不同损失函数下Poisson分布参数的E-Bayes估计及其E-MSE

摘要/Abstract

摘要：

为了度量E-Bayes估计的误差，该文基于E-Bayes估计的定义，引入了E-Bayes估计的E-MSE（expected mean square error）的定义.对Poisson分布的参数，在不同损失函数（包括：平方损失，K-损失，加权平方损失）下分别给出了E-Bayes估计及其E-MSE的表达式.用MonteCarlo方法进行模拟比较提出的估计方法的性能，分析了一个真实数据集并进行了比较，所得结果比较基于E-MSE，结果表明该文提出的方法可行且便于应用.

关键词: Poisson分布, E-Bayes估计, E-MSE, 损失函数, Monte Carlo方法

Abstract:

In order to measure the error of E-Bayesian estimation, this paper the definition of E-MSE(expected mean square error) is introduced based on the definition of E-Bayesian estimation. For parameter of Poisson distribution, under different loss functions (including:squared error loss, K-loss and weighted squared error loss), the formulas of E-Bayesian estimation and formulas of E-MSE are given respectively. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and a real data set have been analysed for illustrative purposes, results are compared on the basis of E-MSE, the results show that the proposed method is feasible and convenient for application.

Key words: Poisson distribution, E-Bayesian estimation, E-MSE, Loss function, Monte Carlo method

中图分类号:

O213.2

韩明. 不同损失函数下Poisson分布参数的E-Bayes估计及其E-MSE[J]. 数学物理学报, 2019, 39(3): 664-673.

Ming Han. E-Bayesian Estimation and Its E-MSE of Poisson Distribution Parameter Under Different Loss Functions[J]. Acta mathematica scientia,Series A, 2019, 39(3): 664-673.

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$n$	20	40	60	80	100
$\widehat\lambda_{EB1}$	0.5074	0.5057	0.5049	0.5022	0.5015
$\widehat\lambda_{EB2}$	0.4873	0.4932	0.4966	0.4960	0.4964
$\widehat\lambda_{EB3}$	0.4636	0.4810	0.4884	0.4898	0.4915
E-MSE$(\widehat\lambda_{EB1})$	0.0250	0.0125	0.0083	0.0062	0.0048
E-MSE$(\widehat\lambda_{EB2})$	0.0256	0.0126	0.0084	0.0063	0.0050
E-MSE$(\widehat\lambda_{EB3})$	0.0274	0.0131	0.0086	0.0064	0.0051

$n$	20	40	60	80	100
$\widehat\lambda_{EB1}$	0.9977	1.0003	0.9988	0.9997	0.9999
$\widehat\lambda_{EB2}$	0.9730	0.9879	0.9905	0.9936	0.9947
$\widehat\lambda_{EB3}$	0.9489	0.9756	0.9823	0.9874	0.9898
E-MSE$(\widehat\lambda_{EB1})$	0.0487	0.0247	0.0165	0.0124	0.0099
E-MSE$(\widehat\lambda_{EB2})$	0.0493	0.0249	0.0166	0.0125	0.0100
E-MSE$(\widehat\lambda_{EB3})$	0.0511	0.0253	0.0168	0.0126	0.0101

$n$	20	40	60	80	100
$\widehat\lambda_{EB1}$	1.9735	1.9882	1.9929	1.9936	1.9941
$\widehat\lambda_{EB2}$	1.9489	1.9758	1.9846	1.9873	1.9891
$\widehat\lambda_{EB3}$	1.9247	1.9635	1.9764	1.9811	1.9841
E-MSE$(\widehat\lambda_{EB1})$	0.0963	0.0491	0.0329	0.0247	0.0197
E-MSE$(\widehat\lambda_{EB2})$	0.0969	0.0492	0.0330	0.0248	0.0198
E-MSE$(\widehat\lambda_{EB3})$	0.0987	0.0497	0.0332	0.0249	0.0199

$n$	20	40	60	80	100
$\widehat\lambda_{EB1}$	3.9248	3.9617	3.9720	3.9787	3.9885
$\widehat\lambda_{EB2}$	3.9003	3.9493	3.9638	3.9725	3.9886
$\widehat\lambda_{EB3}$	3.8760	3.9370	3.9555	3.9663	3.9887
E-MSE$(\widehat\lambda_{EB1})$	0.1915	0.0978	0.0656	0.0494	0.0395
E-MSE$(\widehat\lambda_{EB2})$	0.1921	0.0980	0.0657	0.0495	0.0397
E-MSE$(\widehat\lambda_{EB3})$	0.1939	0.0984	0.0659	0.0496	0.0398

$k$	0	1	2	3	4	5	6	7	8	9	10	11	总数
$n_{k}$	64	171	239	220	155	83	46	20	6	3	0	1	1008