Estimation of parametric functions for multivariate normal distributions with incomplete observations

多元正态分布的统计推断理论在许多多元分析教材中都可以看到。一些专著处理的情况下，不完整或缺失的观察给出。本文主要研究了不完全观测下多元正态分布的统计估计问题。以下是主要结果。(1)利用生态位重叠度量来评价两个种群的相似性。在完整样本的情况下，当样本来自具有未知均值向量和协方差矩阵的多元正态分布时，考虑了Matusita测度的估计问题（Minami和Shimizu， 1999）。推导了Matusita测度估计的渐近方差和偏倚，并比较了三种偏倚减小方法。不完全观测的情况应该在未来进行研究。(2)提出了估算面积雨率第一积矩和协方差的阈值方法（Hossain and Shimizu, 1999）。实证研究表明，对于东京都降水数据，对于陆地和岛屿这两个不同地理位置，第一积矩与超过两个阈值的概率之间存在很强的相关性。以二元对数正态分布为模型，通过最小化渐近归一化方差来选择理论最优阈值。实验结果与理论结果吻合较好。(3)讨论了缺少观测值的生态位为二元正态分布（Minami、Shimizu和Mishra）时Matusita测度的估计问题。考虑了离散参数的最大似然（ML）估计和限制最大似然（REML）估计对Matusita测度的插件估计。仿真研究表明，用REML估计得到的插件估计比用ML估计得到的插件估计具有更小的均方误差。(4)分析了1996年6月- 1999年3月日本国立环境研究所小型激光雷达在筑波连续观测的地基激光雷达数据（Takagiwa等，见）。研究了垂直分布和季节变化。在地面激光雷达测量中，没有探测到较厚的低层云之上的云。结合地面激光雷达和空间激光雷达的统计分析，对云重叠模型进行验证。(5)《宫冈与崎》（1999）和《史密斯与宫冈》（1999）。少

The theory of statistical inference in multivariate normal distributions with complete samples can be seen in many textbooks of multivariate analysis. Some monographs deal with the case when incomplete or missing observations are given. In this research we mainly studied statistical estimation for multivariate normal distributions with incomplete observations. The following are the main results.(1) Measures of niche overlap are used to asses the similarity of two populations. The problem of estimating Matusita's measure when samples from multivariate normal distributions with unknown mean vectors and covariance matrices was considered for the case of complete samples (Minami and Shimizu, 1999). Asymptotic variances and biases of Matusita's measure estimates were derived and three bias reduction methods were compared. The case for incomplete observations should be studied in the future.(2) Threshold methods for estimating area rain-rate first product moment and covariance were proposed … More (Hossain and Shimizu, 1999). Empirical study shows a strong correlation between the first product moment and the, probability of exceeding two threshold values for two geographically different locations, land and island, for Tokyo Metropolitan rain-fall data. Theoretical optimal thresholds under the bivariate lognormal distribution as a model were chosen by minimizing the asymptotic normalized variance. A good agreement between the empirical and theoretical results was observed.(3) The problem of estimating Matusita's measure when the niches are bivariate normal distributions with missing observations was discussed (Minami, Shimizu and Mishra, to appear).The plug-in estimates of Matusita's measure by the Maximum Likelihood (ML) estimates and the Restricted Maximum Likelihood (REML) estimates for dispersion parameters were considered. Simulation study shows that the plug-in estimate by the REML estimation tends to have smaller mean square error than that by the ML estimation.(4) The ground-based lidar data (June 1996-March 1999) continuously observed in Tsukuba with the National Institute for Environmental Studies compact lidar were analyzed (Takagiwa, et al., to appear). The vertical distribution and seasonal variation were studied. In the ground-based lidar measurement, clouds above thick lower clouds are not detected. The cloud overlapping model will be validated by combining the statistical analysis of ground-based lidars and the space lidar.(5) Other are in Miyaoka and Tazaki (1999) and in Smith and Miyaoka (1999). Less