Modifications On The Central Limit Theorem For Independent And Identically Distributed Random Variables

The central limit theorem plays an important role in probability and mathematical statistics. It opens up a new world for research on large sample analysis and has a wide application in medical science, economics, meteorology and so on. In recent years, researchers have made efforts to figure out the convergence rate of the central limit theorem and drawn some conclusions about the asymptotic expansion of distribution functions. In this paper, modifications on the asymptotic expansion of the distribu-tion function of the sum of independent and identically distributed random variables are made. It is shown that modified statistics could better approximately follow normal distribution and accuracy is improved effectively by using this method. Based on the theoretical results, simulation study is conducted for two widely used probability dis-tributions, namely, exponential distribution and Pareto distribution. Results between classical statistic and modified statistics are compared to illustrate that this new method outperforms the traditional one in terms of the estimation accuracy. In addition, the selection of a parameter is also briefly discussed.