基于蒙特卡洛模拟的CLT正态近似收敛速度量化研究

Quantitative Study on the Convergence Rate of CLT Normal Approximation Based on Monte Carlo Simulation

摘要: 中心极限定理是统计推断的理论基础，该定理的收敛速率并非固定不变，与总体分布形态密切相关。本研究选取均匀分布、指数分布、对数正态分布、双峰混合正态分布四类总体分布作为研究对象，借助蒙特卡洛随机模拟，在多个样本量条件下对标准化后的样本均值的抽样分布进行大量重复试验。研究采用Shapiro-Wilk检验、超额峰度、Q-Q图等多种诊断工具，系统评估各分布向标准正态逼近的收敛速度。研究表明：均匀分布收敛最快；双峰混合正态分布虽形态异常，其收敛速度优于右偏的指数分布；对数正态分布在样本量2至10区间偏度与峰度出现反常上升，显示CLT收敛过程存在非单调特征。上述发现呈现了不同总体分布下CLT收敛速度的差异，为大样本统计推断中的样本量选取提供了参考依据。

Abstract: The central limit theorem is the theoretical foundation of statistical inference. The convergence rate of this theorem is not fixed and is closely related to the overall distribution pattern. This study selected four types of population distributions, namely uniform distribution, exponential distribution, logarithmic normal distribution, and bimodal mixed normal distribution, as the research objects. With the help of Monte Carlo random simulation, large-scale repeated experiments were conducted on the sampling distribution of the standardized sample mean under multiple sample size conditions. The study used various diagnostic tools such as the Shapiro-Wilk test, excess kurtosis, and Q-Q plot to systematically evaluate the convergence speed of each distribution towards the standard normal. Research shows that uniform distribution converges the fastest. Although the bimodal mixed normal distribution has an abnormal shape, its convergence speed is better than the right skewed exponential distribution. The logarithmic normal distribution shows an abnormal increase in skewness and kurtosis in the sample size range of 2 to 10, indicating the existence of non monotonic characteristics in the CLT convergence process. The above findings present differences in the convergence speed of CLT under different population distributions, providing a reference for sample size selection in large sample statistical inference.