Variable Selection and Joint Estimation of Mean and Covariance Models with an Application to eQTL Data

Algorithm 2

Variable selection in the precision matrix estimation.

Compute using obtained from Algorithm 1.

Compute by (8).

Combining initial for all with , compute and BIC.

while BIC decreases do

for in do

if no elements of are included in the joint model then

compute Rao statistics for by , .

else

Set a linear regression model with response and predictors ’s whose

corresponding coefficients are already included in the joint model. Here

. Compute Rao statistic for adding one predictor among

whose corresponding ’s are not in this linear model.

end if

end for

Among all ’s not in the joint model, add one with the maximum Rao value.

Denote this by for and .

Update , using (8) as well as and compute BIC.

end while

while BIC decreases do

Compute Wald statistics for all in the current model.

Delete one with the minimum Wald.

Update , using (8) as well as and compute BIC.

end while

The optimal model is chosen by the minimum BIC.