Estimation of Parameters from Incomplete Data
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不完整数据下的参数估计
- 研究问题
- 多元正态样本中部分个体缺少某些变量观测时，哪些特殊数据结构允许显式最大似然估计，其效率如何评价？
- 方法与思路
- 作者研究三元正态不完整数据的一个特殊情形，给出显式最大似然解，推导估计量方差和协方差，并讨论二元特例。
- 材料陈述的贡献
- 摘要称获得特殊情形下的显式估计公式及抽样方差协方差，提供数值示例，并考察仅使用完整数据部分的通常方法的效率。
- 阅读关注点
- 应核对允许显式解的缺失模式、正态与抽样条件、方差推导，以及完整数据方法效率比较的准则和二元化条件。
- 阅读边界
- 仅据版本未注明的索引摘要；特殊缺失结构及缺失机制条件未说明，不能推广至任意缺失数据，也不能推断刊本变化。
原文摘要
THE present note is concerned with a special case of the general problem of obtaining efficient estimators for the parameters of a normal multivariate population when the available sample data are incomplete in the sense that measures on all variables are not available for all individuals in the sample. Such fragmentary data may arise because part of the data are irretrievably lost (e.g., in an archaeological find), or because certain data were purposely not collected. The decision not to measure all individuals in the sample on all variables may be reached because of the cost of measurement, because of limited time, because the measurement of a certain variable alters or destroys the individual measured (e.g., in mental testing, testing explosives), and so forth. The general problem for normal bivariate populations has been treated by Wilks [3]. This treatment has been further generalized to normal multivariate populations by Matthai [2], who deals explicitly with the trivariate case. Unfortunately, the general maximum likelihood equations have proved rather intractable, and no simple formulas for the maximum likelihood estimators are available in the general case, even for a sample from a bivariate population. In the present paper, the problem of estimating the parameters of a normal trivariate population from incomplete data is dealt with in a special case for which explicit solutions to the maximum likelihood equations are readily obtained. This special case is described in the following section. Formulas for the maximum likelihood estimators are given; their application is illustrated by a numerical example. The sampling variances and covariances of the maximum likelihood estimators are derived. An examination is made of the efficiency of the usual methods that utilize only that portion of the data that is complete. The foregoing results are specialized to apply to a commonly encountered bivariate (rather than trivariate) situation.