A robust principal component analysis

A robust principal component analysis for samples from a bivariate distribution function is described. The method is based on robust estimators for dispersion in the univariate case along with a certain linearization of the bivariate structure. Besides the continuity of the functional defining the direction of the suitably modified principal axis, we prove consistency of the corresponding sequence of estimators. Asymptotic normality is established under some additional conditions.Principal component robustness robust estimator for dispersion linearization

Some properties of bivariate empirical hazard processes under random censoring

In Campbell (1982, IMS Lecture Notes--Monograph Series Vol. 2, pp. 243-256, IMS, Hayward, CA) and Campbell and Földes (1982, Proceedings, Internat. Colloq. Nonparametric Statist. Inform., 1980, North-Holland, New York) some asymptotic properties of bivariate empirical hazard processes under random censoring are given. Taking the representation of the empirical hazard process for bivariate randomly censored samples in Campbell, op. cit., as a starting point and restricting attention to strong properties, we obtain a speed of strong convergence for the weighted bivariate empirical hazard processes as well as a speed of strong uniform convergence for bivariate hazard rate estimators. Our approach is based on a local fluctuation inequality for the bivariate hazard process and differs from the martingale methods quite often used in the univariate case.Bivariate randomly censored sample weighted empirical hazard process hazard rate estimator strong convergence

Asymptotic behavior of sample mean direction for spheres

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The almost sure behavior of the oscillation modulus of the multivariate empirical process

Let [omega]n denote the oscillation modulus of the uniform multivariate empirical process, defined as the variation of the process over multi-dimensional intervals with Lebesgue measure not exceeding an [epsilon] (0,1). The a.s. limiting behavior of [omega]n is established for sequences {an} of five different orders of magnitude that constitute an essentially complete spectrum of possibilities. Extensions to processes with underlying d.f. more general than the uniform are indicated. The results may have applications in density estimation and in the theory of multivariate spacings. For related results in the univariate case we refer in particular to Mason, Shorack and Wellner (1983) and Mason (1984), and for a general setting to Alexander (1984).multivariate empirical process oscillation modulus almost sure limits

Curve estimation for mn-decomposable time series including bilinear processes

The speed of a.s. uniform convergence of nonparametric curve estimators for mn-decomposable processes is studied. Such processes include, e.g., linear and bilinear processes. A simple and straghtforward method is used relating this kind of problem to compound empirical processes (i.e., empirical processes with random jumps). Hazard rate estimation under random censoring is also included as an example.compound empirical processes mn-decomposable processes curve estimation hazard rate estimation under random censoring

The order of magnitude of the moments of the modulus of continuity of multiparameter poisson and empirical processes

In this note we derive the exact order of magnitude of the moments of the modulus of continuity for multiparameter Poisson and, almost as a corollary, for multivariate empirical processes.Multiparameter Poisson and empirical processes modulus of continuity exponential bound

Asymptotic estimate of probability of misclassification for discriminant rules based on density estimates

Let X1,..., X1 and Y1,..., Yn be independent random samples from the distribution functions (d.f.) F and G respectively. Assume that F' = f and G' = g. The discriminant rule for classifying and independently sampled observation Z to F if and to G, otherwise where l and n are the estimates of f and g respectively based on a common kernel function and the training X- and Y-samples, are considered optimal in some sense. Let Pf denote the probability measure under the assumption that Z ~ F and set P0 = Pf(f(Z) > g(Z)) and . In this article we have derived the rate at which PN --> P0 as N = l + n --> [infinity], for the situation where l = n, F(x) = TM(x - [theta]2) and G(x) = M(x - [theta]1) for some symmetric d.f. M and parameters [theta]1, [theta]2. We have examined a few special cases of M and have established that the rate of convergence of PN to P0 depends critically on the tail behavior of m = M'.optimal classification rule probability of misclassification kernel function density estimates

Strong uniform convergence of density estimators on compact Euclidean manifolds

In this note the naive estimators for densities on the sphere in Ruymgaart (1989) are generalised to compact smooth submanifolds of a Euclidean space and their convergence behaviour is studied. As special examples are treated Stiefel manifolds (which include both the sphere and the orthogonal groups). Furthermore in the last section some goodness of fit procedures are discussed.Density estimation goodness of fit Stiefel manifold Grasman manifold Euclidean manifolds