An upper bound for the distribution function of a positive definite quadratic form

In this paper the researchers are presenting an upper bound for the distribution function of quadratic forms in normal vector with mean zero and positive definite covariance matrix. They also will show that the new upper bound is more precise than the one introduced by Okamoto [4] and the one introduced by Siddiqui [5]. Theoretical Error bounds for both, the new and Okamoto upper bounds are derived in this paper. For larger number of terms in any given positive definite quadratic form, a rough and easier upper bound is suggested.     

Moments of generalized quadratic forms and distributions of certain multivariate test statistics

Summary Moments and distributions of quadratic forms or quadratic expressions in normal variables are available in literature. Such quadratic expressions are shown to be equivalent to a linear function of independent central or noncentral chi-square variables. Some results on linear functions of generalized quadratic forms are also available in literature. Here we consider an arbitrary linear function of matrix-variate gamma variables. Moments of the determinant of such a linear function are evaluated when the matrix-variate gammas are independently distributed. By using these results, arbitrary non-null moments as well as the non-null distribution of the likelihood ratio criterion for testing the hypothesis of equality of covariance matrices in independent multivariate normal populations are derived. As a related result, the distribution of a linear function of independent matrix-variate gamma random variables, which includes linear functions of independent Wishart matrices, is also obtained. Some properties of generalized special functions of several matrix arguments are used in deriving these results.     

Using power transformations when approximating quantiles

Estimators are obtained tor quantiles of survival distributions. This is accomplished by approximating Lritr distribution of the transtorrneri data, where the transformation used is that of Box and Cox (1964). The normal approximation as in Box and Cox and, in addition, the extreme value approximation are considered. More generally, to use the methods given, the approximating distribution must come from a location-scale family. For some commonly used survival random variables T the performance of the above approximations are evaluated in terms of the ratio of the true quantiles of T to the estimated one, in the long run. This performance is also evaluated for lower quantiles using simulated lognormai, Weibull and gamma data. Several examples are given to illustrate the methodology herein, including one with actual data.     

Distribution approximation and modelling via orthogonal polynomial sequences

A general methodology is developed for approximating the distribution of a random variable on the basis of its exact moments. More specifically, a probability density function is approximated by the product of a suitable weight function and a linear combination of its associated orthogonal polynomials. A technique for generating a sequence of orthogonal polynomials from a given weight function is provided and the coefficients of the linear combination are explicitly expressed in terms of the moments of the target distribution. On applying this approach to several test statistics, we observed that the resulting percentiles are consistently in excellent agreement with the tabulated values. As well, it is explained that the same moment-matching technique can be utilized to produce density estimates on the basis of the sample moments obtained from a given set of observations. An example involving a well-known data set illustrates the density estimation methodology advocated herein.     

Distribution of products of independently distributed pathway random variables

A lot of work has been done on products and ratios of random variables by Provost and his co-workers, see, for example, Provost [S.B. Provost, The exact distribution of the ratio of a linear combination of chi-square variables over the root of a product of chi-square variables, Canad. J. Statist. 14 (1986), pp. 61–67; S.B. Provost, The distribution function of a statistic for testing the equality of scale parameters in two gamma populations, Metrika 36 (1989), pp. 337–345]. Here, we extend this idea by introducing a pathway model. The exact density functions of the products of pathway random variables are obtained using the Mellin transform technique. Their computable series forms are derived. The particular cases of the derived results are shown to be associated with the thermonuclear functions and reaction rate probability integral in the theory of nuclear reaction rate, Krätzel integral in applied analyses and inverse Gaussian density in stochastic processes. Graphical representations of the density functions of the product of random variables for the different values of the pathway parameters are shown. The new probability model is fitted to revenue data.     

Distribution of the quotient of noncentral normal random variables

The Mellin convolution is used to derive in analytical form an exact 3-parameterprobabilitydensity function of the quotient of two noncentral normal random variables. In contrast with the 5-parameter probability density function previously derivedby Fieller (1932) and Hinkley (1969), this 3-parameter probability density function is feasible for computer evaluation of the mean and cumulative distribution function, which are needed, for example, when dealing with estimation and distribution problems in regression analysis and sampling theory. When the normal variables are independent, the probability density function reduces to a 2-parameter function, for which a computer program is operational. An illustrative example is given for one set of parameters when the normal variables are independent, in which themean and functional form of the probability density function are presented, together with a brief tabulation of the probability density function.     

Approximations to noncentral distributions

Two methods for approximating the distribution of a noncentral random variable by a central distribution in the same family are presented. The first consists of relating a stochastic expansion of a random variable to a corresponding asymptotic expansion for its distribution function. The second approximates the cumulant generating function and is used to provide central χ² and gamma approximations to the noncentral χ² and gamma distributions.     

On linear combinations of independent exponential variables

The exact distribution of a linear combination of n indepedent negative exponential random variables , when the coefficients cf the linear combination are distinct and positive , is well-known. Recently Ali and Obaidullah (1982) extended this result by taking the coeff icients to be arbitrary real numbers. They used a lengthy geometric.

al approach to arrive at the result . This article gives a simple derivation of the result with the help of a generalized partial fraction technique. This technique also works when the variables involved are gamma variables with certain types of parameters. Results are presented in a form which can easily be programmed for computational purposes. Connection of this problem t o various problems in different fields is also pointed out.     

Analysis of multivariate repeated measures data with a Kronecker product structured covariance matrix

In this article we consider a set of t repeated measurements on p variables (or characteristics) on each of the n individuals. Thus, data on each individual is a p ×t matrix. The n individuals themselves may be divided and randomly assigned to g groups. Analysis of these data using a MANOVA model, assuming that the data on an individual has a covariance matrix which is a Kronecker product of two positive definite matrices, is considered. The well-known Satterthwaite type approximation to the distribution of a quadratic form in normal variables is extended to the distribution of a multivariate quadratic form in multivariate normal variables. The multivariate tests using this approximation are developed for testing the usual hypotheses. Results are illustrated on a data set. A method for analysing unbalanced data is also discussed.     

A Note on the Characterization of Positive Linnik Laws

Recently, Jayakumar & Pillai (1996) gave an interesting characterization of the positive Linnik laws in terms of the spectrum function of an infinitely divisible law. This paper improves their result and simplifies their proof. It proves another characterization result in terms of the Pareto law. Further, it represents the positive Linnik random variable as a function of independent gamma random variables.     

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.     

Properties of doubly-truncated gamma variables

The truncated gamma distribution has been widely studied, primarily in life-testing and reliability settings. Most work has assumed an upper bound on the support of the random variable, i.e. the space of the distribution is (0,u). We consider a doubly-truncated gamma random variable restricted by both a lower (l) and upper (u) truncation point, both of which are considered known. We provide simple forms for the density, cumulative distribution function (CDF), moment generating function, cumulant generating function, characteristic function, and moments. We extend the results to describe the density, CDF, and moments of a doubly-truncated noncentral chi-square variable.     

On the Simultaneous Estimation of Scale Parameters

This note is an extension of Das Gupta's results (1986) on the estimation of multiparameter gamma distribution. Consider p (p ? 2) independent positive random variables with possibly different scale-parameter densities. For the estimation of the powers of the scale parameters it is shown that the “best multiple estimator” is inadmissible with respect to a large class of weighted quadratic loss functions.     

Density of the quotient of non-negative quadratic forms in normal variables with application to the F-statistic

The density of the quotient of two non-negative quadratic forms in normal variables is considered. The covariance matrix of these variables is arbitrary. The result is useful in the study of the robustness of theF-test with respect to errors of the first and second kind. An explicit expression for this density is given in the form of a proper Riemann-integral on a finite interval, suitable for numerical calculation.     

On the distribution of quadratic forms in normal random variables

This paper provides necessary and sufficient conditions for a quadratic form in singular normal random variables to be distributed as a given linear combination of independent noncentral chi-square variables. Using this result, an extension of Cochran's theorem to quadratic forms of noncentral chi-square variables is derived.     

On some sampling distributions for skew-normal population

The distribution of weighted function of independent skew-normal random variables, which includes the sample mean, is useful in many applications. In this paper, we derive this distribution and study the null distribution of a linear form and a quadratic form. Finally, we discuss some of its applications in control charts, in which the skew-normal model plays a key role.     

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.     

A Viable Alternative to Resorting to Statistical Tables

It is shown in this article that, given the moments of a distribution, any percentage point can be accurately determined from an approximation of the corresponding density function in terms of the product of an appropriate baseline density and a polynomial adjustment. This approach, which is based on a moment-matching technique, is not only conceptually simple but easy to implement. As illustrated by several applications, the percentiles so obtained are in excellent agreement with the tabulated values. Whereas statistical tables, if at all available or accessible, can hardly ever cover all the potentially useful combinations of the parameters associated with a random quantity of interest, the proposed methodology has no such limitation.     

The Distribution of a Ratio of Quadratic Forms in Noncentral Normal Variables

Abstract

An expression for the exact cumulative distribution function of a ratio of quadratic forms in noncentral normal variable is derived in terms of infinite series of top order invariant polynomials.     

The posterior distribution of the fixed and random effects in a mixed-effects linear model

The subject of this paper is Bayesian inference about the fixed and random effects of a mixed-effects linear statistical model with two variance components. It is assumed that a priori the fixed effects have a noninformative distribution and that the reciprocals of the variance components are distributed independently (of each other and of the fixed effects) as gamma random variables. It is shown that techniques similar to those employed in a ridge analysis of a response surface can be used to construct a one-dimensional curve that contains all of the stationary points of the posterior density of the random effects. The “ridge analysis” (of the posterior density) can be useful (from a computational standpoint) in finding the number and the locations of the stationary points and can be very informative about various features of the posterior density. Depending on what is revealed by the ridge analysis, a multivariate normal or multivariate-t distribution that is centered at a posterior mode may provide a satisfactory approximation to the posterior distribution of the random effects (which is of the poly-t form).