Testing for change in the mean via convergence in distribution of sup-functionals of weighted tied-down partial sums processes

Abstract

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

\({Z_n}\left( t \right): = \left( {{S_{\left\lceil {\left( {n + 1} \right)t} \right\rceil }} - \left[ {\left( {n + 1} \right)t} \right]{S_n}/n} \right)/\sqrt n \)

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.