In specific practical problems, structures with ordered and inequality constraints find extensive applications in multiple fields. When estimating parameters in such cases, one often encounters restrictions imposed by ordered constraints. For instance, in pharmaceutical testing, researchers may impose constraints on the dosage of a particular drug, necessitating consideration of ordered constraints. Therefore, ordered constraint mean estimation holds significant practical value in these applications. In previous literature, the focus on mean estimation under ordered constraints has been mainly on 2 and 3 multivariate normal populations. This paper extends this focus to the estimation problem of multivariate normal population means under simple partial order constraints. It introduces a new estimate , based on the PAVA algorithm when the population covariance matrix is known, and proves its consistent superiority over the unordered maximum likelihood estimate . Finally, the effectiveness of the proposed estimation method is validated through simulation experiments and compared with the maximum likelihood estimation method.
Key words
multivariate normal populations /
simple order restriction /
maximum likelihood estimation /
risk function /
PAVA algorithm
{{custom_keyword}} /
{{custom_sec.title}}
{{custom_sec.title}}
{{custom_sec.content}}
References
[1] Shi Ningzhong. Maximum Likelihood Estimation of means and variance from normal populations under simultaneous order restrictions. Multivariate Anal.,1994, ( 49) : 282-294
[2] Shi Ningzhong, Jiang Hua. Maximum Likelihood Estimation of isotonic normal means with unknown variance. Multivariate Anal., 1998, (64): 183-195
[3] Ma Tiefeng, Wang Songgui. Estimation of means of multivariate normal populations with order restriction. Journal of Multivariate Analysis, 2010, 101(3): 594-602
[4] 罗平,李树有.三个多元正态总体在简单半序约束下均值估计.统计研究,2013,30(03):101-105, DOI: 10.19343/j.cnki.11-1302/c.2013.03.015(Luo Ping, Li Shuyou. Estimation of means of three multivariate normal populations with simple order restriction. Statistical Research, 2013, 30(3): 101-105, DOI: 10.19343/j.cnki.11-1302/c.2013.03.015)
[5] 罗平,李树有.三个多元正态总体在简单半序约束下均值估计-基于协方差阵未知.应用数学学报,2015, 38(6):1136-1144(Luo Ping, Li Shuyou. Estimation of means of more multivariate normal populations based on the restriction of unknown covariance matrices. Acta Mathematica Applicatae Sinica, 2015, 38(6): 1136- 1144)
[6] 李树有.正态总体参数在序约束下的估计与检验.博士论文,吉林大学,2007(Li Shuyou. Estimation and Test for Parameters of Normal Populations under Order Restrictions. Doctoral Dissertation of Jilin University, 2007)
[7] Li Zhiguo. Some problems in statistical inference under order restrictions. Doctoral Dissertation of the University of Michigan, 2008
[8] Ganguly A, Mitra D, Kundu D. Order Restricted Inference for Adaptive Progressively Censored Competing Risks Data. arXiv preprint arXiv:2205.03550, 2022
[9] González J C, van Delden A, de Waal T. Assessment of the effect of constraints in a new multivariate mixed method for statistical matching. Computational Statistics Data Analysis, 2023, 177: 107569
[10] Garg N, Misra N. Isotonic Regression Estimators For Simultaneous Estimation of Order Restricted Location/Scale Parameters of a Bivariate Distribution: A Unified Study. arXiv preprint arXiv: 2301.00690, 2023
[11] Ghosh S, Chaudhuri S. Maximum Likelihood under constraints: Degeneracies and Random Critical Points. arXiv preprint arXiv:1910.01396, 2019.
[12] López-Lopera A F, Bachoc F, Durrande N, et al. Finite-dimensional Gaussian approximation with linear inequality constraints. SIAM/ASA Journal on Uncertainty Quantification, 2018, 6(3): 1224- 1255
[13] 国冰.基于bootstrap方法序约束下正态总体均值、方差的区间估计.黑龙江科学,2016,7(23):23-24(Guo Bing. Intervalestimation of mean and variance of normal population based onbootstrap algorithm. Heilongjiang Science, 2016, 7(23): 23-24)
[14] 史宁中.保序回归与最大似然估计.应用概率统计,1993(02):203-215(Shi Ningzhong. Isotonic Regression and the Maximum Likelihood Estimation. Chinese Journal of Applied Probability and Statistics, 1993(02): 203-215)
[15] 李树有,史宁中,张宝学.正态总体均值与标准差比在序约束下的广义p-值检验.应用概率统计,2009, 25(01):77-85(Li Shuyou, Shi Ningzhong, Zhang Baoxue. Testing Ratios of Means to Standard Deviations from Normal Populations under Order Restrictions with Generalized p Values. Chinese Journal of Applied Probability and Statistics, 2009, 25(01): 77-85)
[16] 宋海燕.序约束下参数模型的统计推断.博士论文,吉林大学,2005(Song Haiyan. Order Restricted Statistical Inference of Parametrer Models. Doctoral Dissertation of Jilin University, 2005)
[17] 赵春雪,李树有,宓颖.多个Marshall-Olkin Fréchet分布总体参数在序约束下的极大似然估计.东北师大学报(自然科学版),2018,50(03):43-47,DOI:10.16163/j.cnki.22-1123/n.2018.03.009(Zhao Chunxue, Li Shuyou, Mi Ying. Maximum likelihood estimation of parameters for multiple Marshall-lkin Frechet distribution population under theorder restriction. Journal of Northeast Normal University (Natural Science Edition), 2018, 50(03): 43-47, DOI:10.16163/j.cnki.22-1123/n.2018.03.009)
[18] 杜宇静.序贯k-out-of-n系统在序约束下参数的估计和算法.应用概率统计,2018,34(01):75-83(Du Yujing. Order restricted estimation of parameters and algorithm for sequential k-out-of-n systems with covariates. Chinese Journal of Applied Probability and Statistics, 2018, 34(01): 75-83)
[19] Hu Xiaomi. On an ad hoc test for order restricted multivariate normal means. Communications in Statistics-theory and Methods, 2016, 45(5): 1501-1507
[20] Betcher J, Peddada S D. Statistical inference under order restrictions in analysis of covariance using a modified restricted maximum likelihood estimator. Sankhya. Series B., [Methodological.], 2009, 71(1): 79
{{custom_fnGroup.title_en}}
Footnotes
{{custom_fn.content}}