On some distributions arising from certain generalized sampling schemes

With the notion of success in a series of trials extended tD refer to a run of like outcomes, several new distributions are obtained as the result of sampling from an urn without replacement. or with additional replacements., In this context, the hy-pergeometric, negative hypergeometric, logarithmic series, generalized Waring, Polya and inverse Polya distributions are extended and their properties are studied     

On some inflated generalized discrete distributions

Generalized discrete distributions such as the double Poisson and the double binomial family of Lagrange distributions are considered when the probabilities are inflated by a constant λ (0 < λ < 1). In each of the above cases, the effect of inflation on the variance is discussed. Also, the Bayesian estimate of inflation as well as those of the parameters are attempted. A maximum likelihood method is also suggested.     

Characterization of a class cf discrete distributions by properties of their moment distributions

The concept of a moment distribution is here applied to the class of factorial series distributions introduced by the author (197U). Moment distributions turn out to have a simple interpretation in this case and often take on a simple form. A few examples are given to illustrate this fact     

Multivariate discrete exponential family of distributions and their properties

The paper generalizes the univariate discrete exponential family of distributions to the multivariate situation, and this generalization includes the multivariate power series distributions, the multivariate Lagrangian distributions, and the modified multivariate power-series distributions. This provides a unified approach for the study of these three classes of distributions. We obtain recurrence relations for moments and cumulants, and the maximum likelihood estimation for the discrete exponential family. These results are applied to some multivariate discrete distributions like the Lagrangian Poisson, Lagrangian (negative) multinomial, logarithmic series distributions and multivariate Lagrangian negative binomial distribution.     

A wider class of lagrange distributions of the second kind

Starting with the second Lagrange expansion, with f(z) and g(z) as two probability generating functions defined on non-negative integers, Janardan and Rao( 1983) introduced a new class of discrete distributions called the Lagrange Distributions (LD₂) of the second kind. In this note, this class of LD₂ distributions is widened by removing the restriction that the functions f(z) and g(z) be probability generation functions. It is also shown that the class of modified power series distributions is a subclass of LD₂.     

Invariance of linearity of regression and related characterizations of some classical discrete distributions

Rao (1963) introduced what we call an additive damage model. In this model, original observation is subjected to damage according to a specified probability law by the survival distribution. In this paper, we consider a bivariate observation with second component subjected to damage. Using the invariance of linearity of regression of the first component on the second under the transition of the second component from the original to the damaged state, we obtain the characterizations of the Poisson, binomial and negative binomial distributions within the framework of the additive damage model.     

Correlation between the numbers of two types of children in a family with the mpsd for the family size

The family size (sibship size) N is regarded as an integer-valued random variable having a Modified Power Series distribution (MPSD) of Gupta (1974). The family produces two types of children, with probabilities p and q (p+q =1) . It is proved that the correlation between the numbers B and C of these children is positive or negative according as the function log f(θ) is convex or concave with respect to the function g(θ), (see Section 2). This condition is a simple and a natural extension of the one given by Rao et al (1973). Several examples are discussed to illustrate the result.     

Functional equations in characterizations of discrete distributions ev rao-rubin condition and its variants

The solutions of Cauchy functional equation and its special cases, studied by various authors while characterizing discrete distributions by Rao-Rubin condition and its variants, axe discussed in this survey paper. Certain new results on the solutions of generalizations of these functional equations are also mentioned     

A Characterization For Gpsd Signals In Additive Noise

In this paper we consider a convoluted generalized power series distibution and characterize the distributions by soiutions to system of differential equations. Characterization results are derived for Poisson, binomial, geometric and Pascal (negative binomial) as special cases and later unified with Samaniego [1976, 1980] and Samaniego and Gong [1979]

receiveddate="Oct1985" reviseddate="Jun1986"     

On discrete weighted distributions and their use in model choice for observed data

This paper provides a brief structural perspective of discrete weighted distributions in theory and practice.. It develops a unified view of previous work involving univariate and bivariate models with some new results pertaining to mixtures, form-invariance and Bayesian inference     

On some distributions arising in inverse cluster sampling

In this paper, distributions of items sampled inversely in clusters are derived. In particular, negative binomial type of distributions are obtained and their properties are studied. A logarithmic series type of distribution is also defined as a limiting form of the obtained generalized negative binomial distribution.     

Inverse moments of discrete distributions

In this paper an expression for the inverse moment of order r is given for the truncated binomial and Poisson distributions. This enables one to obtain inverse moments in a finite series. Some applications and multivariate generalizations are also given. The method also enables one to obtain relations between inverse moments and factorial moments and distributions of sums of variables.     

Effect of length-biased sampling on the modeling error

In statistical data analysis, the choice of an appropriate model is a very important factor. An inappropriate model leads to a different kind of error in the analysis. This error has been called by C. R. Rao as type III error or modeling error as opposed to type I and type II errors in statistical inference.In This paper we Study the relative errors in Incurred by Erroneously Assuming the Distribution of the Family Size N as P(n) While in fact it is the Length-biased (Weighted) Version of P(n).An Analytical Expression for the Relative Error,When the Distribution of N Belongs to the Class of Modified Power Series Distributions, is Derived. More Specifically, the Effect of length-biasing on the Relative Error is Investigated, When N Follows a Generalized Poisson Distribution. These Results are Compared With the Case When N Follows a Poisson Distribution.     

On a class of bivariate mixed Sarmanov distributions

Multivariate distributions are more and more used to model the dependence encountered in many fields. However, classical multivariate distributions can be restrictive by their nature, while Sarmanov's multivariate distribution, by joining different marginals in a flexible and tractable dependence structure, often provides a valuable alternative. In this paper, we introduce some bivariate mixed Sarmanov distributions with the purpose to extend the class of bivariate Sarmanov distributions and to obtain new dependency structures. Special attention is paid to the bivariate mixed Sarmanov distribution with Poisson marginals and, in particular, to the resulting bivariate Sarmanov distributions with negative binomial and with Poisson‐inverse Gaussian marginals; these particular types of mixed distributions have possible applications in, for example modelling bivariate count data. The extension to higher dimensions is also discussed. Moreover, concerning the dependency structure, we also present some correlation formulas.     

A new generalized two-sided class of distributions with an emphasis on two-sided generalized normal distribution

The ordinary-G class of distributions is defined to have the cumulative distribution function (cdf) as the value of the cdf of the ordinary distribution F whose range is the unit interval at G, that is, F(G), and it generalizes the ordinary distribution. In this work, we consider the standard two-sided power distribution to define other classes like the beta-G and the Kumaraswamy-G classes. We extend the idea of two-sidedness to other ordinary distributions like normal. After studying the basic properties of the new class in general setting, we consider the two-sided generalized normal distribution with maximum likelihood estimation procedure.     

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.     

Some bivariate families of lagrangian probability distributions

The bivariate Lagrange expansion, given by Poincare (1986), has been explained and slightly modified which gives bivariate Lagrangian probability models. A generalized bivariate Lagrangian Poisson distribution with six parameters has been obtained and studied. Also, the bivariate Lagrangian binomial, bivariate Lagrangian negative binomial and bivariate Lagrangian logarithmic series distribution have been obtained.     

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.     

On certain moment ratios of a class of discrete distributions

Two moment ratios are proposed for their uses In discriminating member among a family of dlscrete distributions and In approximating a member by another one Approximation of the Generalized Poisson( Borel–Tanner) and Neyman Type A distributions by the negatlve blnomial are also given.