New development of statistical estimation of parameters with linear inequalities restriction

本文首先考虑两个有序方差未知的正态分布的共同均值估计问题。我们给出了一个广泛的估计类，其中包括Nair（1982）和Elfessi等（1992）提出的估计，并证明了随机估计优于不考虑方差序限制的估计，包括Graybill和Deal（1959）给出的一个当未知方差是有序的时，我们给出了两个有序均值的广义个体估计。我们发现，在估计的均值与较大的方差，估计不考虑方差的顺序限制随机主导的估计，考虑到这两个顺序限制的建议类。然而，在估计具有较小方差的均值方面，即使在均方误差方面也不可能有类似的改进。我们还证明了两个有序均值同时估计问题的一个控制结果。在皮特曼准则下讨论了上述问题，并证明了皮特曼准则的有效性。

We first consider the problem of estimating the common mean of two normal distributions with unknown ordered variances. We give a broad class of estimators which includes the estimators proposed by Nair (1982) and Elfessi et al. (1992) and show that the estimators stochastically dominate the estimators which do not take into account the order restriction on variances, including the one given by Graybill and Deal (1959).Then we propose a broad class of individual estimators of two ordered means when unknown variances are ordered. We show that in estimating the mean with larger variance, estimators which do not take into account the order restriction on variances are stochastically dominated by the proposed class of estimators which take into account both order restrictions. However, in terms of estimating the mean with smaller variance, similar improvement is not possible even in terms of mean squared error. We also show a domination result in the simultaneous estimation problem of two ordered means. Further, improving upon the unbiased estimators of the two means is also discussed.We also discuss the above problems under Pitman closeness criterion and confirm the validity of Pitman closeness criterion.