Thu Pham-Gia

Determination of exact sample sizes in the Bayesian estimation of the difference of two proportions

Using generalized hypergeometric functions in several variables in a Bayesian context, we compute the exact minimum double-sample size (n 1 , n 2 ) required in the Bernoulli sampling of two independent populations, so that the expected length (or the maximum length) of the highest posterior density credible interval of P = P 1 - P 2 is less than a preset quantity, where P 1 and P 2 are two independent proportions. This precise and computer-intensive approach permits the treatment of this Bayesian sample size determination problem under very general hypotheses and also provides a relationship between the minimal values of n 1 and n 2 . Similar results are derived in an applied Bayesian decision theory context, with a quadratic loss function, and the criteria used are now the posterior risk, the Bayes risk and the expected value of sample information.

Statistical Discrimination Analysis Using the Maximum Function

The maximum of k functions defined on R n , n ≥ 1, by f max (x) = max{f 1 (x),…, f k (x)}, ∀ x ∈ R n , can have important roles in Statistics, particularly in Classification. Through its relation with the Bayes error, which is the reference error in classification, it can serve to compute numerical bounds for errors in other classification schemes. It can also serve to define the joint L1-distance between more than two densities, which, in turn, will serve as a useful tool in Classification and Cluster Analyses. It has a vast potential application in digital image processing too. Finally, its versatile role can be seen in several numerical examples, related to the analysis of Fishers classical iris data in multidimensional spaces.

System Stress-Strength Reliability: The Multivariate Case

Present day complex systems with dependence between their components require more advanced models to evaluate their reliability. We compute the reliability of a system consisting of two subsystems S 1, and S2 connected in series, where the reliability of each subsystem is of general stress-strength type, defined by R1 = P(A TX > BTY). A & B are column-constant vectors, and strength X & stress Y are multigamma random vectors, i.e. (X, Y) ~ MG (alpha, beta), where alpha and beta are k-dimensional constant vectors. A Bayesian approach is adopted for R2 = P(B TW > 0), where W is multinormal, i.e. W ~ MN(mu, T), with the mean vector mu, and the precision matrix T having a joint s-normal-Wishart prior distribution. Final computations are carried out by simulation, an approach which plays a major role in this article. The results obtained show that the approach adopted can deal effectively with the dependence between components of X & Y

Bounds for the Bayes Error in Classification: A Bayesian Approach Using Discriminant Analysis

We study two of the classical bounds for the Bayes error Pe, Lissack and Fu’s separability bounds and Bhattacharyya’s bounds, in the classification of an observation into one of the two determined distributions, under the hypothesis that the prior probability χ itself has a probability distribution. The effectiveness of this distribution can be measured in terms of the ratio of two mean values. On the other hand, a discriminant analysis-based optimal classification rule allows us to derive the posterior distribution of χ, together with the related posterior bounds of Pe.

Bayesian decision criteria in the presence of noises under quadratic and absolute value loss functions

SummaryMisclassifications, or noises, in the sampling stage of a Bayesian scheme can seriously affect the values of decision criteria such as the Bayes Risk and the Expected Value of Sample Information. This problem does not seem to be much addressed in the existing literature. In this article, using an approach based on hypergeometric functions and numerical computation, we study the effects of these noises under the two most important loss functions: the quadratic and the absolute value. A numerical example illustrates these effects in a representative case, using both loss functions, and provides additional insights into the general problem.

Saturation and an iterative construction process

Let {j}Kj=1∞ be a sequence of kernels, let {aj}j=1∞ be a sequence of positive numbers, and let f0 be a measurable function. Setting J0 = f0, we study the convergence in Lp(1 ⩽ p ⩽ 2 and p = ∞) of the sequence of singular integrals {Jn}n = 1∞ defined inductively by Jn(x) = (an(2π)12∝−∞∞Jn−1(x − t)nK(ant)dt, x ϵ R. n nThe convergence of {Jn} in L∞ finds an application in Bray-Mandelbrojts “repeated averaging” construction concerning a non quasi-analytic class of functions.