Relation of modified power series distributions to lagrangian probability distributions

The class of Lagrangian probability distributions ‘LPD’, given by the expansion of a probability generating function f‘t’ under the transformation u = t/g‘t’ where g ‘t’ is also a p.g.f., has been substantially widened by removing the restriction that the defining functions g ‘t’ and f‘t’ be probability generating functions. The class of modified power series distributions defined by Gupta ‘1974’ has been shown to be a sub-class of the wider class of LPDs     

M31. Least squares solutions over interval restrictions

In this article, the three-parameter I.G. distribution is standardized with zero mean and unit variance. The third standard moment α₃ is employed as the shape parameter. Tables of the cumulative probability function are given as a function of the standardized variate z, and of the shape parameter, α₃. Various comparisons are made with the lognormal, Weibull, and gamma distributions.     

Some theoretical results regarding the polygonal distribution

Polygonal distributions are a class of distributions that can be defined via the mixture of triangular distributions over the unit interval. We demonstrate that the densities of polygonal distributions are dense in the class of continuous and concave densities with bounded second derivatives. Furthermore, we prove that polygonal density functions provide O(g^{? 2}) approximations (where g is the number of triangular distribution components), in the supremum distance, to any density function from the hypothesized class. Parametric consistency and Hellinger consistency results for the maximum likelihood (ML) estimator are obtained. A result regarding model selection via penalized ML estimation is proved.     

An analysis of quantile measures of kurtosis: center and tails

The consequences of substituting the denominator Q ₃(p)  −  Q ₁(p) by Q ₂  −  Q ₁(p) in Groeneveld’s class of quantile measures of kurtosis (γ ₂(p)) for symmetric distributions, are explored using the symmetric influence function. The relationship between the measure γ ₂(p) and the alternative class of kurtosis measures κ₂(p) is derived together with the relationship between their influence functions. The Laplace, Logistic, symmetric Two-sided Power, Tukey and Beta distributions are considered in the examples in order to discuss the results obtained pertaining to unimodal, heavy tailed, bounded domain and U-shaped distributions. The authors thank the referee for the careful review.     

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.     

Some Utilizations of Lambert W Function in Distribution Theory

We introduce a new class of positive infinitely divisible probability laws calling them 𝔏_γ distributions. Their cumulant-generating functions (cgf) are expressed in terms of the principal branch of the Lambert W function. The probability density functions (pdfs) of 𝔏_γ laws are bounded resembling pdf of a Lévy stable distribution. The exponential dispersion model constructed starting from an 𝔏_γ distribution admits the inverse Gaussian approximation. The natural exponential family constructed starting from an 𝔏_γ distribution constitutes the reciprocal of the natural exponential family generated by a spectrally negative stable law with α = 1. We derive new results on 𝔏_γ laws and the related exponential dispersion models, including their convolution and scaling closure properties. We generate another exponential dispersion model starting from an exponentially compounded 𝔏_γ law. This distribution emerges in the Poisson mixture representation of a generalized Poisson law. We extend the Poisson approximation for the scaled Neyman type A exponential dispersion model. We derive saddlepoint-type approximations for some of these exponential dispersion models. The role of the Lambert W function is emphasized.     

On tail-ordering and comparison of failure rates

In this paper, we have studied some implications between tail-ordering (also known as dispersive ordering) and failure rate ordering (also called TP₂ ordering) of two probability distribution functions. Based on independent random samples from these distributions, a class of distribution-free tests has been proposed for testing the null hypothesis that the two life distributions are identical against the alternative that one failure rate is uniformly smaller than the other. The tests have good efficiencies as compared to their competitors.     

A characterization of invariant power-series abundance distributions

Consider a population the individuals in which can be classified into groups. Let y, the number of individuals in a group, be distributed according to a probability function f(y;ø_o) where the functional form f is known. The random variable y cannot be observed directly, and hence a random sample of groups cannot be obtained. Consider a random sample of N individuals from the population. Suppose the N individuals are distributed into S groups with x₁, x₂, …, x_S representatives respectively. The random variable x, the number of individuals in a group in the sample, will be a fraction of its population counterpart y, and the distributions of x and y need not have the same functional form. If the two random variables x and y have the same functional form for their distributions, then the particular common distribution is called an invariant abundance distribution. The paper provides a characterization of invariant abundance distributions in the class of power-series distributions.     

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.     

The 3 F 2 with Complex Parameters as Generating Function of Discrete Distribution

A new discrete family of probability distributions that are generated by the ₃ F ₂ function with complex parameters is presented. Some of the properties of this new family are studied as well as methods of estimation for its parameters. It affords considerable flexibility of shape which turns the distribution into an appropriate candidate for modeling data that cannot be adequately fitted by classical families with fewer parameters. Finally, three examples in the fields of Agriculture and Education are included in order to show the versatility and utility of this distribution.     

On optimal tests for separate hypotheses and conditional probability integral transformations

Consider the problem of testing the composite null hypothesis that a random sample X₁,…,X_n is from a parent which is a member of a particular continuous parametric family of distributions against an alternative that it is from a separate family of distributions. It is shown here that in many cases a uniformly most powerful similar (UMPS) test exists for this problem, and, moreover, that this test is equivalent to a uniformly most powerful invariant (UMPI) test. It is also seen in the method of proof used that the UMPS test statistic Is a function of the statistics U₁,…,U_n?k obtained by the conditional probability integral transformations (CPIT), and thus that no Information Is lost by these transformations, It is also shown that these optimal tests have power that is a nonotone function of the null hypothesis class of distributions, so that, for example, if one additional parameter for the distribution is assumed known, then the power of the test can not lecrease. It Is shown that the statistics U₁, …, U_n?k are independent of the complete sufficient statistic, and that these statistics have important invariance properties. Two examples at given. The UMPS tests for testing the two-parameter uniform family against the two-parameter exponential family, and for testing one truncation parameter distribution against another one are derived.     

A new class of probability distributions via cosine and sine functions with applications

ABSTRACT

In this paper, we introduce a new class of (probability) distributions, based on a cosine-sine transformation, obtained by compounding a baseline distribution with cosine and sine functions. Some of its properties are explored. A special focus is given to a particular cosine-sine transformation using the exponential distribution as baseline. Estimations of parameters of a particular cosine-sine exponential distribution are performed via the maximum likelihood estimation method. A simulation study investigates the performances of these estimates. Applications are given for four real data sets, showing a better fit in comparison to some existing distributions based on some goodness-of-fit tests.     

New class of location-parameter discrete probability distributions and their characterizations

A new class of location-parameter discrete probability distributions (LDPD) has been defined where the population mean is the location parameter. It has been shown that some single parameter discrete distributions do not belong to this class and all discrete probability distributions belonging to this class can be characterized by their variances only. Expressions are given for the first four central moments and a recurrence formula for higher central moments has been obtained. Eight theorems are given to characterize the various distributions in the LDPD class.     

Application domains for the Delta method

The Delta method uses truncated Lagrange expansions of statistics to obtain approximations to their distributions. In this paper, we consider statistics Y=g(μ+X), where X is any random vector. We obtain domains 𝒟 such that, when μ∈𝒟, we may apply the distribution derived from the Delta method. Namely, we will consider an application on the normal case to illustrate our approach.     

Partial differential equations for hypergeometric functions 3F2 of matrix argument

Many multivariate non-null distributions and moment formulas can be expressed in terms of hypergeometric functions _pF_q of matrix arqument. Muirhead [6] and Constantine and Muirhead [2] gave partial differential equations for the functions of ₂F₁ of one argument matrix and two argument matrices, respectively. Such differential equations have been used to obtain asymptotic expansions of the functions (Muirhead [7], [8], [9], Sugiura [10]). The purpose of this paper is to derive partial differential equations for the functions ₃F₂ (a₁ a₂, a₃; b₁, b₂, R) and ₃F₂ (a₁, a₂, a₃; b₁, b₂; R, S). Differential equations for ₂F₂ are also obtained.     

Discrete generalized exponential distribution of a second type

In this paper, we shall attempt to introduce another discrete analogue of the generalized exponential distribution of Gupta and Kundu [Generalized exponential distributions, Aust. N. Z. J. Stat. 41(2) (1999), pp. 173–188], different to that of Nekoukhou et al. [A discrete analogue of the generalized exponential distribution, Comm. Stat. Theory Methods, to appear (2011)]. This new discrete distribution, which we shall call a discrete generalized exponential distribution of the second type (DGE₂(α, p)), can be viewed as another generalization of the geometric distribution. We shall first study some basic distributional and moment properties, as well as order statistics distributions of this family of new distributions. Certain compounded DGE₂(α, p) distributions are also discussed as the results of which some previous lifetime distributions such as that of Adamidis and Loukas [A lifetime distribution with decreasing failure rate, Statist. Probab. Lett. 39 (1998), pp. 35–42] follow as corollaries. Then, we will investigate estimation of the parameters involved. Finally, we will examine the model with a real data set.     

ESTIMATING THE PARAMETERS OF NORMAL AND LOGISTIC DISTRIBUTIONS FROM CENSORED SAMPLES1

Let g(z) be the ratio of the ordinate and the probability integral of the distribution of a variate z. The relation g(z)～+βz is used to derive (i) estimators μ_r and s?_r of the parameters of a truncated normal distribution and (ii) estimators μ_c and s?_c of the mean and standard deviation of a logistic distribution from doubly censored samples. The variances and eovariances of these estimators are obtained. They are shown to be nearly as efficient as the maximum likelihood estimators and easier to compute.     

Phase Type Distributions with Finite Support

This research is motivated by the fact that many random variables of practical interest have a finite support. For fixed a < b, we consider the distribution of a random variable X = (a + Ymod(b ? a)), where Y is a phase type (PH) random variable. We demonstrate that as we traverse for Y the entire set of PH distributions (or even any subset thereof like Coxian that is dense in the class of distributions on [0, ∞)), we obtain a class of matrix exponential distributions dense in (a, b). We call these Finite Support Phase Type Distributions (FSPH) of the first kind. A simple example shows that though dense, this class by itself is not very efficient for modeling; therefore, we introduce (and derive the EM algorithms for) two other classes of finite support phase type distributions (FSPH). The properties of denseness, connection to Markov chains, the EM algorithm, and ability to exploit matrix-based computations should all make these classes of distributions attractive not only for applied probability but also for a much wider variety of fields using statistical methodologies.     

Limit Distributions of Random Sums of Z+-Valued Random Variables

In this article, we formulate a transfer theorem in terms of probability generating functions and discuss two approaches to limit distributions of random sums of Z ₊-valued random variables. We then develop Z ₊-valued N-ID and ?-ID laws.     

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.