Strong Gaussian approximations of product-limit and Quantile Processes for Strong mixing and censored data

TL;DR

This paper develops strong Gaussian approximations for the product-limit and quantile processes under censored dependent data, enabling precise probabilistic analysis and laws of the iterated logarithm.

Contribution

It introduces novel strong Gaussian approximation techniques for these processes in dependent censored data models, with explicit convergence rates.

Findings

Established strong Gaussian approximations with rate O((log n)^(-λ))

Derived laws of the iterated logarithm for the product-limit process

Provided theoretical foundations for analyzing censored dependent data

Abstract

In this paper, we consider the product-limit quantile estimator of an unknown quantile function under a censored dependent model. This is a parallel problem to the estimation of the unknown distribution function by the product-limit estimator under the same model. Simultaneous strong Gaussian approximations of the product-limit process and product-limit quantile process are constructed with rate $O((\log n)^{-\lambda})$ for some $\lambda>0,$. The strong Gaussian approximation of the product-limit process is then applied to derive the laws of the iterated logarithm for product-limit process.