THE MONTE CARLO EVALUATION OF ORTHANT PROBABILITIES FOR MULTIVARIATE NORMAL DISTRIBUTIONS

The calculation of multivariate normal orthant probabilities is practically impossible when the number of variates is greater than five or six, except in very special cases. A transformation of the integral is obtained which enables quite accurate Monte Carlo estimates to be obtained for a fairly high number of dimensions, particularly if control variates are used.     

SOME SIMPLE APPROXIMATIONS FOR THE DOUBLY NONCENTRAL z DISTRIBUTION

We give two simple approximations for evaluating the cumulative probabilities of the doubly noncentral z distribution. These can easily be used for evaluating the cumulative probabilities of the doubly noncentral F distribution as well. We compare our results with those obtained by Tiku (1965) using series expansion. An industrial situation where a quality characteristic of interest follows the doubly noncentral z distribution is also cited. However, in this case the exact probabilities could be calculated using results on the ratio of two normal variables.     

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.     

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 CHARACTERIZING SOME BIVARIATE DISCRETE DISTRIBUTIONS

The bivariate (correlated) Poisson, binomial, negative binomial and logarithmic distributions are characterized by the conditional distribution of one random vector on the other and a regression function of the second random vector on the first. Characterizations of the above distributions when their component random variables are independent are also included.     

DISPERSION CONTROL FOR MULTIVARIATE PROCESSES – SOME COMPARISONS

Tang & Barnett (1996) present some techniques for monitoring and controlling the dispersion of multivariate normal processes based on subgroup data. The current paper compares the proposed techniques and various competing procedures. Simulation results indicate that the proposed techniques are superior to existing procedures.     

FINITE MIXTURE OF CERTAIN DISTRIBUTIONS

ABSTRACT

A two parameter extended form of standard gamma function is suggested which provide extra flexibility to the density function over positive range. A finite mixture of beta distribution is defined by using the suggested extended form. The shape of density function of extended gamma distribution and also that of finite mixture of beta distribution for various values of the parameters are shown. Inverted distribution of extended gamma and that of finite mixture of beta distribution are given.     

SERIES REPRESENTATION OF DOUBLY NONCENTRAL t DISTRIBUTION BY USE OF THE MELLIN INTEGRAL TRANSFORM

ABSTRACT

The Mellin integral transform is widely used to find the distributions of products and quotients of independent random variables defined over the positive domain. But it is hardly used to derive the distributions defined over both positive and negative values of the random variables. In this paper, the Mellin integral transform is applied to obtain the doubly noncentral t density and its distribution function in convergent series forms.     

ON THE NONCENTRAL DISTRIBUTION OF A RANDOM MATRIX USEFUL IN CLASSIFICATION THEORY

To prove the optimality properties of the maximum likelihood (and also minimum distance) discriminant rule Rogers (1980, p. 98) embeds the maximum likelihood discriminant function in a Cauchy-Schwartz inequality. This embedding procedure of Rogers (1980) may be used to derive a new distribution for Anderson's (1958) classification statistic.     

NONPARAMETRIC ESTIMATION OF MULTIVARIATE CONVEX-TRANSFORMED DENSITIES

We study estimation of multivariate densities p of the form p(x) = h(g(x)) for x ∈ ?(d) and for a fixed monotone function h and an unknown convex function g. The canonical example is h(y) = e(-y) for y ∈ ?; in this case, the resulting class of densities [Formula: see text]is well known as the class of log-concave densities. Other functions h allow for classes of densities with heavier tails than the log-concave class.We first investigate when the maximum likelihood estimator p? exists for the class P(h) for various choices of monotone transformations h, including decreasing and increasing functions h. The resulting models for increasing transformations h extend the classes of log-convex densities studied previously in the econometrics literature, corresponding to h(y) = exp(y).We then establish consistency of the maximum likelihood estimator for fairly general functions h, including the log-concave class P(e(-y)) and many others. In a final section, we provide asymptotic minimax lower bounds for the estimation of p and its vector of derivatives at a fixed point x(0) under natural smoothness hypotheses on h and g. The proofs rely heavily on results from convex analysis.     

AN ANALYSIS OF CLASSICAL AND TIE-BREAKER TENNIS

Expressions for the probability that a player wins a game, set or match of classical or tie-breaker tennis are obtained. Also, expressions for the distribution, the mean and the variance of the number of points in a game, set or match of classical or tie-breaker tennis are obtained. These results explain the long matches often previously observed in classical tennis between two players each with very effective serves on fast grass-court surfaces. The methodology used to derive these expressions could be used to obtain corresponding expressions for other "nested-type" scoring systems. For example, tennis is 3-nested (game, set, match) whereas squash is only 2-nested (game, match).     

A NEW FAMILY OF NON-NEGATIVE DISTRIBUTIONS

We introduce a new, flexible family of distributions for non‐negative data, defined by means of a quantile function. We describe some properties of this family, and discuss several methods for estimating the parameters. The distribution is applied to an example from environmental engineering.     

MULTIVARIATE GENERALIZED POLYA DISTRIBUTION OF ORDER k

ABSTRACT

An order k (or cluster) generalized Polya distribution and a multivariate generalized Polya-Eggenberger one where derived in (Sen, K.; Jain, R. Cluster Generalized Negative Binomial Distribution. In Probability Models and Statistics, A. J. Medhi Festschrift on the Occasion of his 70th Birthday; Borthakur, A.C. et al., Eds.; New Age International Publishers: New Delhi, 1996; 227–241 and Sen, K.; Jain, R. A Multivariate Generalized Polya-Eggenberger Probability Model-First Passage Approach. Communications in Statistics—Theory and Methods 1997, 26, 871–884). Presently, both distributions are generalized to a multivariate generalized Polya distribution of order k by means of an appropriate sampling scheme and a first passage event. This new distribution includes as special cases new multivariate Polya and inverse Polya distributions of order k and the multivariate generalized negative binomial distribution of the same order derived recently in (Tripsiannis, G.A.; Philippou, A.N.; Papathanasiou, A.A. Multivariate Generalized Distributions of Order k. Medical Statistics Technical Report #41: Democritus University of Thrace, Greece, 2001). Limiting cases are considered and applications are indicated.     

HIGH-DIMENSIONAL PARAMETRIC MODELLING OF MULTIVARIATE EXTREME EVENTS

Multivariate extreme events are typically modelled using multivariate extreme value distributions. Unfortunately, there exists no finite parametrization for the class of multivariate extreme value distributions. One common approach is to model extreme events using some flexible parametric subclass. This approach has been limited to only two or three dimensions, primarily because suitably flexible high-dimensional parametric models have prohibitively complex density functions. We present an approach that allows a number of popular flexible models to be used in arbitrarily high dimensions. The approach easily handles missing and censored data, and can be employed when modelling componentwise maxima and multivariate threshold exceedances. The approach is based on a representation using conditionally independent marginal components, conditioning on positive stable random variables. We use Bayesian inference, where the conditioning variables are treated as auxiliary variables within Markov chain Monte Carlo simulations. We demonstrate these methods with an application to sea-levels, using data collected at 10 sites on the east coast of England.     

ESTIMATION OF MULTIVARIATE SHANNON ENTROPY USING MOMENTS

Three new entropy estimators of multivariate distributions are introduced. The two cases considered here concern when the distribution is supported by a unit sphere and by a unit cube. In the former case, the consistency and the upper bound of the absolute error for the proposed entropy estimator are established. In the latter one, under the assumption that only the moments of the underlying distribution are available, a non‐traditional estimator of the entropy is suggested. We also study the practical performances of the constructed estimators through simulation studies and compare the estimators based on the moment‐recovered approaches with their counterparts derived by using the histogram and k th nearest neighbour constructions. In addition, one worked example is briefly discussed.