Edward C. van der Meulen

On stochastic properties of m-spacings

Abstract Let X be a nonnegative and continuous random variable having the probability density function (pdf) f(.). Let X k:n (k=1,2,…,n) denote the kth order statistic based on n independent observations on X and, for a given positive integer m (⩽n) , let D k,n (m) =X k+m−1:n −X k−1:n , k=1,2,…,n−m+1 , denote the successive (overlapping) spacings of gap size m (to be referred as m-spacings); here X0:n≡0. It is shown that if f(.) is log convex, then the pdf of corresponding simple (gap size one) spacings D k,n (1) , k=1,2,…,n , are also log convex. It is also shown that the m-spacings D k,n (m) , k=1,2,…,n−m+1 , preserve the log concavity of the parent pdf f(.). Under the log convexity of the parent pdf f(.), we further show that, for k=1,2,…,n−m, D k,n (m) is smaller than Dk+1,n(m) in the likelihood ratio ordering and that, for a fixed 1⩽k⩽n−m+1 and n⩾k+m−1, D k,n+1 (m) is smaller than Dk,n(m) in the likelihood ratio ordering. Finally, we show that if X has a decreasing failure rate then, for k=1,2,…,n−m, D k,n (m) is smaller than Dk+1,n(m) in the failure rate ordering and that, for a fixed 1⩽k⩽n−m+1 and n⩾k+m−1, D k,n+1 (m) is smaller than Dk,n(m) in the failure rate ordering.

Density-free convergence properties of various estimators of entropy

Abstract Let ƒ(x) be a probability density function, x∈Rd. The Shannon (or differential) entropy is defined as H(ƒ)=−∫ƒ(x) log ƒ(x) d x . In this paper we propose, based on a random sample X1,…, Xn generated from ƒ, two new nonparametric estimators for H(ƒ) . Both entropy estimators are histogram-based in the sense that they involve a histogram-based desntiy estimator ƒ n . We prove their a.s. consistency with the only condition on ƒ that H(ƒ) is finite.

Some Reflections On The Interference Channel

This paper provides a short overview of the advances on the interference channel, which is a well-known channel in multiuser information theory, that have appeared in the literature since 1976. Several open problems are defined.

On Universal Noiseless Source Coding for Infinite Source Alphabets

We show that there is a universal noiseless source code for the class of all coun-tably infinite memoryless sources for which a fixed given uniquely decodable code has finite expected codeword length. This source code is derived from a class of distribution estimation procedures which are consistent in expected information divergence.

Distribution Estimates Consistent in χ2-Divergence

For a class of histogram based distribution estimators it is proved the consistency in χ2-divergence and expected χ2-divergence. For one of these estimators, introduced formerly by Barron, is also evaluated the rate of consistency in the expected χ2-divergence. These results are stronger than similar results formerly established in the literature for the total variation and Kullbacks discrimination information, which are divergences dominated by the χ2-divergence.

Entropy-Based Random Number Evaluation

SYNOPTIC ABSTRACTPrevious work has shown how to test a simple hypothesis of uniformity on the interval (0, 1) by using spacings-based estimates of entropy. In this paper we use Monte Carlo methods to extend previous tables of critical points and power for such entropy tests to the large sample sizes likely to be desirable when evaluating the output of one or more random number generators. A comparison with asymptotic critical points and power is made. The results are used to evaluate a number of commonly used random number generators, which are of importance in such areas as bootstrapping. At least one random number generator is found unsuitable for use. Since a generator cycling on .00, .01, .02, …, .99 (to more digits) could have a sample entropy of nearly zero, this test is appropriate only for generators that pass other extensive testing, such as the TESTRAND tests (e.g., see Karian and Dudewicz (1991)).

A Consistent Goodness of Fit Test Based on the Total Variation Distance

In this paper we propose a consistent test for testing a simple hypothesis versus a composite alternative. The simple hypothesis consists of a continuous distribution v, while the composite alternative is the set of probability measures μ such that μ is in total variation T(μ,ν) at least distance δ < 0 away from ν. For this hypothesis testing problem no uniformly consistent test exists. The proof of the consistency result implies a way to restrict the alternative in order to have uniform consistency, and thus uniform exponential consistency.

Mutual information, variation, and Fano's inequality

Some upper and lower bounds are obtained for the maximum of the absolute value of the difference between the mutual information |I(X; Y) − I(X′; Y′)| of two pairs of discrete random variables (X, Y) and (X′, Y′) via the variational distance between the probability distributions of these pairs. In particular, the upper bound obtained here substantially generalizes and improves the upper bound of [1]. In some special cases, our upper and lower bounds coincide or are rather close. It is also proved that the lower bound is asymptotically tight in the case where the variational distance between (X, Y) and (X′ Y′) tends to zero.

On estimation of the common mean of k (≥2) normal populations with order restricted variances

For estimating the common mean of k ([greater-or-equal, slanted] 2) normal populations with order restricted unknown variances, using the theory of isotomic regression, we propose a simple estimator which is better than the usual Graybill-Deal estimator in terms of stochastic dominance and the Pitman measure of closeness.

Synchronization of firing times in a stochastic neural network model with excitatory connections

We investigate a finite, stochastic, completely neural network model with excitatory couplings. The dynamics of the moments of firing in the net is described by a Markov chain. We derive exponential bounds for its transition probabilities. Moreover, the exponential fast synchronization of the moments of firing is proved. The results are illustrated by computer simulations.