分層不等機率抽樣之迴歸參數估計的比較分析

A Comparative Analysis among Estimators of Regression Coefficients under Stratified Sampling with Unequal Probability

陳建銘(Chien-Ming Chen)

指導教授 ： 許玉雪

國立臺北大學/商學院/統計學系/碩士(2008年)

摘要

抽樣調查設計隨著資訊需求的增加日益複雜，然若參數的估計方法無法配合所使用的抽樣調查設計，則統計推論結果將難以達到預期的精確度。彙整過去文獻中配合複雜抽樣調查設計之迴歸分析方法，有四種:(1)最小平方法、(2)分層加權最小平方法、(3)機率加權最小平方法及(4)Quasi- Aitken機率加權最小平方法。過去的研究發現：(1)在不等機率抽樣時，一般常用的最小平方法所得到的迴歸參數估計式並非不偏估計式；(2)機率加權最小平方法可以改善不偏性，然卻增加估計的變異程度。本文主要研究目的為，比較分析各種迴歸參數估計式在分層不等機率抽樣下的表現，並找出不偏且估計變異較小的迴歸參數估計式。研究方法係採用Monte Carlo模擬方式模擬比較最小平方法、分層加權最小平方法、機率加權最小平方法及Quasi-Aitken機率加權最小平方法在分層不等機率抽樣下的表現。研究結果顯示機率加權最小平方法與Quasi-Aitken機率加權最小平方法的迴歸參數估計式皆具有不偏性，其中Quasi-Aitken機率加權最小平方法的迴歸參數估計式的變異最小。

關鍵字

分層抽樣 ； 迴歸分析 ； 不等機率抽樣 ； Monte Carlo模擬 ； 最小平方法 ； 分層加權最小平方法 ； 機率加權最小平方法 ； Quasi-Aitken機率加權最小平方法

並列摘要

This paper aims to compare the estimators of regression coefficients under stratified sampling with unequal probability based upon a Monte Carlo approach. Recently, regression analysis has become popular with complex surveys. This paper intends to compare alternative estimators for regression coefficients under a complex survey. The alternative estimators used in this paper include least squares estimator, stratified weighted least squares estimator, probability weighted least squares estimator, and Quasi-Aitken weighted least squares estimator. Least squares methods which ignore population structure and sampling design could give seriously misleading results. There are two findings summarized from previous studies: (1) the least squares estimator is a common choice of researchers, but under an unequal probability design, the estimator is biased, (2) the probability weighted estimator is consistent but may have a large variance. Monte Carlo approach is used in this paper to compare the efficiency of the four estimators of regression coefficients based upon bias, variance, and MSE. The simulation results show that probability weighted least squares estimator and Quasi-Aitken weighted least squares estimator are unbiased estimators of regression coefficients. The simulation results also find that the Quasi-Aitken weighted least squares estimator has a smaller asymptotic variance than least squares estimator.

並列關鍵字

Regression Analysis ； Stratified sampling ； PPS ； Stratified weighted least squares estimator ； Probability weighted least squares estimator ； Quasi-Aitken weighted least squares estimator ； Monte Carlo simulation

參考文獻

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