Randomized Empirical Processes and Confidence Bands via Virtual Resampling

02/12/2018
by   Miklós Csörgő, et al.
0

Let X,X_1,X_2,... be independent real valued random variables with a common distribution function F, and consider {X_1,...,X_N }, possibly a big concrete data set, or an imaginary random sample of size N≥ 1 on X. In the latter case, or when a concrete data set in hand is too big to be entirely processed, then the sample distribution function F_N and the the population distribution function F are both to be estimated. This, in this paper, is achieved via viewing {X_1,...,X_N } as above, as a finite population of real valued random variables with N labeled units, and sampling its indices {1,...,N } with replacement m_N:= ∑_i=1^N w_i^(N) times so that for each 1≤ i ≤ N, w_i^(N) is the count of number of times the index i of X_i is chosen in this virtual resampling process. This exposition extends the Doob-Donsker classical theory of weak convergence of empirical processes to that of the thus created randomly weighted empirical processes when N, m_N →∞ so that m_N=o(N^2).

READ FULL TEXT
research
08/09/2023

On a variance dependent Dvoretzky-Kiefer-Wolfowitz inequality

Let X be a real-valued random variable with distribution function F. Set...
research
05/03/2019

A Constructive Proof of a Concentration Bound for Real-Valued Random Variables

Almost 10 years ago, Impagliazzo and Kabanets (2010) gave a new combinat...
research
12/19/2020

The level crossings of random sums

Let {a_j}_j = 0^N and {b_j}_j = 0^N be sequences of mutually independent...
research
11/28/2021

A Simple Necessary Condition For Independence of Real-Valued Random Variables

The standard method to check for the independence of two real-valued ran...
research
10/11/2019

The density of complex zeros of random sums

Let {η_j}_j = 0^N be a sequence of independent, identically distributed ...
research
11/04/2022

Significance improvement by randomized test in random sampling without replacement

This paper studies one-sided hypothesis testing under random sampling wi...
research
04/24/2022

On Stute's representation for a class of smooth, possibly data-adaptive empirical copula processes

Given a random sample from a continuous multivariate distribution, Stute...

Please sign up or login with your details

Forgot password? Click here to reset