ON CHARACTERIZING DISTRIBUTIONS VIA LINEARITY OF REGRESSION FOR ORDER STATISTICS

Let X₁X_n be a random sample from an absolutely continuous distribution with the corresponding order statistics X_1:n≤X_2:n≤X_n:n. A complete solution of the problem, posed in 1967 by T. Ferguson, of determining the distribution by linearity of regression of X_k+2:n with respect to X_k:n is given. The only possible distributions are of the exponential, power and Pareto type. A linear regression relation for exponents of order statistics is also considered.     

BIVARIATE DISTRIBUTION WITH TRUNCATED POISSON MARGINAL DISTRIBUTIONS

ABSTRACT

A bivariate distribution, whose marginal distributions are truncated Poisson distributions, is developed as a product of truncated Poisson distributions and a multiplicative factor. The multiplicative factor takes into account the correlation, either positive or negative, between the two random variables. The distributional properties of this model are studied and the model is fitted to a real life bivariate data.     

PROBABILITY APPROXIMATIONS FOR DIVISIBLE DISCRETE DISTRIBUTIONS

If an integer-valued random variable can be represented as a sum of independent random variables, then powerful tools exist to derive approximations to its distribution. We apply this idea to examples in some of which it is not clear how to give a physical interpretation to the independent sum-mands. We consider bounds on the accuracy of single term approximations, Edgeworth expansions and saddlepoint approximations for both individual probabilities and cumulative probabilities.     

A TEST OF INDEPENDENCE FOR BIVARIATE SYMMETRIC STABLE DISTRIBUTIONS

A test of independence in symmetric bivariate stable distributions is constructed using the empirical characteristic function as a test statistic. A particular class of distributions considered in detail is X = U + V, Y=V+W where U, V, and W are mutually independent symmetric stable distributions with the same index.     

ASPECTS OF CORRELATION IN BIVARIATE POISSON DISTRIBUTIONS AND PROCESSES

This paper reviews some interesting but scattered results that are known about correlation in bivariate Poisson distributions and processes and presents some new results. A particular concern in both contexts is with results and examples relating to negative correlation.     

ON ASYMPTOTIC OPTIMALITY OF SOME TESTS FOR EXPONENTIAL DISTRIBUTIONS

The probability density function (pdf) of a two parameter exponential distribution is given by f(x; p, s?) =s?^-1 exp {-(x - ρ)/s?} for x≥ρ and 0 elsewhere, where 0 < ρ < ∞ and 0 < s?∞. Suppose we have k independent random samples where the ith sample is drawn from the ith population having the pdf f(x; ρ_i, s?_i), 0 < ρ_i < ∞, 0 < s?_i < s?_i < and f(x; ρ, s?) is as given above. Let X_i1 < X_i2 <… < X_iri denote the first r_i order statistics in a random sample of size n_i, drawn from the ith population with pdf f(x; ρ_i, s?_i), i = 1, 2,…, k. In this paper we show that the well known tests of hypotheses about the parameters ρ_i, s?_i, i = 1, 2,…, k based on the above observations are asymptotically optimal in the sense of Bahadur efficiency. Our results are similar to those for normal distributions.     

CHARACTERISATIONS OF SOME INCOME DISTRIBUTIONS BASED ON MULTIPLICATIVE DAMAGE MODELS

This paper characterises the Pareto and scaled beta distributions within the context of multiplicative damage and generating models. The results obtained allow the destructive or generating mechanism to have more general distributions on (0,1) than the beta(e, l) distribution considered by several authors, thus generalising some recent results. Errors in earlier work are mentioned.     

ON CLASSIFICATION OF A BIVARIATE BINARY OBSERVATION

Classification of a bivariate binary observation into one of the two possible groups requires the estimation of the joint cell probabilities under each of the two groups. Two widely used approaches for the estimation of such joint cell probabilities are: [1] Seber, G. A. F. 1984. Multivariate Observations New York: John Wiley and Sons. [Crossref] , [Google Scholar] kernel based non-parametric approach; and [2] McLachlan, G. J. 1992. Discriminant Analysis and Statistical Pattern Recognition New York: John Wiley and Sons. [Crossref] , [Google Scholar] multinomial distribution based cell counts approach. In these traditional approaches, the joint cell probabilities are estimated without making any assumptions about the structural forms for these probabilities. Consequently, it is not clear, how these traditional approaches take into account the correlation that may exist between the 2-dimensional binary observations. In this paper, we model the cell probabilities by a suitable bivariate binary distribution which accommodates the correlation in a natural way, and examine the effect of this type of modelling in classifying a new correlated binary observation into one of the two groups. This is done by comparing the probability of misclassification yielded by the proposed model based approach with those of the kernel as well as multinomial distribution based approaches. It is shown through a simulation study that the probabilities of misclassification for the model based approach are substantially smaller than those of the other two approaches. We illustrate the use of the proposed model based approach in classification by analyzing a combined data from two epidemiological surveys of 6–11 year old children conducted in Connecticut, the New Haven Child Survey (NHCS) and the Eastern Connecticut Child Survey (ECCS).     

MINIMUM QUADRATIC DISTANCE ESTIMATION FOR A PARAMETRIC FAMILY OF DISCRETE DISTRIBUTIONS DEFINED RECURSIVELY

The distribution function of a random sum can easily be computed iteratively when the distribution of the number of independent identically distributed elements in the sum is itself defined recursively. Classical estimation procedures for such recursive parametric families often require specific distributional assumptions (e.g. Poisson, Negative Binomial). The minimum distance estimator proposed here is an estimator within a larger parametric family. The estimator is consistent, efficient when the parametric family is truncated, and can be made either robust or asymptotically efficient when the parametric family has infinite range. Its asymptotic distribution is derived. A brief illustration with Automobile Insurance data is included.     

A BIVARIATE BINOMIAL DISTRIBUTION AND SOME APPLICATIONS1

This study is concerned with the joint distribution of the total numbers of occurrences of binary characters A and B, given three independent samples in which both characters, A but not B, and B but not A, are observed. The distribution function is given; its conditional distributions and regression functions are found; bounds on certain joint probabilities are established; and conditions for bivariate Poisson and Gaussian limits are studied. An application yields the joint distribution of sign statistics for the pair-wise comparison of treatments with a control.     

SOME STATISTICAL DISTRIBUTIONS OF USE IN ANALYSING BIRD COUNT DATA1

The following report arose from an enquiry by a zoologist, J. M. Cullen, concerning various possible ways of analysing bird count data. He was particularly interested in two questions: (i) What is the best way to estimate abundances of various species?, and (ii) What evidence is there that birds occur randomly (in particular, as a Poisson process) as opposed to in flocks or according to some mutual repulsion process?     

ON ESTIMATION FOLLOWING SELECTION FROM NONREGULAR DISTRIBUTIONS

Independent random samples are drawn from k (≥ 2) populations, having probability density functions belonging to a general truncation parameter family. The populations associated with the smallest and the largest truncation parameters are called the lower extreme population (LEP) and the upper extreme population (UEP), respectively. For the goal of selecting the LEP (UEP), we consider the natural selection rule, which selects the population corresponding to the smallest (largest) of k maximum likelihood estimates as the LEP (UEP), and study the problem of estimating the truncation parameter of the selected population. We unify some of the existing results, available in the literature for specific distributions, by deriving the uniformly minimum variance unbiased estimator (UMVUE) for the truncation parameter of the selected population. The conditional unbiasedness of the UMVUE is also checked. The cases of the left and the right truncation parameter families are dealt with separately. Finally, we consider an application to the Pareto probability model, where the performances of the UMVUE and three other natural estimators are compared with each other, under the mean squared error criterion.     

ON CONVERGENCE RATES FOR NONPARAMETRIC POSTERIOR DISTRIBUTIONS

Rates of convergence of Bayesian nonparametric procedures are expressed as the maximum between two rates: one is determined via suitable measures of concentration of the prior around the “true” density f₀, and the other is related to the way the mass is spread outside a neighborhood of f₀. Here we provide a lower bound for the former in terms of the usual notion of prior concentration and in terms of an alternative definition of prior concentration. Moreover, we determine the latter for two important classes of priors: the infinite–dimensional exponential family, and the Pólya trees.     

ON TESTS OF SYMMETRY FOR DISCRETE POPULATIONS

In this paper problems of tests of symmetry about the origin with discrete samples are considered. Recently Vorli?ková established the asymptotic normality of linear rank statistics and signed rank statistics in [5] and [6]. Here we propose statistics which are conditionally the sum of independent variables, including the locally most powerful tests for a one sided one parameter family. Their asymptotic distributions are derived under the null hypothesis and the contiguous rounding off location alternatives. We propose four types of signed rank tests and investigate their properties.