On some ridge regression estimators: a nonparametric approach

This paper considers the R-estimation of the parameters of a multiple regression model when the design matrix is ill-conditioned. Accordingly, we introduce the ridge regression (RR) modification to the usual R-estimators and consider five RR R-estimators when it is suspected that the regression parameters may belong to a linear subspace of the parameter space. The regions of optimality of the proposed estimators are determined based on the quadratic risks. Asymptotic relative efficiency tables and risk graphs are provided for the numerical and graphical comparisons of the five estimators.

[1]  Shalabh.  Pitman Closeness Comparison of Least Squares and Stein-Rule Estimators in Linear Regression Models with Non-Normal Disturbances , 2001 .

[2]  C. Stein,et al.  Estimation with Quadratic Loss , 1992 .

[3]  A. K. Md. Ehsanes Saleh,et al.  Optimum Critical Value for Pre-Test Estimator , 2006 .

[4]  Mohammad Arashi,et al.  A note on classical Stein-type estimators in elliptically contoured models , 2010 .

[5]  J. Neyman,et al.  Mathematical Statistics and Probability , 1962 .

[6]  D. S. Tracy,et al.  Ridge regression using scrambled responses , 1999 .

[7]  N. Benda,et al.  Pre-test estimation and design in the linear model☆ , 1996 .

[8]  Calyampudi Radhakrishna Rao,et al.  Linear Statistical Inference and its Applications , 1967 .

[9]  J. Jurecková,et al.  Nonparametric Estimate of Regression Coefficients , 1971 .

[10]  IMPROVING THE 'HKB' ORDINARY TYPE RIDGE ESTIMATOR , 2001 .

[11]  A. K. Md. Ehsanes Saleh,et al.  Theory of preliminary test and Stein-type estimation with applications , 2006 .

[12]  Edward C. Malthouse,et al.  Ridge regression and direct marketing scoring models , 1999 .

[13]  Rand R. Wilcox,et al.  The statistical implications of pre-test and Stein-rule estimators in econometrics , 1978 .

[14]  A. K. Md. Ehsanes Saleh,et al.  Preliminary test ridge regression estimators with student’s t errors and conflicting test-statistics , 2004 .

[15]  Calyampudi R. Rao,et al.  Linear Statistical Inference and Its Applications. , 1975 .

[16]  Mohammad Arashi,et al.  Using improved estimation strategies to combat multicollinearity , 2011 .

[17]  Judith A. Giles.  Pre-testing for linear restrictions in a regression model with spherically symmetric disturbances☆ , 1991 .

[18]  B. M. Kibria,et al.  Effect of W, LR, and LM Tests on the Performance of Preliminary Test Ridge Regression Estimators , 2003 .

[19]  Shalabh.  Improved Estimation in Measurement Error Models Through Stein Rule Procedure , 1998 .

[20]  P. Sen,et al.  Nonparametric Methods in General Linear Models. , 1986 .

[21]  A. E. Hoerl,et al.  Ridge Regression: Applications to Nonorthogonal Problems , 1970 .

[22]  A. E. Hoerl,et al.  Ridge regression: biased estimation for nonorthogonal problems , 2000 .

[23]  F. N. David,et al.  LINEAR STATISTICAL INFERENCE AND ITS APPLICATION , 1967 .

[24]  T. A. Bancroft,et al.  ANALYSIS AND INFERENCE FOR INCOMPLETELY SPECIFIED MODELS INVOLVING THE USE OF PRELIMINARY TEST(S) OF SIGNIFICANCE , 1964 .

[25]  Mohammad Arashi,et al.  Improved variance estimation under sub-space restriction , 2009, J. Multivar. Anal..

[26]  Marvin H. J. Gruber.  Improving Efficiency by Shrinkage: The James--Stein and Ridge Regression Estimators , 1998 .

[27]  P. K. Sen,et al.  A Note on Asymptotically Distribution Free Tests for Subhypotheses in Multiple Linear Regression , 1973 .

[28]  S. M. M. Tabatabaey,et al.  Estimation of the location parameter under LINEX loss function: multivariate case , 2010 .

[29]  Pranab Kumar Sen,et al.  Nonparametric methods in general linear models , 1987 .

[30]  T. A. Bancroft,et al.  On Pooling Means when Variance is Unknown , 1968 .

[31]  Kazuhiro Ohtani.  A Comparison of the Stein-Rule and Positive-Part Stein-Rule Estimators in a Misspecified Linear Regression Model , 1993 .

[32]  Eshetu Wencheko,et al.  Estimation of the signal-to-noise in the linear regression model , 2000 .

[33]  ModelsEdward C. Malthouse.  Shrinkage Estimation and Direct Marketing Scoring , 1998 .

[34]  Nityananda Sarkar,et al.  A new estimator combining the ridge regression and the restricted least squares methods of estimation , 1992 .

[35]  A. K. Md. Ehsanes Saleh,et al.  Performance of some new preliminary test ridge regression estimators and their properties , 1993 .

[36]  D. Gibbons.  A Simulation Study of Some Ridge Estimators , 1981 .

[37]  T. A. Bancroft,et al.  On Biases in Estimation Due to the Use of Preliminary Tests of Significance , 1944 .

[38]  A. Saleh,et al.  ON SOME R-AND M-ESTIMATORS OF REGRESSION PARAMETERS UNDER UNCERTAIN RESTRICTION , 1989 .