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

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• 作者：Yu. V. Martsynyuk
• 关键词：nonparametric test for change in the mean ; “no ; change in the mean” null hypothesis ; “at most one change in the mean” alternative hypothesis ; sup ; functional of weighted tied ; down partial sums processes ; standard Wiener process ; Brownian bridge
• 刊名：Mathematical Methods of Statistics
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：25
• 期：3
• 页码：219-232
• 全文大小：701 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Statistics
  Statistical Theory and Methods
  Russian Library of Science
  
• 出版者：Allerton Press, Inc. distributed exclusively by Springer Science+Business Media LLC
• ISSN：1934-8045
• 卷排序：25

文摘

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.Keywordsnonparametric test for change in the mean“no-change in the mean” null hypothesis“at most one change in the mean” alternative hypothesissup-functional of weighted tied-down partial sums processesstandard Wiener processBrownian bridge2000 Mathematics Subject Classificationprimary 60F1762G10secondary 60G15References1.D. Chibisov, “Some Theorems on the Limiting Behavior of Empirical Distribution Functions”, Selected Transl. Math. Statist. Probab. 6, 147–156 (1964).Google Scholar2.M. Csörgő, “A Glimpse of the Impact of Pál Erdős on Probability and Statistics”, Canad. J. Statist. 30, 493–556 (2002).MathSciNetCrossRefMATHGoogle Scholar3.M. Csörgő, S. Csörgő, L. Horváth, and D. M. Mason, “Weighted empirical and quantile processes”, Ann. Probab. 14, 31–85 (1986).MathSciNetCrossRefMATHGoogle Scholar4.M. Csörgőand L. 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