Nader Ebrahimi

Bayesian approach to life testing and reliability estimation using asymmetric loss function

Abstract In this paper we derive Bayes estimators of the mean lifetime and the reliability function in the exponential life testing model. The loss functions used are asymmetric to reflect that, in most situations of interest, overestimating is more harmful than underestimating. A number of prior distributions have been considered and Bayesian estimates have been compared with corresponding estimates with squared error loss function.

Ordering univariate distributions by entropy and variance

This paper examines the role of variance and entropy in ordering distributions and random prospects. There is no universal relation between entropy and variance orderings of distributions. But we place their relationship in the context of a stronger ordering relation known as dispersion ordering. Further, some conditions are identified under which variance and entropy order similarly when continuous variables are transformed. We also analyze parametric changes which do not disturb the agreement between these rankings. The results are conveniently tabulated in terms of distribution parameters.

Two-year remission and subsequent relapse in children with newly diagnosed epilepsy.

Summary:  Purpose: Although remission is the ultimate measure of seizure control in epilepsy, and epilepsy syndrome should largely determine this outcome, little is known about the relative importance of syndrome versus other factors traditionally examined as predictors of remission or of relapse after remission. The purpose of this study was to examine remission and relapse with respect to the epilepsy syndrome and other factors traditionally considered with respect to seizure outcome.

Two measures of sample entropy

In many statistical studies the entropy of a distribution function is of prime interest. This paper proposes two estimators of the entropy. Both estimators are obtained by modifying the estimator proposed by Vasicek (1976). Consistency of both estimators is proved, and comparisons have been made with Vasiceks estimator and its generalization proposed by Dudewicz and Van der Meulen (1987). The results indicate that the proposed estimators have less bias and have less mean squared error than Vasiceks estimator and its generalization

Information properties of order statistics and spacings

We explore properties of the entropy, Kullback-Leibler information, and mutual information for order statistics. The probability integral transformation plays a pivotal role in developing our results. We provide bounds for the entropy of order statistics and some results that relate entropy ordering of order statistics to other well-known orderings of random variables. We show that the discrimination information between order statistics and data distribution, the discrimination information among the order statistics, and the mutual information between order statistics are all distribution free and are computable using the distributions of the order statistics of the samples from the uniform distribution. We also discuss information properties of spacings for uniform and exponential samples and provide a large sample distribution-free result on the entropy of spacings. The results show interesting symmetries of information orderings among order statistics.

Information Distinguishability with Application to Analysis of Failure Data

Abstract In maximum entropy (ME) modeling, the information discrepancy between two distributions is measured in terms of their entropy difference. In discrimination information statistics the information discrepancy between two distributions is measured in terms of the Kullback-Leibler function (i.e., relative entropy or cross-entropy). This article presents an equivalence between Kullback–Leibler functions and entropy differences involving an ME distribution. Based on this equivalence, the concept of information discrimination (ID) distinguishability is introduced as a unifying framework for the two methods of measuring information discrepancy between distributions. Applications of ID distinguishability as diagnostics for examining robustness of parametric procedures and sensitivity of nonparametric statistics across parametric families of distributions is proposed. The equivalence result facilitates estimation of Kullback-Leibler functions in terms of entropy estimates. Application of the ID distinguish...

Residual entropy and its characterizations in terms of hazard function and mean residual life function

A direct approach to measure uncertainty in the residual life time distribution has been initiated by Ebrahimi (1996, Sankhya Ser. A 58, 48-57) and explored further by Ebrahimi and Pellerey (1995) and Ebrahimi and Kirmani (1996). In this paper, some new properties of the proposed measure in connection to order statistics and record values are derived. The generalized Pareto distribution has been widely used in the literature. We have also given several characterizations of this distribution in terms of the proposed measure.

On the statistical analysis of the number of errors remaining in a software design document after inspection

Sometimes, complex software systems fail because of faults introduced in the requirements and design stages of the development process. Reviewing documents related to requirements and design by several reviewers can remove some of these faults, but often a few remain undetected until the software is developed. In this paper, we propose a procedure leading to the estimation of the number of faults which are not discovered. The main advantage of our procedure is that we do not need the standard assumption of independence among reviewers.

Some results on ordering of survival functions through uncertainty

A direct approach to measure uncertainty in the residual life time distribution has been initiated by Ebrahimi (1995) and explored further by Ebrahimi and Pellerey (1995). In this paper, some new properties of the proposed measure are derived and the impact of aging characteristics on the uncertainty of residual life studied. The results obtained help in establishing the uncertainty ordering between survival functions. Further, conditions for decreasing (increasing) uncertainty of residual life are also given.

A measure of discrimination between two residual life-time distributions and its applications

A measure of discrepancy between two residual-life distributions is proposed on the basis of Kullback-Leibler discrimination information. Properties of this measure are studied and the minimum discrimination principle is applied to obtain the proportional hazards model.