are also presented. The maximum likelihood estimator (MLE) for the shape parameter is derived. Also, estimators based on order statistics for the shape parameter are obtained. Finally, comparisons between the estimators are made based on simulation study.