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262 Introduction to Robust Estimation and Hypothesis Testing 6.8.2 Comparing Robust Generalized Variances Robust generalized variances can be compared as well. A percentile bootstrap appears to avoid type I errors above the nominal level. But situations are encountered where the actual level can be substantially smaller than the nominal level. Corrections are available in some situations (Wilcox, 2006e), which are used by the R function described in the next section, but no details are given here. 6.8.3 R function gvar2g The R function gvar2g(x, y, nboot = 100, DF = T, eop = 1, est = skipcov, alpha = 0.05, cop = 3, op = 1, MM = F, SEED = T) compares two independent groups based on a robust version of the generalized variance. By default, the OP covariance matrix is used in conjunction with Carling's modification of the boxplot rule. Setting MM=T, a MAD-median rule is used. If DF=T, and if the sample sizes are equal, the function reports an adjusted critical p-value, assuming that the goal is to have a type I error probability equal to .05, the argument est=skipcov, and that other conditions are met. Otherwise, no adjusted critical value is reported. For information about the arguments op, cop, and eop, see the R function skipcov. 6.9 Multivariate Density Estimators This section outlines two multivariate density estimators that will be used when plotting data. The first is based on a simple extension of the expected frequency curve described in Chapter 3 and the other is a multivariate analog of the adaptive kernel density estimator. An extensive discussion of multivariate density estimation goes beyond the scope of this book, but some indication of the method used here, when plotting data, seems warranted. The strategy behind the expected frequency curve is to determine the proportion of points that are close to X i . There are various ways this might be done and here a method based on the MVE covariance matrix is used. Extant results suggest this gives a reasonable first approximation of the shape of a distribution in the bivariate case, but there are many alternative methods for determining which points are close to X i , and virtually nothing is known about their relative merits for the problem at hand. Here, the point X i is said to be close to X i if (X i - X i ) M -1 (X i - X i ) h, where M is the MVE covariance matrix described in Section 6.3.1, and h is the span. Currently, h = 0.8 seems to be a good choice for most situations. Letting N i represent the www.elsevierdirect.com