82 – Computer Experiments, Statistical Engineering, and Applications in Physical Sciences
Statistical Applications of CLT for Dependent Data
Martial Longla
University of Mississippi
This paper presents a survey of central limit theorems for dependent data, emphasizing on cases when the dependence is not quanti ed. New central limit theorems are provided for some time series examples with better properties than known results (when mixing is assumed). The given results are used to estimate the mean and provide con dence intervals for the mean of several populations. Several statistical models are considered and tests are provided to show the importance of the results. Some of these theorems use the CLT of Kipnis and Varadhan for reversible Markov chains and other results use the Lindeberg's condition for arrays of independent data. The use of a smoothing kernel allows us to prove a theorem that provides con dence intervals for an ARFIMA model without explicit use of the fractional dierence parameter or its estimate. Several setups are used to illustrate the use of the results. While developing these concepts, we use simulations to show that even when some assumptions of the theorems are violated, the estimators that are proposed still perform well on large samples. We provide some comparisons that can help applied statisticians and encourage the use of these methods. Several statistical models are considered.