For multivariate data, the halfspace depth function can be seen as a natural and affine equivariant generalization of the univariate empirical cdf. For any multivariate data set, we show that the resulting halfspace depth function completely determines the empirical distribution. We do this by...

We consider independent pairs (X1, [Sigma]1), (X2, [Sigma]2), ..., (Xn, [Sigma]n), where each[Sigma]iis distributed according to some unknown density functiong([Sigma]) and, given[Sigma]i=[Sigma],Xihas conditional density functionq(x|[Sigma]) of the Wishart type. In each pair the...

LetA1,A2, ..., ANandB1, B2, ..., BMbe two sequences of events, and let[nu]N(A) and[nu]M(B) be the number of thoseAiandBj, respectively, that occur. We prove that Bonferroni-type inequalities forP([nu]N(A)[greater-or-equal, slanted]u,[nu]M(B)[greater-or-equal, slanted]v),...

This paper derives the prediction distribution of future responses from the linear model with errors having an elliptical distribution with known covariance parameters. For unknown covariance parameters, the marginal likelihood function of the parameters has been obtained and the prediction...

We investigate some geometrical aspects of the discriminant functions of the kindfp(x)=[summation operator]pk=1 ck[tau](a'kx) for suitable constantsak, ckwhere[tau]is a sigmoidal transformation. This function is realized by a multilayer perceptron with one hidden layer. These results are...

In a number of problems, interest is centered on only a few of the coefficients of the multiple linear regression model, while the remaining parameters are treated as nuisance parameters. At the same time, the experimenter is interested in estimating the parameters robustly. We propose a new...

We consider the classical fixed-size confidence region estimation problem for the mean vector[mu]in theNp([mu], [Sigma]) population where [Sigma] is unknown but positive definite. We write[lambda]1for the largest characteristic root of [Sigma] and assume that[lambda]1is simple. Moreover, we...

Based upon the idea of construction of data driven smooth tests for composite hypotheses presented in Inglotet al.(1997) and Kallenberg and Ledwina (1997), two versions of data driven smooth test for bivariate normality are proposed. Asymptotic null distributions are derived, and consistency of...

Let[mu]be a Gaussian measure (say, onRn) and letK,L[subset, double equals]Rnbe such thatKis convex,Lis a "layer" (i.e.,L={x: a[less-than-or-equals, slant]<x, u>[less-than-or-equals, slant]b} for somea, b[set membership, variant]Randu[set membership, variant]Rn), and the centers of mass (with...</x, u>

Radial positive definite functions are of importance both as the characteristic functions of spherically symmetric probability distributions, and as the correlation functions of isotropic random fields. The Euclid's hat functionhn(||x||),x[set membership, variant]n, is the self-convolution of an...