A new family of generalized distributions

Kumaraswamy [Generalized probability density-function for double-bounded random-processes, J. Hydrol. 462 (1980), pp. 79–88] introduced a distribution for double-bounded random processes with hydrological applications. For the first time, based on this distribution, we describe a new family of generalized distributions (denoted with the prefix ‘Kw’) to extend the normal, Weibull, gamma, Gumbel, inverse Gaussian distributions, among several well-known distributions. Some special distributions in the new family such as the Kw-normal, Kw-Weibull, Kw-gamma, Kw-Gumbel and Kw-inverse Gaussian distribution are discussed. We express the ordinary moments of any Kw generalized distribution as linear functions of probability weighted moments (PWMs) of the parent distribution. We also obtain the ordinary moments of order statistics as functions of PWMs of the baseline distribution. We use the method of maximum likelihood to fit the distributions in the new class and illustrate the potentiality of the new model with an application to real data.     

A generalization for two-sided power distributions and adjusted method of moments

A new rich class of generalized two-sided power (TSP) distributions, where their density functions are expressed in terms of the Gauss hypergeometric functions, is introduced and studied. In this class, the symmetric distributions are supported by finite intervals and have normal shape densities. Our study on TSP distributions also leads us to a new class of discrete distributions on {0, 1, …, k}. In addition, a new numerical method for parameter estimation using moments is given.     

Generalized polya eggenberger family of distributions and its relation to lagrangian katz family

Janardan (1973) introduced the generalized Polya Eggenberger family of distributions (GPED) as a limiting distribution of the generalized Markov-Polya distribution (GMPD). Janardan and Rao (1982) gave a number of characterizing properties of the generalized Markov-Polya and generalized Polya Eggenberger distributions. Here, the GPED family characterized by four parameters, is formally defined and studied. The probability generating function, its moments, and certain recurrence relations with the moments are provided. The Lagrangian Katz family of distributions (Consul and Famoye (1996)) is shown to be a sub-class of the family of GPED (or GPED ₁ ) as it is called in this paper). A generalized Polya Eggenberger distribution of the second kind (GPED ₂ ) is also introduced and some of it's properties are given. Recurrence relations for the probabilities of GPED ₁ and GPED ₂ are given. A number of other structural and characteristic properties of the GPED ₁ are provided, from which the properties of Lagrangian Katz family follow. The parameters of GMPD ₁ are estimated by the method of moments and the maximum likelihood method. An application is provided.     

Recurrence relations for single and product moments of k-th record values from pareto,generalized pareto and burr distributions

In this note we give recurrence relations satisfied by single and product momenrs of k-th upper-record values from the Pareto, generalized Pareto and Burr distributions. From these relations one can obtain all the single and product moments of all k-th record values and at the same time all record values ( k=1). Moreover, we see that the single and product moment of all k-th record values from these distributions can be exprrssed in terms of the moments of the minimal statistic of a k-sample from the exponential distribution may be deduced by letting the shape parameter deptend to 0. At the end we give characterizations of the three distributions considered. These results generalize, among other things, those given by Balakrishnan and Abuamllah (1994).     

Order statistics from non-identical power function random variables

In this paper, we derive some recurrence relations satisfied by the single and the product moments of order statistics arising from n independent and non-identically distributed power function random variables. These recurrence relations will enable one to compute all the single and the product moments of all order statistics in a simple recursive manner. The results for the multiple-outlier model are deduced as special cases. The results are further generalized to the case of truncated power function random variables.     

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.     

An approximation of the inverse moments of the positive hypergeometric distribution

The negative moments of the positive hyper geometric distribution are often approximated by the inverse of the positive moments of this distribution. In this paper, a suitable approximation to the positive hypergeometric distribution is used to obtain the negative moments.     

Relationships Between Central Moments and Cumulants,with Formulae for the Central Moments of Gamma Distributions

Abstract

Simple expressions are presented that relate cumulants to central moments without involving moments about the origin. These expressions are used to obtain recursive formulae for the central moments of the gamma distribution, with exponential and chi-square distributions as special cases.     

Negative moments of a modified power series distribution and bias of the maximum likelihood estimator

The class of Modified Power Series distributions (MPSD) containing Lagrangian Poisson (LPD) (Consul and Jain, 1973) and Lagrangian binomial distributions (LBD) (Jain and Consul, 1971) was studied by Gupta (1974). We investigate the problem of finding the negative momentsE[X^-r ], of displaced and decapitated Modified Power Series Distributions. We derive the relationship between rand (r-1) negative moments. The negative moments of the decapitated and displaced LPD are obtained. These results are, then, used to find the exact amount of bias in the ML estimators of the parameters in the LPD and the LBD. We have also given the variances of the ML estimator and the minimum variance unbiased estimator of the parameter in the LPD.     

On approximating the central and noncentral multivariate gamma distributions

In this paper a finite series approximation involving Laguerre polynomials is derived for central and noncentral multivariate gamma distributions. It is shown that if one approximates the density of any k nonnegative continuous random variables by a finite series of Laguerre polynomials up to the (n₁, …, n_k)th degree, then all the mixed moments up to the order (n₁, …, n_k) of the approximated distribution equal to the mixed moments up to the same order of the random variables. Some numerical results are given for the bivariate central and noncentral multivariate gamma distributions to indicate the usefulness of the approximations.     

Niveaudurchagangszeiten zur charakterisierung sequentieller schätzverfahren

Let θ be a parameter of a homogenous additive stochastic process. In order to get an unbiased and efficient estimator for a function h(v) one has often to use sequential procedures. In this paper we consider processes of the socalled exponential class. We study level crossing times, which characterize certain sequential estimations. It is shown that the family of level crossing times for an increasing sequence of levels is also a process of the exponential class. The density function of the one-dimensional probability distributions of this new process is given Examples and applications conclude the paper.     

A class of maximum-entropy multivariate distributions

This paper characterizes a class of multivariate distributions that includes the multinormal and is contained in the exponential family. The wide range of possible applications of these distributions is suggested by some of hte characteristics germane to them: First, they maximize Shannon's entropy among all distributions that have finite moments of given orders. As such, they constitute a class of distributions that includes the multinormal and some likely alternatives. Second, they can exhibit several modes, and, further-more, they do so with a relatively small number of parameters (compared to mixtures of multinormals). Third, they are the stationary distributions of certain diffusion processes. Fourth, they approximate, near the multinormal, the multivariate Pearson family. And fifth, the maximum likelihood estimators of their population moments are the sample moments. Two possible methods of estimating the distributions are studied in this paper: maximum likelihood estimation, and a fast procedure that can be used to find consistent estimators of the parameters via sample moments. A FORTTAN subroutine that implements the latter method is also provided.     

Multivariate lagrange distributions and a subfamiliy

Second order moments about its means, i.e. the variances and covari-ances for multivariate Lagrange distributions are derived in a matrix form. A subfamily of multivariate Lagrange distributions which can be characterized as the distributions of customers served in a busy period in queues with some conditions are considered. Theorems about their probability functions, one of which is a multivariate generalization of a formula by Takà cs(1989). are given and the means and second order moments about its means are considered. As an example, a multivariate Borel-Tanner distribution is derived.     

The unit-inverse Gaussian distribution: A new alternative to two-parameter distributions on the unit interval

A new two-parameter distribution over the unit interval, called the Unit-Inverse Gaussian distribution, is introduced and studied in detail. The proposed distribution shares many properties with other known distributions on the unit interval, such as Beta, Johnson S_B, Unit-Gamma, and Kumaraswamy distributions. Estimation of the parameters of the proposed distribution are obtained by transforming the data to the inverse Gaussian distribution. Unlike most distributions on the unit interval, the maximum likelihood or method of moments estimators of the parameters of the proposed distribution are expressed in simple closed forms which do not need iterative methods to compute. Application of the proposed distribution to a real data set shows better fit than many known two-parameter distributions on the unit interval.     

Transmuted Weibull distribution: Properties and estimation

In this article, we investigate the potential usefulness of the three-parameter transmuted Weibull distribution for modeling survival data. The main advantage of this distribution is that it has increasing, decreasing or constant instantaneous failure rate depending on the shape parameter and the new transmuting parameter. We obtain several mathematical properties of the transmuted Weibull distribution such as the expressions for the quantile function, moments, geometric mean, harmonic mean, Shannon, Rényi and q-entropies, mean deviations, Bonferroni and Lorenz curves, and the moments of order statistics. We propose a location-scale regression model based on the log-transmuted Weibull distribution for modeling lifetime data. Applications to two real datasets are given to illustrate the flexibility and potentiality of the transmuted Weibull family of lifetime distributions.     

Computational comparison of the general weighted moments estimators of the scale parameter of a Pareto distribution with a known shape parameter with other estimators based on a multiply type II-censored sample

Wu et al. [Computational comparison for weighted moments estimators and BLUE of the scale parameter of a Pareto distribution with known shape parameter under type II multiply censored sample, Appl. Math. Comput. 181 (2006), pp. 1462–1470] proposed the weighted moments estimators (WMEs) of the scale parameter of a Pareto distribution with known shape parameter on a multiply type II-censored sample. They claimed that some WMEs are better than the best linear unbiased estimator (BLUE) based on the exact mean-squared error (MSE). In this paper, the general WME (GWME) is proposed and the computational comparison of the proposed estimator with the WMEs and BLUE is done on the basis of the exact MSE for given sample sizes and different censoring schemes. As a result, the GWME is performing better than the best estimator among 12 WMEs and BLUE for all cases. Therefore, GWME is recommended for use. At last, one example is given to demonstrate the proposed GWME.     

Moments of Order Statistics—From Even to Odd Sample Sizes

Abstract

A method is demonstrated to compute the complete set of first moments of order statistics for an arbitrary distribution, given only the first moments of the maximal order statistics either for all even sample sizes, or for all odd samples sizes.     

GMM tests for the Katz family of distributions

Generalized method of moments (GMM) is used to develop tests for discriminating discrete distributions among the two-parameter family of Katz distributions. Relationships involving moments are exploited to obtain identifying and over-identifying restrictions. The asymptotic relative efficiencies of tests based on GMM are analyzed using the local power approach and the approximate Bahadur efficiency. The paper also gives results of Monte Carlo experiments designed to check the validity of the theoretical findings and to shed light on the small sample properties of the proposed tests. Extensions of the results to compound Poisson alternative hypotheses are discussed.     

Characterizing parameters of multivariate elliptical distributions*

This paper defines new parameters characterizing multivariate elliptical distributions. Mardia's coefficient of multivariate kurtosis is shown to be essentially one of these parameters. A simple relation is established between centered multivariate product moments and second moments of the variables. The general results are verified on the contaminated normal distribution as an example.     

Lower confidence bounds as precision measure for truncated processes

Process capability index C_p has been the most popular one used in the manufacturing industry to provide numerical measures on process precision. For normally distributed processes with automatic fully inspections, the inspected processes follow truncated normal distributions. In this article, we provide the formulae of moments used for the Edgeworth approximation on the precision measurement C_p for truncated normally distributed processes. Based on the developed moments, lower confidence bounds with various sample sizes and confidence levels are provided and tabulated. Consequently, practitioners can use lower confidence bounds to determine whether their manufacturing processes are capable of preset precision requirements.