hi all a technical question for those bright statisticians. my question involves ridge regression. definition: n=sample size of a data set X is the matrix of data with , say p variables Y is the y matrix i.e the response variable Z(i,j) = ( X(i,j)- xbar(j) / [ (n-1)^0.5* std(x(j))] Y_new(i)=( Y(i)- ybar(j) ) / [ (n-1)^0.5* std(Y(i))] (note that i have scaled the Y matrix as well) k is the ridge constant the ridge estimate for the betas is = inverse(Z'Z+kI)*Z'Y_new=W*Z'Y_new the associated variance covariance matrix sigma*W*(Z'Z)*W where sigma is the residual variance based on the transformed variables if we transform the variables back to the original variables the beta estimates are now: beta(j)= std(y)*betaridge(j)/std(x(j)) but what is the covariance matrix of these estimates??? i know that this might not be the correct forum for this question, but since i know that many users are statisticians i know that i will get an informed response.
If I'm not mistaken, you only need to know that if V is the covariance matrix of a random vector X, then the covariance of the linear transformation AX + b is AVA'. Substitute betahat for X, and figure out what A is and you're set. (b is 0 in your case.) Andy> From: Clark Allan > > hi all > > a technical question for those bright statisticians. > > my question involves ridge regression. > > definition: > > n=sample size of a data set > > X is the matrix of data with , say p variables > > Y is the y matrix i.e the response variable > > Z(i,j) = ( X(i,j)- xbar(j) / [ (n-1)^0.5* std(x(j))] > > Y_new(i)=( Y(i)- ybar(j) ) / [ (n-1)^0.5* std(Y(i))] (note > that i have > scaled the Y matrix as well) > > k is the ridge constant > > the ridge estimate for the betas is = > inverse(Z'Z+kI)*Z'Y_new=W*Z'Y_new > > the associated variance covariance matrix sigma*W*(Z'Z)*W > where sigma is > the residual variance based on the transformed variables > > if we transform the variables back to the original variables the beta > estimates are now: beta(j)= std(y)*betaridge(j)/std(x(j)) > > but what is the covariance matrix of these estimates??? > > i know that this might not be the correct forum for this question, but > since i know that many users are statisticians i know that i > will get an > informed response. >
hello all some help required once again! does anyone recall the equations for the following ridge constants? 1. hoerl and kennard (1970) 2. hoerl, kennard and baldwin (1975) 3. lawless and wang could you also specify whether or not one has to transform the X and Y variables. if so , how and in which cases. a worked example with a data set would be most helpful. thanking you in advance *** Allan