Vol 25, No 3 (2016)

The first aim of this paper is to generalize the online estimator of a regression function introduced by Révész [26, 27] to the multivariate framework. Similarly to the univariate framework, the study of the convergence rate of the multivariate Révész’s estimator requires a tedious condition connecting the stepsize of the algorithm and the unknown value of the density of the regressor variable at the point at which the regression function is estimated. The second aim of this paper is to apply the averaging principle of stochastic approximation algorithms to remove this tedious condition.

In this paper, we carry out a theoretical study of the correlation coefficients between exponential order statistics and their monotonicity properties. Then, aided by a Monte Carlo simulation study, we make some conjectures about the correlation structure of order statistics from a larger class of distributions.

Let X₁,..., X_n, n > 1, be nondegenerate independent chronologically ordered realvalued observables with finite means. Consider the “no-change in the mean” null hypothesis H₀: X₁,..., X_n is a randomsample on X with Var X <∞. We revisit the problem of nonparametric testing for H₀ versus the “at most one change (AMOC) in the mean” alternative hypothesis H_A: there is an integer k*, 1 ≤ k* < n, such that EX₁ = · · · = EX_k* ≠ EX_k*+1 = ··· = EX_n. A natural way of testing for H₀ versus H_A is via comparing the sample mean of the first k observables to the sample mean of the last n - k observables, for all possible times k of AMOC in the mean, 1 ≤ k < n. In particular, a number of such tests in the literature are based on test statistics that are maximums in k of the appropriately individually normalized absolute deviations Δ_k = |S_k/k - (S_n - S_k)/(n - k)|, where S_k:= X₁ + ··· + X_k. Asymptotic distributions of these test statistics under H₀ as n → ∞ are obtained via establishing convergence in distribution of supfunctionals of respectively weighted |Z_n(t)|, where {Z_n(t), 0 ≤ t ≤ 1}_n≥1 are the tied-down partial sums processes such that

if 0 ≤ t < 1, and Z_n(t):= 0 if t = 1. In the present paper, we propose an alternative route to nonparametric testing for H₀ versus H_A via sup-functionals of appropriately weighted |Z_n(t)|. Simply considering max_1ɤk<n Δk as a prototype test statistic leads us to establishing convergence in distribution of special sup-functionals of |Z_n(t)|/(t(1 - t)) under H₀ and assuming also that E|X|^r < ∞ for some r > 2. We believe the weight function t(1 - t) for sup-functionals of |Z_n(t)| has not been considered before.