基于均方误差最小准则下总体均值估计的 4×4 最优排序集抽样设计及其性质

数学物理学报 ›› 2026, Vol. 46 ›› Issue (6): 2317-2328.

基于均方误差最小准则下总体均值估计的 4×4 最优排序集抽样设计及其性质

符梦娟, 陈望学^*(), 王得吉, 曹潜

吉首大学数学与统计学院湖南吉首 416000

收稿日期:2025-07-28 修回日期:2026-01-23 出版日期:2026-12-26 发布日期:2026-08-14
通讯作者: 陈望学 E-mail:chenwangxue2015@163.com
基金资助:
国家自然科学基金(12261036);国家自然科学基金(11901236)

4×4 Optimal Ranked Set Sampling Design for Estimation of Population Mean and its Properties under the Criterion of Minimum Mean Squared Error

Mengjuan Fu, Wangxue Chen^*(), Deji Wang, Qian Cao

College of Mathematics and Statistics, Jishou University, Hunan Jishou 416000

Received:2025-07-28 Revised:2026-01-23 Online:2026-12-26 Published:2026-08-14
Contact: Wangxue Chen E-mail:chenwangxue2015@163.com
Supported by:
NSFC(12261036);NSFC(11901236)

摘要/Abstract

摘要：

平衡排序集抽样 (RSS) 适用于总体分布完全未知时的统计推断, 但如果已有总体分布的某些信息时, 使用某些准则就可以进一步提高统计推断的效率. 已有文献在均方误差 (MSE )最小准则下, 针对正态分布、均匀分布和指数分布找到了总体均值估计的 3×3 最优 RSS 设计. 该文给出了 4×4 RSS 设计下总体均值估计的 MSE 及其 35 种设计. 基于 MSE 最小准则下,给出了寻找 4×4 最优 RSS 设计的一般结论. 为了提高估计的精度, 进一步研究了可循环 4×4 最优 RSS 设计下总体均值估计的 MSE. 针对正态分布、均匀分布和指数分布, 基于 MSE 最小准则下, 给出了这些分布的 4×4 最优 RSS 设计. 进一步研究了这些分布在可循环 4×4 最优 RSS 设计下总体均值估计的 MSE, 研究结果表明在可循环 4×4 最优 RSS 设计下这些分布中总体均值估计的 MSE 随着循环的增大不断减小.

关键词: 排序集抽样, 均方误差, 最优排序集抽样设计

Abstract:

Balanced ranked set sampling (RSS) is suitable for statistical inference when the population distribution is completely unknown, but if some information about the population distribution is available, the efficiency of statistical inference can be further improved by using certain criteria. Under the criterion of minimum mean square error (MSE), existing literature has found the 3×3 optimal RSS designs for the normal distribution, uniform distribution, and exponential distribution. This paper presents the MSE of estimation of population mean under 4×4 RSS designs and its 35 designs. Based on the criterion of minimum MSE, general conclusions for finding the 4×4 optimal RSS design is given. To improve the accuracy of estimation, the MSE of estimation of population mean under the cyclic 4×4 optimal RSS design is further studied. For the normal distribution, uniform distribution, and exponential distribution, the 4×4 optimal RSS designs under the criterion of minimum MSE are derived. By further studying the MSE of estimation of population mean under the cyclic 4×4 optimal RSS design for these distributions, the research results show that the MSE of estimation of population mean for these distributions under the cyclic 4×4 optimal RSS design continuously decreases with the increase of cycles.

Key words: ranked set sampling, mean squared error, optimal ranked set sampling design

中图分类号:

O212.2

符梦娟, 陈望学, 王得吉, 曹潜. 基于均方误差最小准则下总体均值估计的 4×4 最优排序集抽样设计及其性质[J]. 数学物理学报, 2026, 46(6): 2317-2328.

Mengjuan Fu, Wangxue Chen, Deji Wang, Qian Cao. 4×4 Optimal Ranked Set Sampling Design for Estimation of Population Mean and its Properties under the Criterion of Minimum Mean Squared Error[J]. Acta mathematica scientia,Series A, 2026, 46(6): 2317-2328.

图/表 4

参考文献 14

[1]	McIntyre G A. A method for unbiased selective sampling using ranked sets. Australian Journal of Agricultural Research, 1952, 3(4): 385-390 doi: 10.1071/AR9520385
[2]	Yang R, Chen W X. Maximum likelihood estimator of the shape parameter under simple random sampling and moving extremes ranked set sampling. Statistics andProbability Letters, 2025, 226: 110465
[3]	Dai W C, Chen W X, Zhao H L. Estimation of the population mean under imperfect simple Z ranked set sampling. Statistical Papers, 2025, 66(4): 81 doi: 10.1007/s00362-025-01707-9
[4]	Chen M, Chen W X, Deng C H. Some new results on parameter estimation of the exponential-Poisson distribution in ranked set sampling. Applied Mathematics-A Journal of Chinese Universities Series B, 2025, 40(2): 413-428
[5]	汪寒, 陈望学. Bilal 分布中参数的修正极大似然估计及其抽样性质. 系统科学与数学, 2025, 45(2): 639-650 doi: 10.12341/jssms23681
	Wang H, Chen W X. Modified maximum likelihood estimator and sampling properties of estimator for Bilal distribution. Journal of Systems Science and Mathematical Sciences, 2025, 45(2): 639-650 doi: 10.12341/jssms23681
[6]	姜杰, 陈望学, 汪寒. 不同类型排序集抽样设计下 Topp-Leone 分布中参数的极大似然估计及其性质. 应用数学学报, 2025, 48(2): 294-304 doi: 10.20142/j.cnki.amas.202401043
	Jiang J, Chen W X, Wang H. Maximum likelihood estimator of the parameter of Topp-Leone distribution and its properties under different types of ranked set sampling. Acta Mathematicae Applicatae Sinica, 2025, 48(2): 294-304 doi: 10.20142/j.cnki.amas.202401043
[7]	Takahasi K, Wakimoto K. On unbiased estimates of the population mean based on the sample stratified by means of ordering. Annals of the Institute of Statistical Mathematics, 1968, 20(1): 1-31 doi: 10.1007/BF02911622
[8]	Chen Z H, Bai Z D. Ranked Set Sampling:Theory and Applications. New York: Springer, 2004
[9]	Kaur A, Patil G P, Taillie C. Unequal allocation models for ranked set sampling with skew distributions. Biometrics, 1997, 53: 123-130 doi: 10.2307/2533102
[10]	Chen W X, Tian Y, Xie M Y. Maximum likelihood estimator of the parameter for a continuous one-parameter exponential family under the optimal ranked set sampling. Journal of Systems Science and Complexity, 2017, 30(6): 128-141
[11]	Chen W X, Tian Y, Xie M Y. The global minimum variance unbiased estimator of the parameter for a truncated parameter family under the optimal ranked set sampling. Journal of Statistical Computation and Simulation, 2018, 88(17): 3399-3414 doi: 10.1080/00949655.2018.1520233
[12]	张棚, 陈望学. 最优排序集抽样下正态分布中总体方差的优良估计及其性质. 系统科学与数学, 2026, 46(6): 2048-2063 doi: 10.12341/jssms240884
	Zhang P, Chen W X. Some properties of optimal estimation of population variance for normal distribution under optimal ranked set sampling. Journal of Systems Science and Mathematical Sciences, 2026, 46(6): 2048-2063 doi: 10.12341/jssms240884
[13]	Li M M, Chen W X. $3$×$3$ optimal ranked set sampling design with $k$ cycles and best linear invariant estimators of the parameters for normal distribution. Statistics and Probability Letters, 2025, 224: 110455
[14]	Al-Saleh M F. Steady-state ranked set sampling and parametric estimation. Journal of Statistical Planning and Inference, 2004, 123(1): 83-95 doi: 10.1016/S0378-3758(03)00139-3