James D. Broffitt

Generalized Linear and Quadratic Discriminant Functions Using Robust Estimates

Abstract Two new methods of constructing robust linear and quadratic discriminant functions are introduced. The first is a generalization of Fishers procedure for finding a linear discriminant function. It places less weight on those observations that are far from the overlapping regions of the two populations. The second new method substitutes M-estimates of the means and the covariance matrices into the usual expressions for the linear and quadratic discriminant functions. Monte Carlo results indicate lower misclassification probabilities for these schemes compared to Fishers linear discriminant function in cases of heavy-tailed or contaminated distributions.

Distribution-Free Partial Discriminant Analysis

Abstract A distribution-free rank procedure is proposed for use in partial discrimination problems involving two populations. It is shown that this procedure can be applied with virtually any discriminant function. Moreover, the discriminant function may be selected after observing the samples on which it is to be based. Using Monte Carlo methods the rank procedure is compared with a normal theory and a tolerance region procedure. The rank procedure was the only one that adequately controlled the probabilities of misclassification while maintaining relatively small probabilities of not classifying an observation.

Discriminant Analysis Based on Ranks

Abstract A model-free rank procedure is proposed for the two-population discrimination problem that enables the practitioner to better control the balance between the two probabilities of misclassification. The method is applied to the discriminant functions resulting from normal assumptions and also to an adaptive one which is a weighted average of the linear and quadratic discriminant functions, where the weights are determined from the data. A Monte Carlo study shows that the rank method can greatly improve the balance between the two misclassification probabilities while keeping their average comparatively small.

A Bayes estimator for ordered parameters and isotonic Bayesian graduation

Abstract The problem is to estimate the parameters θ 1 ..., θk under the order restriction θ 1 < ... < θk . A prior distribution is specified such that the probability of the set {(θ 1, ..., θk ): θ 1< ... <θk } is one. This guarantees that the mean of the posterior distribution (Bayes estimator) satisfies the order restriction. This technique, which produces a smoothing effect, is applied to the problem of estimating mortality rates over consecutive age intervals. In this application the result may be viewed as an isotonic Bayesian graduation. A real data example is provided.

A Power Approximation for the Chi-Square Goodness-of-Fit Test: Simple Hypothesis Case

Abstract The limiting distribution of the chi-square goodness-of-fit statistic Tn under alternatives is noncentral chi-square if the alternative probabilities approach the null probabilities at an appropriate rate as n → ∞. It is shown that for fixed alternative probabilities, the limiting distribution of (Tn − μ n )/σ n is standard normal. Both of these asymptotic results can be used to approximate the power of the goodness-of-fit test. Numerical comparisons between these two approximations indicate that for large values of the true power, the normal approximation is best, but for moderate values of power, the chi-square approximation is best.

Robust estimation in growth curve models

The growth curve model introduced by potthoff and Roy 1964 is a general statistical model which includes as special cases regression models and both univariate and multivariate analysis of variance models. The methods currently available for estimating the parameters of this model assume an underlying multivariate normal distribution of errors. In this paper, we discuss tw robst estimators of the growth curve loction and scatter parameters based upon M-estimation techniques and the work done by maronna 1976. The asymptotic distribution of these robust estimators are discussed and a numerical example given.

Isotonic Bayesian Graduation with an Additive Prior

Assume the mortality rate at age x + j−1 is q x +j−1 = 1− exp(−θj), j = l,…,k. Isotonic Bayesian graduation provides a Bayes estimator of θ 1 ,… ,θ k (and consequently, q x ,…,q x +k•i) under the assumption θ 1 < … < θ k . This is accomplished by specifying a prior distribution for which P(Θ1 < … < Θk) = 1. In a previous paper the prior was defined by where Y 1 …,Y k are independent. In this paper the Bayes estimator is developed using the prior The advantages are an easier specification of the prior parameters and shorter computational time.

Zero Correlation, Independence, and Normality

Abstract Two easy classroom examples are given of marginally normal random variables that are dependent but have zero correlation. One example is similar in nature to Andersons well-known illustration that marginal normality does not imply joint normality.

An Example of the Large Sample Behavior of the Midrange

Let X(1) ? . < X(n) be the order statistics based on a random sample from one member of the family {fo(x) = h(x-0); -00 < 0 < oo }. Assume h is a known density function which is symmetric about zero, so that 0 is the median of fo (x). In mathematical statistics courses the most commonly studied estimators of 0 seem to be the median (X), mean (X), and midrange (i). Under mild conditions we have, from standard asymptotic results, the large sample approximations

A Special Distributional Result for Bilinear Forms

Abstract Necessary and sufficient conditions are given such that a quadratic form has moment-generating function E[exp ( tU′BU)] = (1 – t 2)–r/4 for |t| < 1 with U ∼ Nk (μ, Σ) and Σ positive definite. An important corollary gives conditions under which the bilinear form X′AY involving two different multivariate normal random vectors (of not necessarily the same dimensions) has the same distribution as the sum of independent random variables, each having the LaPlace (double exponential) distribution.