Asymptotic Representations of the Multivariate Empirical Distribution Function and Applications

Often, many complicated statistics used as estimators or test statistics take the form of the (multivariate) empirical distribution function evaluated at a random vector (V_n). Denote such statistics by S_n. This paper describes methods for the study of various asymptotic properties of S_n. First, under minimal assumptions, a weak asymptotic representation for S_n is derived. This result may be used to show the asymptotic normality of S_n. Second, under slightly more stringent regularity conditions, an almost sure representation of S_n, with suitable order (as.) of the remainder term is studied and then a law of the iterated logarithm for S_n, is derived. In this context, strong convergence results from a sequential point of view are also studied. Finally, weak convergence to a Brownian motion process is established. As an application, we show the limiting normality of S_n, for a random number of summands.     

Cross-cumulants measure for independence

We propose a measure for independence of group of random variables, given by a sum of cross-cumulants of a given order n  . A similar measure was known for the case of fourth-order cross-cumulants from the JADE algorithm for ICA (independent component analysis). We derive a formula for its calculation using cumulant tensors. In the case n=4$n=4$ our formula allows efficient calculation of this measure, using cumulant matrices. Much attention is devoted to the case of six-order cross-cumulants, aiming to show that this measure can be calculated using again cumulant matrices.     

IMPROVING UPON THE BEST INVARIANT ESTIMATOR IN MULTIVARIATE LOCATION PROBLEMS

We are concerned with estimators which improve upon the best invariant estimator, in estimating a location parameter θ. If the loss function is L(θ - a) with L convex, we give sufficient conditions for the inadmissibility of δ₀(X) = X. If the loss is a weighted sum of squared errors, we find various classes of estimators δ which are better than δ₀. In general, δ is the convolution of δ₁ (an estimator which improves upon δ₀ outside of a compact set) with a suitable probability density in R^p. The critical dimension of inadmissibility depends on the estimator δ₁ We also give several examples of estimators δ obtained in this way and state some open problems.     

LINEAR MINIMAX ESTIMATORS FOR ESTIMATING A FUNCTION OF THE PARAMETER

In this note we provide sufficient conditions for the minimaxity of linear estimators of the form aX+b in the one-parameter exponential family for estimating a differentiable function g(θ) with normalized quadratic loss. We provide some examples which show that the natural estimator X is minimax in estimating a function of the parameter (different from the mean).