欢迎来到OAJRC平台！

期刊栏目

投稿

审稿

期刊目次

加入编委

期刊订阅

添加您的邮件地址以接收即将发行期刊数据：

当前位置：国际应用数学进展 > 2026年 8卷 (2)期

Open Access Article

Advances in International Applied Mathematics. 2026; 8: (2) ; 1-8 ; DOI: 10.12208/j.aam.20260010.

Optimality analysis of parameter estimation for common distributions under a unified framework: A teaching supplement for mathematical statistics courses
统一框架下常见分布的参数估计及其优良性分析 ——数理统计课程的教学补充

作者： 陈银 *

扬州大学数学学院江苏扬州

*通讯作者：陈银,单位：扬州大学数学学院江苏扬州；

发布时间: 2026-08-30 总浏览量: 32

PDF 全文下载引用本文

摘要

参数估计的优良性是数理统计学中的核心问题，直接关系到统计推断的可靠性与决策质量。在统一框架下，本文逐一推导了常见分布（二项分布，泊松分布，正态分布，指数分布）的参数的最大似然估计（MLE）和矩估计（ME）这两种估计量的解析表达式，并基于无偏性准则和相合性准则对其进行了系统比较。主要结果表明：二项分布和泊松分布中参数的MLE与ME均为无偏且相合；正态分布的参数μ的MLE与ME均为无偏且相合的，参数的MLE与ME为相合但有偏的；指数分布的参数λ的MLE与ME是相合但有偏的。本研究为常见分布参数估计方法的选择提供了理论依据，亦可供数理统计教学参考。

关键词： 二项分布；泊松分布；正态分布；指数分布；无偏性；有效性；相合性

Abstract

The optimality properties of parameter estimation are central topic in mathematical statistics as it directly affects the reliability of statistical inference and the quality of decision-making. Within a unified framework, this paper derives step by step the formulas for both the maximum likelihood estimators (MLE) and the moment estimators (ME) of the parameters of several common distribution (binomial, Poisson, normal, and exponential distributions) and systematically compares them based on the criteria of unbiasedness and consistency. The main results show that for the binomial and Poisson distributions, both MLE and ME of their parameters are unbiased and consistent; for the normal distribution, MLE and ME of the parameter μ are unbiased and consistent, whereas those of the parameter are consistent but biased; for the exponential distribution, MLE and ME of the parameter λ is consistent but biased. This study provides a theoretical basis for choosing parameter estimation methods for common distributions and may also serve as a reference for the teaching of mathematical statistics.

Key words： Binomial distribution; Poisson distribution; Normal distribution; Exponential distribution; Unbiasedness; Efficiency; Consistency

参考文献 References

[1] 茆诗松, 程依明, 濮晓龙, 概率论与数理统计教程. 北京:高等教育出版社, (2019).

[2] 陈希孺, 倪国熙, 数理统计学教程. 安徽:中国科学技术大学出版社, (2009).

[3] Cramér, H., Mathematical Methods of Statistics. Princeton University Press. (1999).

[4] Fan, J., Li, R. Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association, 96(456), 1348-1360. (2001).

[5] Fisher, R. A., On an absolute criterion for fitting frequency curves. Messenger of Mathematics, 41, 155-160, (1912).

[6] Fisher, R. A., On the mathematical foundations of theoretical statistics. Philosophical Transactions of the Royal Society of London, Series A, 222, 309-368, (1922).

[7] Fisher, R. A., Theory of statistical estimation. Mathematical Proceedings of the Cambridge Philosophical Society, 22(5), 700-725, (1925).

[8] Hájek, J., A characterization of limiting distributions of regular estimates. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 14(4), 323-330, (1970).

[9] Huber, P. J., Robust estimation of a location parameter. The Annals of Mathematical Statistics, 35(1), 73-101. (1964).

[10] Lehmann, E. L., Scheffé, H., Completeness, similar regions, and unbiased estimation. Sankhyā: The Indian Journal of Statistics, 10(4), 305-340, (1950)

[11] Lehmann, E. L., Scheffé, H. Completeness, similar regions, and unbiased estimation: Part II. Sankhyā: The Indian Journal of Statistics, 15(3), 219-236, (1955).

[12] Pearson, K., Contributions to the mathematical theory of evolution. Philosophical Transactions of the Royal Society of London, Series A, 185, 71-110, (1894).

[13] Rao, C. R., Information and accuracy attainable in the estimation of statistical parameters. Bulletin of the Calcutta Mathematical Society, 37, 81-91, (1945).

[14] Tibshirani, R., Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society: Series B, 58(1), 267-288, (1996).

[15] Wald, A., Note on the consistency of the maximum likelihood estimate. The Annals of Mathematical Statistics, 20(4), 595-601, (1949)

[16] Zhang, C. H., Zhang, S. S., Confidence intervals for low dimensional parameters in high dimensional linear models. Journal of the Royal Statistical Society: Series B, 76(1), 217-242, (2014).

[17] Zou, H., The adaptive lasso and its oracle properties. Journal of the American Statistical Association, 101(476), 1418-1429, (2006).

引用本文

陈银, 统一框架下常见分布的参数估计及其优良性分析 ——数理统计课程的教学补充[J]. 国际应用数学进展, 2026; 8: (2) : 1-8.

入驻平台

回到顶部

搜索文章: