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Large Sample Properties of Ridge Estimators of Regression Parameters in Linear Regression Models with Missing Data

Author: DongDong
Tutor: QinYongSong
School: Guangxi Normal University
Course: Probability Theory and Mathematical Statistics
Keywords: Missing Data Linear model Ridge Estimate Consistency Asymptotic normality MAR missing mechanism
CLC: O212.1
Type: Master's thesis
Year: 2010
Downloads: 84
Quote: 2
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Abstract

In many practical problems, due to a variety of man-made or other unknown factors, are likely to lead to a large number of missing data generated widespread phenomenon of missing data, for example, in the field of public opinion surveys, market research, medical research, as well as socio-economic research in recent years , statistical inference of missing data situations has become a hot research field in the statistical community today. usual statistical methods often can not be directly applied in the case of missing data, the necessary data, missing data processing method common Complete-Case and fill method, Complete-Case missing data items to delete, and then on the remaining items constitute the complete sample in accordance with the usual statistical methods of statistical inference, fill divided into fixed fill they are necessary to complement the missing values, then get the complete sample, in accordance with the usual statistical methods of statistical inference. linear models have a strong practical background in the medical, biological, economic and random fill method, data analysis in the financial, environmental science and engineering fields has been more widely used, the linear model parameter estimation theory and methods, the method of least squares to occupy the center of the foundation of the economy, but when the design matrix X degradation or close degradation, least squares estimation is far from ideal, some scholars have proposed a new estimation method - ridge estimation method, ridge estimation can be used to solve statistical inference problems in the design matrix close to degradation. Hoerl and Kennard (Ridge regression biased estimation for non-orthogonal problems [J]. Tech-nometrics, 1970, 12: 55-57.) 1970 Ridge estimated β (k) = (S kI)? 1X Y is used to improve the minimum squares, where k is estimated gt; 0, S = XX, X and Y, respectively, for the design variables and the response variable matrix, I is the identity matrix. ridge estimation research and applications have been received wide attention and has become the most the impact of a biased estimate. the ridge estimate the theoretical study of the early results, see Hoerl and Kennard (Ridge \squared error of ridge regression [J]. J Roy Statist Soc B, 1976, 38: 248-259.) paper, ridge estimation to theoretical system summarize visible Wang Songgui (linear model theory and its application [M]. Hefei : Anhui Education Press, 1987; linear model Introduction [M] Beijing: Higher Education Press, 2004.) book, they give a sufficient condition for a series of ridge estimate better than least squares estimation. Dai Jian Hua ( ridge estimate the conditions better than least squares estimation [J] Mathematical Statistics and Applied Probability, 1994, 9 (2): 53-58.) to discuss the ridge estimate the significance of the mean square error than least squares estimation, given the ridge estimate better than least squares estimation of the necessary conditions and more general sufficient condition; Wong Kai Ying (regression coefficient Ridge estimated Consistency [J] Mathematical Statistics and Applied Probability, 1987, 3 (1): 42-51 .) Ridge estimated Strong Consistency r order Consistency, and some limit properties based on ridge regression error estimated ordinary least squares estimates under the same conditions exactly the same large sample properties. scholars ridge estimate do improvements to expect reduced the mean square error to improve the precision of the estimates, the ridge estimate article further improved, and improve the precision of the estimates are in varying degrees. constraints linear model in As Zheng Changguang, (linear constraints estimated [J]. Applied Probability and Statistics, 1986, 2 (1): 5-12.) put it, the parameter β constrained least squares estimation of β?'s mean square error under certain conditions, can become very large, so the effect is not ideal, which prompted people to look for in the class of β biased estimate a reasonable estimate to improved β? Lei celebrate regression coefficients (linear model ridge estimate Consistency [J]. Guangxi Normal University, 1999, 10 (1): 21-24.) bring along inferior-Rβ = 0 constraints linear regression model ridge estimation of coefficients strong, weak consistency The mean square Consistency get the necessary and sufficient conditions for weak consistency and the strong consistency sufficient condition; Shi Jianhong (Ridge and constrained regression coefficients of the linear regression model estimated [J]. Shanxi Normal University (Natural Science ), 2001, 15 (4): 10-16.) Qi inferior Rβ = 0 constraints linear regression model, a new class of Ridge and the estimated β? (k) = (kW I)? 1β?, proved that β? (k) is better than the constrained least squares estimation of the parameter β under certain regularity conditions and excellent guidelines, and discuss the estimated allowable agricultural scenery, Wan-Rong Liu, Li Minghui (non-homogeneous regression coefficients of the linear regression model equality constraints Ridge Estimation [J]. Sichuan Normal University (Natural Science), 2007, 30 (6): 721-725.) Nonhomogeneous Rβ = r constraints raised under the constraints of a linear regression model Ridge estimated estimated statistical properties, and discuss its relationship with constrained least squares estimation parameters Ridge and proved that under certain regularity conditions and excellent guidelines estimated better than the constrained least squares estimation. often generate missing data in real life, but the missing data Linear Regression coefficient ridge estimate statistical inference problems have not been examined. fixed design Chapter completely under a linear model with linear constraints, the case of incomplete data missing in the response variable, using three different processing methods dealing with missing data, namely the use of observed data, deterministic top-up to get the \the estimated, randomness complement \and asymptotic normality. randomized design with linear constraints linear model, the case of incomplete data missing in the response variable, using three different processing methods for handling missing data in the third chapter, namely complete data using observed on deterministic top-up \arbitrary linear function of the regression coefficients estimated proved strong, weak consistency and asymptotic normality. features of this paper are reflected in the following two aspects: 1 MAR missing mechanism to study under a fixed design with line Large sample properties estimated by linear regression coefficient Ridge constraint on the missing response variables, the use of three different missing data handling methods given three ridge estimation of regression coefficients estimated proved estimated Strong, Weak Consistency , proved that any linear function of the regression coefficients estimated that the strong, weak consistency and asymptotic normality .2 MAR missing mechanism, researchers randomized design with linear constraints linear regression coefficient Large sample properties of ridge estimation of the missing response variables, the use of three different types of missing data processing method gives three estimated regression coefficient ridge estimate proved strong, weak consistency estimated proved regression coefficients any linear function of the estimated strong, weak consistency and asymptotic normality.

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