Irwin Guttman

Statistical inference for Pr(Y < X): The normal case

This article examines statistical inference for Pr(Y < X), where X and Y are independent normal variates with unknown means and variances. The case of unequal variances is stressed. X can be interpreted as the strength of a component subjected to a stress Y, and Pr(Y < X) is the components reliability. Two approximate methods for obtaining confidence intervals and an approximate Bayesian probability interval are obtained. The actual coverage probabilities of these intervals are examined by simulation.

Bayesian Inference for Masked System Lifetime Data

Estimating component and system reliabilities frequently requires using data from the system level. Because of cost and time constraints, however, the exact cause of system failure may be unknown. Instead, it may only be ascertained that the cause of system failure is due to a component in a subset of components. This paper develops methods for analysing such masked data from a Bayesian perspective. This work was motivated by a data set on a system unit of a particular type of IBM PS/2 computer. This data set is discussed and our methods are applied to it

Care and Handling of Univariate Outliers in the General Linear Model to Detect Spuriosity—A Bayesian Approach

We deal with the situation covered by the univariate general linear model, that is, it is intended that n observations be generated in accordance with the usual model y = Xβ + e however, it is feared that k of the observations are spurious, that is, not generated in the manner intended, so that for an unknown set of k distinct integers, say (i 1, … ik ), a subset of the first n integers, we have, specifically, that ytj , = x tj ′, β + a j , + ∊ tj , where in general x t ′ denotes the t-th row of X, and where (a 1, …, a k ), so called shift parameters, are such that – ∞ < a j , < ∞. In this paper, we discuss the posterior distribution of β, when indeed it is assumed a priori that any given set of k observations has uniform probability I/( n k ) of being spurious. The properties of the posterior of β are discussed, and the results used in an example using data generated from a response surface design. Ad hoc procedures are discussed for gaining information on k, when k is unknown. These ad hoc procedures ar...

An index of rotatability

An index of rotatability is suggested that will enable the experimenter to obtain an immediate appreciation of the overall shape of specified variance contours for symmetrical second-order designs. Values of the index are tabulated for the central composite designs for two to eight factors. Three designs of Roquemore (1976) are assessed via the index. Comparisons are made with an entirely different index suggested by Khuri (1988). It is concluded that both indexes are useful and sensibly consistent.

Effect of correlation on the estimation of a mean in the presence of spurious observations

Abstract : This paper examines the effect of various correlation structures of observations on rules for estimating a mean which are designed to quard against the possibility of spurious observations (that is, observations generated in a manner not intended). The premium and protection of these rules are evaluated and discussed for the equi-correlation case and for the case of an autoregressive process of first order. It is shown that the premium and protection of a given rule which is designed for the estimator of a general mean mu when spuriosity may exist and when the observations are independent, lacks robustness to departures from independence. It is also shown that in moderate sized samples a spurious observation could seriously bias the usual estimator of the autoregressive coefficient alpha. One application of these results is in the case of a first order autoregressive model which can be used to represent many time series data encountered in business and economics.

Confidence limits for stress-strength models with explanatory variables

A lower confidence bound is obtained for Pr(Y > X|z 1, z 2), where X and Y are independent normal variables, with explanatory variables z 1 and z 2, respectively. For equal residual variances, an exact solution is obtained, but for the unequal variance case, an approximate lower confidence bound is developed. Examples of the use of these procedures are given.

Shortest confidence and prediction intervals for the log-normal

We provide the shortest prediction interval for X, and the shortest confidence interval for the median of X, when X has the log-normal distribution for both the case a2, the variance of log X, known and unknown. Tables are given to assist the practitioner in constructing these intervals. A real-life example is provided to illustrate the results.

Analysis of Outliers with Adjusted Residuals

Many statistical procedures designed to guard against the occurrence of outliers or spurious observations in normal theory are based upon examining the magnitude of the residuals. A major difficulty involved is caused by the fact that the residuals are correlated. It is shown that one way to avoid such difficulty is to adjust the residuals using information from an auxiliary experiment so that the adjusted residuals become uncorrelated. For the problem of making inferences about the unknown mean of a normal population N(μ, σ2) with known σ2, this leads to a set of estimation procedures by which the observation(s) associated with the largest adjusted residual(s) in magnitude will be excluded. Certain properties of the procedures are discussed and exact numerical results are given for the cases of one and two spurious observations. Generalization to the case of unknown variance and to the general linear model is also given.

Bayesian Analysis of Stochastically Ordered Distributions of Categorical Variables

Abstract This article considers a finite set of discrete distributions all having the same finite support. The problem of interest is to assess the strength of evidence produced by sampled data for a hypothesis of a specified stochastic ordering among the underlying distributions and to estimate these distributions subject to the ordering. We present a Bayesian approach that is an alternative to using the posterior probability of the hypothesis and the Bayes factor in favor of the hypothesis. We develop computational methods for the implementation of Bayesian analyses. We analyze examples to illustrate inferential and computational developments. The methodology used for testing a hypothesis is seen to apply to a wide class of problems in Bayesian inference and has some distinct advantages.

Optimal collapsing of mixture distributions in robust recursive estimation

Several authors have discussed Kalman filtering procedures using a mixture of normals as a model for the distributions of the noise in the observation and/or the state space equations. Under this model, resulting posteriors involve a mixture of normal distributions, and a “collapsing method” must be found in order to keep the recursive procedure simple. We prove that the Kullback-Leibler distance between the mixture posterior and that of a single normal distribution is minimized when we choose the mean and variance of the single normal distribution to be the mean and variance of the mixture posterior. Hence, “collapsing by moments” is optimal in this sense. We then develop the resulting optimal algorithm for “Kalman filtering” for this situation, and illustrate its performance with an example.