Lawrence M. Leemis

A COMPARISON OF APPROXIMATE INTERVAL ESTIMATORS FOR THE BERNOULLI PARAMETER

The goal of this paper is to compare the accuracy of two approximate confidence interval estimators for the Bernoulli parameter p. The approximate confidence intervals are based on the normal and Poisson approximations to the binomial distribution. Charts are given to indicate which approximation is appropriate for certain sample sizes and point estimators.

Computing the distribution of the product of two continuous random variables

We present an algorithm for computing the probability density function of the product of two independent random variables, along with an implementation of the algorithm in a computer algebra system. We combine this algorithm with the earlier work on transformations of random variables to create an automated algorithm for convolutions of random variables. Some examples demonstrate the algorithms application.

Survival Distributions Satisfying Benford's Law

Abstract Hill stated that “An interesting open problem is to determine which common distributions (or mixtures thereof) satisfy Benfords law …”. This article quantifies compliance with Benfords law for several popular survival distributions. The traditional analysis of Benfords law considers its applicability to datasets. This article switches the emphasis to probability distributions that obey Benfords law.

Univariate Distribution Relationships

Probability distributions are traditionally treated separately in introductory mathematical statistics textbooks. A figure is presented here that shows properties that individual distributions possess and many of the relationships between these distributions.

Computing the cumulative distribution function of the Kolmogorov-Smirnov statistic

Abstract We present an algorithm for computing the cumulative distribution function of the Kolmogorov–Smirnov test statistic D n in the all-parameters-known case. Birnbaum (1952, J. Amer. Statist. Assoc. 47, 425–441), gives an n -fold integral for the CDF of the test statistic which yields a function defined in a piecewise fashion, where each piece is a polynomial of degree n . Unfortunately, it is difficult to determine the appropriate limits of integration for computing these polynomials. Our algorithm performs the required integrations in a manner that avoids calculating the same integrals repeatedly, resulting in shorter computation time. It can be used to compute the entire CDF or just a portion of the CDF, which is more efficient for finding a critical value or a p -value associated with a hypothesis test. If the entire CDF is computed, it can be stored in memory so that various characteristics of the distribution of the test statistic (e.g., moments) can be calculated. To date, critical tables have been approximated by various techniques including asymptotic approximations, recursive formulas, and Monte Carlo simulation. Our approach yields exact critical values and significance levels. The algorithm has been implemented in a computer algebra system.

Burn-in models and methods : a review

Abstract Burn-in is a technique used to increase the quality of components and systems delivered to a consumer by using the item under normal or accelerated environmental conditions prior to shipment. If a burn-in procedure is effective, items that are delivered to the consumer are superior to those that would have been delivered without burn-in. The measure by which items are judged to be superior is defined by the objective of the burn-in procedure (e.g., maximum mean residual life or maximum probability of mission success after burn-in). This paper reviews the burn-in literature which considers different aspects of the problem, but lacks a structure that relates them to one another.

APPL: A Probability Programming Language

Statistical packages have been used for decades to analyze large datasets or to perform mathematically intractable statistical methods. These packages are not capable of working with random variables having arbitrary distributions. This article presents a prototype probability package named APPL (A Probability Programming Language) that can be used to manipulate random variables. Examples illustrate its use. A current version of the software can be obtained by contacting the third author at leemis@math.wm.edu.Statistical packages have been used for decades to analyze large datasets or to perform mathematically intractable statistical methods. These packages are not capable of working with random variables having arbitrary distributions. This article presents a prototype probability package named APPL (A Probability Programming Language) that can be used to manipulate random variables. Examples illustrate its use. A current version of the software can be obtained by contacting the third author at .

Variate generation for accelerated life and proportional hazards models with time dependent covariates

Variate generation algorithms for lifetimes when survival models incorporate time dependent covariates are presented. These algorithms are closed form for special cases of the function that links the covariate values to the survivor distribution. These algorithms are illustrated by several examples.

Spectral analysis in quality control: a control chart based on the periodogram

Variations in manufacturing processes are expected to fluctuate around a constant level called the process mean. This article is concerned with the development of a control chart that detects cycles in the process mean. The new control chart, the spectral control chart, is based on the periodogram. The value plotted at each point in time is the ratio of the largest periodogram ordinate to the average of all ordinates. When an observation falls above the prescribed control limit, an out-of-control signal is given. The spectral control chart is compared with the Shewhart and geometric moving average charts. Data obtained from Monte Carlo simulations show that the spectral control chart is superior for detecting cyclic variations but that it is not effective for detecting shifts in the process mean. Thus the spectral control chart should be used along with existing control charts so that both shifts and cycles in the process mean can be detected.

Minimum Kolmogorov–Smirnov test statistic parameter estimates

We present and implement an algorithm for computing the parameter estimates in a univariate probability model for a continuous random variable that minimizes the Kolmogorov–Smirnov test statistic. The algorithm uses an evolutionary optimization technique to solve for the estimates. Several simulation experiments demonstrate the effectiveness of this approach.