Characterizations of Populations Using Regression Properties
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AI 根据所提供的论文材料归类：摘要讨论以回归性质刻画概率分布的通用理论，未涉及所列分布效应或其他经济计量专门问题。
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利用回归性质刻画总体分布
- 研究问题
- 能否通过一个统计量对另一个统计量的多项式回归性质，刻画独立同分布随机变量所来自的总体分布？
- 方法与思路
- 作者由三次统计量对线性统计量的三次回归，导出特征函数的三阶非线性微分方程，并研究其系数条件。
- 材料陈述的贡献
- 摘要称获得正态、伽马、二项等多种分布的刻画定理，涵盖此前二次及常数回归结果，并概述高阶推广。
- 阅读关注点
- 应核对统计量的定义、矩条件、方程解的唯一性与特征函数合法性，并区分已证明的三次结果和推广提纲。
- 阅读边界
- 仅据 OpenAlex 索引摘要，稿件版本未注明；具体系数条件和证明未提供，高阶情形仅被描述为方法提纲。
原文摘要
In the present work, we study characterizations of populations obtained by using regression properties of one statistic on another. We extend some of the results of Lukacs and Laha by considering the cubic regression (polynomial regression of order 3) of a cubic statistic $S$ on a linear one $L$. This assumption of cubic regression is used to derive a third order non-linear differential equation in the characteristic function $h(t)$ of a set of $n$ independently and identically distributed random variables. The coefficients in this differential equation represent certain fixed relationships between the coefficients of the statistic $S$ and the regression coefficients. For appropriate choices of the coefficients in the fundamental differential equation, the resulting equation can be solved to yield the characteristic function of a particular distribution. In this way, we are able to obtain a series of characterization theorems for each of a variety of populations including Normal, Gamma, Binomial, Poisson, Geometric and several others. Moreover, all of the results obtained by Lukacs and Laha for characterizing populations using quadratic and constant regression are shown to be special cases of the theorems obtained in the present work. Finally, in the last section, we present an outline of a technique which can be used to study any general $r \leqq m$th order polynomial regression of any $m$th order statistic on a linear one. The approach used to generate the $m$th order differential equation is indicated and a method for determining the appropriate conditions on the coefficients is discussed.