Exchangeably Weighted Bootstraps of General Markov U-Process

Abstract

We explore an exchangeably weighted bootstrap of the general function-indexed empirical U-processes in the Markov setting, which is a natural higher-order generalization of the weighted bootstrap empirical processes. As a result of our findings, a considerable variety of bootstrap resampling strategies arise. This paper aims to provide theoretical justifications for the exchangeably weighted bootstrap consistency in the Markov setup. General structural conditions on the classes of functions (possibly unbounded) and the underlying distributions are required to establish our results. This paper provides the first general theoretical study of the bootstrap of the empirical U-processes in the Markov setting. Potential applications include the symmetry test, Kendall’s tau and the test of independence.

Weak Convergence of the Conditional Set-Indexed Empirical Process for Missing at Random Functional Ergodic Data

“…To establish Theorem 3, we can rely on Theorem 2 of [96] (see also Refs. [10,13,15]). It is sufficient to demonstrate that, for all constant L > 0, as n tends to infinity:…”

“…[1-3] for exhaustive, self-contained texts with a variety of statistical applications) together under mixing conditions and independent identically distributed framework, in the setting of independents variables [4] characterized modulo measurability, the classes C of sets for which the Glivenko-Cantelli theorem holds, we may also cite Refs. [5][6][7][8][9][10][11][12][13][14][15]. Under various mixing conditions, empirical processes based on dependent data have been investigated; for instance, the authors of Ref.…”

On the weak convergence and the uniform-in-bandwidth consistency of the general conditional $U$-processes based on the copula representation: multivariate setting

“…An important contribution is given in the paper [56] where a much stronger form of consistency holds, namely, uniform in t and in bandwidth consistency (i.e., h n , h n ∈ [a n , b n ] where a n < b n → 0 at some specific rate) of r (m) n (φ, t; h n ). In addition, uniform consistency is also established over φ ∈ F for a suitably restricted class F , for recent references see [31,32,34,35,36,37,132]. The main tool in their result is the use of the local conditional U process investigated in [75].…”

On Weak Convergence of the Bootstrap Copula Empirical Process with Random Resample Size

“…In order to draw conclusions about the features of the underlying population, the bootstrap relies on the assumption that a given sample is statistically equivalent to the population as a whole, refer to [23]. For recent reference on the subject, refer to [24][25][26][27][28]. The primary purpose of writing this paper is that, to the best of our knowledge, the results presented here provide a response to a previously understudied topic.…”