Calvin L. Williams

Smooth Estimation of the Reliability Function

Problems with censored data arise quite frequently in reliability applications. Estimation of the reliability function is usually of concern. Reliability function estimators proposed by Kaplan and Meier (1958), Breslow (1972), are generally used when dealing with censored data. These estimators have the known properties of being asymptotically unbiased, uniformly strongly consistent, and weakly convergent to the same Gaussian process, when properly normalized. We study the properties of the smoothed Kaplan-Meier estimator with a suitable kernel function in this paper. The smooth estimator is compared with the Kaplan-Meier and Breslow estimators for large sample sizes giving an exact expression for an appropriately normalized difference of the mean square error (MSE) of the two estimators. This quantifies the deficiency of the Kaplan-Meier estimator in comparison to the smoothed version. We also obtain a non-asymptotic bound on an expected ℒ1-type error under weak conditions. Some simulations are carried out to examine the performance of the suggested method.

Explorations in statistics: the analysis of change

Learning about statistics is a lot like learning about science: the learning is more meaningful if you can actively explore. This tenth installment of Explorations in Statistics explores the analysis of a potential change in some physiological response. As researchers, we often express absolute change as percent change so we can account for different initial values of the response. But this creates a problem: percent change is really just a ratio, and a ratio is infamous for its ability to mislead. This means we may fail to find a group difference that does exist, or we may find a group difference that does not exist. What kind of an approach to science is that? In contrast, analysis of covariance is versatile: it can accommodate an analysis of the relationship between absolute change and initial value when percent change is useless.

Multivariate goodness-of-fit tests based on statistically equivalent blocks

Various nonparametric procedures are known for the goodness-of-fit test in the univariate case. The distribution-free nature of these procedures does not extend to the multivariate case. In this paper, we consider an application of the theory of statistically equivalent blocks(SEB)to obtain distribution-free procedures for the multivariate case. The sample values are transformed to random variables which are distributed as sample spacings from a uniform distribution on [0, 1], under the null hypothesis. Various test statistics are known, based on the spacings, which are used for testing uniformity in the univariate case. Any of these statistics can be used in the multivariate situation, based on the spacings generated from the SEB. This paper gives an expository development of the theory of SEB and a review of tests for goodness-of-fit, based on sample spacings. To show an application of the SEB, we consider a test of bivariate normality.

RELATIVE DIFFERENCE IN DIVERSITY BETWEEN POPULATIONS

An entropy is conceived as a functional on the space of probability distributions. It is used as a measure of diversity (variability) of a population. Cross entropy leads to a measure of dissimilarity between populations. In this paper, we provide a new approach to the construction of a measure of dissimilarity between two populations, not depending on the choice of an entropy function, measuring diversity. The approach is based on the principle of majorization which provides an intrinsic method of comparing the diversities of two populations. We obtain a general class of measures of dissimilarity and show some interesting properties of the proposed index. In particular, it is shown that the measure provides a metric on a probability space. The proposed measure of dissimilarity is essentially a measure of relative difference in diversity between two populations. It satisfies an invariance property which is not shared by other measures of dissimilarity which are used in ecological studies. A statistical application of the new method is given.

A clinical application of expert system methodology

In recent years, several expert systems have been developed for practical applications in applied statistical methodologies. Existing expert systems in statistics have explored several areas, e.g. the determination of appropriate statistical tests, regression analysis, and determination of the ‘best’ experimental design for industrial screening experiments. We present here the DESIGN EXPERT which is a prototype expert system for the design of complex statistical experiments. It is intended for scientific investigators and statisticians who must design and analyze complex experiments, e.g. multilevel medical experiments with nested factors, repeated measures, and both fixed and random eflects. This system is ‘expert’ in the sense that it is capable of the following:(i) recognize specific types of complex experimental designs, based on the application of inference rules to non-technical information supplied by the user; (ii) encode the obtained and inferred information in a flexible general-purpose internal...

Design expert: An expert system application to clinical investigations☆

Abstract In recent years several expert systems have been developed for practical applications in applied statistical methodologies. Existing expert systems in statistics have explored several areas, e.g., the determination of appropriate statistical tests, regression analysis, and determination of the “best” experimental design for industrial screening experiments. The DESIGN EXPERT, a prototype expert system for the design of complex statistical experiments is presented here. It is intended for scientific investigators and statisticians who must design and analyze complex experiments, e.g., multi-level medical experiments with nested factors, repeated measures, and both fixed and random effects. This system is “expert” in the sense that it is able to (i) recognize specific types of complex experimental designs, based on the application of inference rules to nontechnical information supplied by the user; (ii) encode the obtained and inferred information in a flexible general-purpose internal representation, for use by other program modules; (iii) generate analysis of variance tables for the recognized design and an appropriate Biomedical Computer Programs runfile for data analysis, using the encoded information. DESIGN EXPERT can recognize randomized block designs, including lattice designs within embedded Latin Squares, crossover designs, split plots, nesting, repeated measures, and covariates. It is written in an experimental programming language developed specifically for research in Artificial Intelligence.

AN ADAPTIVE PROCEDURE FOR GOODNESS-OF-FIT BASED ON SAMPLE SPACINGS

In the univariate case the problem of testing the null hypothesis that the sample is drawn from a specified continuous distribution is reduced to the problem of testing the null hypothesis that the sample is drawn from a uniform distribution on (0, 1), through the application of the probability integral transformation. There are a number of tests based on sample spacings, which have been considered in the literature, for testing uniformity. Generally, the choice of a test should depend on the specification of the alternative distribution under consideration. However, where there is little indication of the alternative distribution, as in the nonparametric setting, it is often difficult to choose from the alternative tests. In this paper we propose an adaptive procedure for testing the null hypothesis based on the sample spacings. The procedure first selects an appropriate test from among a given number of tests. The selected test is associated with the smallest p-value, where the p-values of the given tests are computed from the sample. Some numerical results are given on the performance of the proposed adaptive procedure, from a study based on simulated data.