不同损失函数下Poisson分布参数的E-Bayes估计及其E-MSE

Abstract

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

CLC Number:

O213.2

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.

Trendmd

Figures/Tables 8

References 17

1	Ali S , Aslam M , Kazmi S M A . A study of the effect of the loss function on Bayes estimate, posterior risk and hazard function for Lindley distribution. Applied Mathematical Modelling, 2013, 37: 6068- 6078 doi: 10.1016/j.apm.2012.12.008
2	Ando T , Zellner A . Hierarchical Bayesian analysis of the seemingly unrelated regression and simultaneous equations models using a combination of direct monte carlo and importance sampling techniques. Bayesian Analysis, 2010, 5 (1): 65- 96 doi: 10.1214/10-BA503
3	Osei F B , Duker A A . Hierarchical Bayesian modeling of the space-time diffusion patterns of cholera epidemic in Kumasi, Ghana. Statistica Neerlandica, 2011, 65: 84- 100 doi: 10.1111/j.1467-9574.2010.00475.x
4	韩明. 贝叶斯统计学及其应用. 上海: 同济大学出版社, 2015
	Han Ming . Bayesian Statistics and Its Applications. Shanghai: Tongji Univ Press, 2015
5	Han Ming . E-Bayesian estimation and hierarchical Bayesian estimation of failure rate. Applied Mathematical Modelling, 2009, 33: 1915- 1922 doi: 10.1016/j.apm.2008.03.019
6	Han Ming . E-Bayesian estimation of the reliability derived from binomial distribution. Applied Mathematical Modelling, 2011, 35: 2419- 2424 doi: 10.1016/j.apm.2010.11.051
7	Han Ming . The E-Bayesian and hierarchical Bayesian estimations of Pareto distribution parameter under different loss functions. Journal of Statistical Computation and Simulation, 2017, 87: 577- 593 doi: 10.1080/00949655.2016.1221408
8	Jaheen Z F , Okasha H M . E-Bayesian estimation for the Burr type XⅡ model based on type-2 censoring. Applied Mathematical Modelling, 2011, 35: 4730- 4737 doi: 10.1016/j.apm.2011.03.055
9	Okasha H M , Wang J . E-Bayesian estimation for the geometric model based on record statistics. Applied Mathematical Modelling, 2016, 40: 658- 670 doi: 10.1016/j.apm.2015.05.004
10	Yousefzadeh F . E-Bayesian and hierarchical Bayesian estimations for the system reliability parameter based on asymmetric loss function. Communications in Statistics-Theory and Methods, 2017, 46: 1- 8 doi: 10.1080/03610926.2014.968736
11	苏兵. Poisson分布参数的几种估计. 临沂师范学院学报, 2003, 25 (6): 21- 23 doi: 10.3969/j.issn.1009-6051.2003.06.007
	Su Bing . Several estimates of Poisson distribution parameter. Journal of Linyi Normal University, 2003, 25 (6): 21- 23 doi: 10.3969/j.issn.1009-6051.2003.06.007
12	韦莹莹, 韦程东, 薛婷婷. Q-对称熵损失下Poisson分布参数倒数的估计. 广西师范学院学报(自然科学版), 2007, 24 (2): 33- 38 doi: 10.3969/j.issn.1002-8743.2007.02.008
	Wei Yingying , Wei Chengdong , Xue Tingting . The estimation of reciprocal of distribution parameters for Poisson distribution under Q-symmetric loss function. Journal of Guangxi Teachers Education University (Natural Science Edition), 2007, 24 (2): 33- 38 doi: 10.3969/j.issn.1002-8743.2007.02.008
13	韦程东. 贝叶斯统计分析及其应用. 北京: 科学出版社, 2015
	Wei C D . Bayesian Statistical Analysis and Its Application. Beijing: Science Press, 2015
14	韩明. Poisson分布参数的E-Bayes估计及其性质. 数学物理学报, 2016, 36A (5): 1010- 1016
	Han Ming . The E-Bayesian estimation of the parameter for Poisson distribution and its properties. Acta Math Sci, 2016, 36A (6): 1010- 1016
15	韩明. 多层先验分布的构造及其应用. 运筹与管理, 1997, 6 (3): 31- 40
	Han Ming . The structure of hierarchical prior distribution and its applications. Chinese Operations Research and Management Science, 1997, 6 (3): 31- 40
16	Berger J O . Statistical Decision Theory and Bayesian Analysis. New York: Springer-Verlag, 1985
17	Han Ming. E-Bayesian estimation and its E-MSE under the scaled squared error loss function, for exponential distribution as example. Communications in Statistics-Simulation and Computation.[2018-02-25]. https://doi.org/10.1080/03610918.2018.1425444

$n$	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

E-Bayesian Estimation and Its E-MSE of Poisson Distribution Parameter Under Different Loss Functions

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

Figures/Tables 8

References 17

Related Articles 10

Metrics

Comments

Recommended 1

$c$	1	3	5	7	9	极差
$\widehat{\lambda}_{EB1}$	2.8225089	2.8197148	2.8169281	2.8141487	2.8113765	0.0111324
$\widehat{\lambda}_{EB2}$	2.8220131	2.8192195	2.8164332	2.8136543	2.8108827	0.0111304
$\widehat{\lambda}_{EB3}$	2.8215173	2.8187242	2.8159385	2.8131600	2.8103889	0.0111284
E-MSE$(\widehat{\lambda}_{EB1})$	0.0027987	0.0027932	0.0027877	0.0027822	0.0027767	2.20e-005
E-MSE$(\widehat{\lambda}_{EB2})$	0.0027990	0.0027934	0.0027879	0.0027824	0.0027769	2.21e-005
E-MSE$(\widehat{\lambda}_{EB3})$	0.0027997	0.0027942	0.0027886	0.0027832	0.0027777	2.20e-005

[1]	Liu Huan, Wu Qiongli, Cournède Paul-Henry. Research on the Effects of Different Sampling Algorithm on Sobol Sensitivity Analysis [J]. Acta mathematica scientia,Series A, 2018, 38(2): 372-384.
[2]	Han Ming. The E-Bayesian Estimation of the Parameter for Poisson Distribution and Its Properties [J]. Acta mathematica scientia,Series A, 2016, 36(5): 1010-1016.
[3]	HAN Ming. Properties of E-Bayesian Estimation for the Reliability Derived form Binomial Distribution [J]. Acta mathematica scientia,Series A, 2013, 33(1): 62-70.
[4]	HAN Ming. E-Bayesian Estimation of Failure Probability in the Case of Data Only One Failure [J]. Acta mathematica scientia,Series A, 2011, 31(2): 577-583.
[5]	ZHOU Zai-Ying. Admissibility of Matrix Nonhomogeneous Linear Estimators in Multivariate Linear Models [J]. Acta mathematica scientia,Series A, 2010, 30(6): 1621-1628.
[6]	Zhang Shangli;Qin Hong. Universal Admissibility of Linear Estimators in Growth Curve Model with Respect to Restricted Parameter Sets [J]. Acta mathematica scientia,Series A, 2008, 28(3): 523-529.
[7]	Han Ming. Expected Bayesian Estimation of Failure Probability and Its Character [J]. Acta mathematica scientia,Series A, 2007, 27(3): 488-495.
[8]	BAI Peng. Kullback Leibler's Divergence between Two Multivariatet Distributions [J]. Acta mathematica scientia,Series A, 2005, 25(5): 694-709.
[9]	WEI Li-Li, ZHANG Wen-Xiu. Bayesian Procedure in Classification Problem With Linear Expon ential Hazard Function [J]. Acta mathematica scientia,Series A, 2003, 23(4): 436-443.
[10]	Xu Xingzhong Wu Qiguang. Linear Admissible Estimators of Regression Coefficient Under Balanced Loss [J]. Acta mathematica scientia,Series A, 2000, 20(4): 468-473.