Serge B. Provost

Quadratic forms in random variables : theory and applications

Textbook for a one-semester graduate course for students specializing in mathematical statistics or in multivariate analysis, or reference for theoretical as well as applied statisticians, confines its discussion to quadratic forms and second degree polynomials in real normal random vectors and matr

The exact distribution of indefinite quadratic forms in noncentral normal vectors

The exact density of the difference of two linear combinations of independent noncentral chi-square variables is obtained in terms of Whittakers function and expressed in closed forms. Two distinct representations are required in order to cover all the possible cases. The corresponding expressions for the exact distribution function are also given.

On sums of independent gamma eandom yariames

This paper gives the exact probability density function of a sum of independent gamma random variables in terms of infinite series. This representation is obtained by means of the technique of the inverse MELLIN transform and is used to derive the distri¬bution function of a ratio of sums of independent gamma random variables whose numera¬tor and denominator may have variables in common.

Predictive Inference from a Two-Parameter Rayleigh Life Model Given a Doubly Censored Sample

This article is concerned with making predictive inference on the basis of a doubly censored sample from a two-parameter Rayleigh life model. We derive the predictive distributions for a single future response, the ith future response, and several future responses. We use the Bayesian approach in conjunction with an improper flat prior for the location parameter and an independent proper conjugate prior for the scale parameter to derive the predictive distributions. We conclude with a numerical example in which the effect of the hyperparameters on the mean and standard deviation of the predictive density is assessed.

The exact density function of the ratio of two dependent linear combinations of chi-square variables

A computable expression is derived for the raw moments of the random variableZ=N/D whereN=Σ1nmiXi+Σn+1smiXi,D=Σn+1sliXi+Σs+1rniXi, and theXis are independently distributed central chi-square variables. The first four moments are required for approximating the distribution ofZ by means of Pearson curves. The exact density function ofZ is obtained in terms of sums of generalized hypergeometric functions by taking the inverse Mellin transform of theh-th moment of the ratioN/D whereh is a complex number. The casen=1,s=2 andr=3 is discussed in detail and a general technique which applies to any ratio having the structure ofZ is also described. A theoretical example shows that the inverse Mellin transform technique yields the exact density function of a ratio whose density can be obtained by means of the transformation of variables technique. In the second example, the exact density function of a ratio of dependent quardratic forms is evaluated at various points and then compared with simulated values.

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 Hermitian quadratic forms in elliptically contoured random vectors

Computable representations of the probability density function and the cumulative distribution function of Hermitian quadratic forms in spherically distributed vectors are obtained from a stochastic decomposition of central quadratic forms into the product of two independent quantities: the square of the norm of the vectors and a linear combination of the components of a Dirichlet random vector. The expressions derived for the Gaussian case appear to lend themselves more readily to numerical evaluation than those currently available in the literature. The distribution of noncentral Hermitian quadratic forms is obtained via a certain mixture representation of the density function of elliptically contoured random vectors. A numerical application to the periodogram is proposed for the case of spherically distributed residuals.

The Sampling Distribution of the Serial Correlation Coefficient

SYNOPTIC ABSTRACTA methodology for the derivation of the cumulants of the lag- k serial covariance is proposed for the case of a Gaussian white-noise process. Explicit representations of the first five moments of the serial correlation coefficient are then given. Alternate representations which are expressed in terms of trigonometric functions are also derived. The first four moments are used to approximate the density of the serial correlation by means of Pearson curves. A computable representation of the exact distribution function is also given. The exact, the approximate and the simulated distributions are compared in a numerical example.

On Craig's theorem and its generalizations

Abstract This article proposes accessible and detailed derivations of Craigs theorem and its generalizations — including criteria for the independence of second degree polynomials in noncentral and singular normal vectors. The main steps in the proof of Craigs theorem as well as the approach used for the generalizations are believed to be original. An alternate proof which is self-contained is proposed. Related results involving combinations of linear, quadratic and bilinear forms are discussed, and an example is provided.

The moment generating function of a bivariate gamma-type distribution

A bivariate gamma-type density function involving a confluent hypergeometric function of two variables is being introduced. The inverse Mellin transform technique is employed in conjunction with the transformation of variable technique to obtain its moment generating function, which is expressed in terms of generalized hypergeometric functions. Its cumulative distribution function is given in closed form as well. Many distributions such as the bivariate Weibull, Rayleigh, half-normal and Maxwell distributions can be obtained as limiting cases of the proposed gamma-type density function. Computable representations of the moment generating functions of these distributions are also provided.