A Mixture of g-priors for Variable Selection when the Number of Regressors Grows with the Sample Size

We consider variable selection problem in linear regression using mixture of $g$-priors. A number of mixtures are proposed in the literature which work well, especially when the number of regressors $p$ is fixed. In this paper, we propose a mixture of $g$-priors suitable for the case when $p$ grows with the sample size $n$. We study the performance of the method based on the proposed prior when $p=O(n^b),~0<b<1$. Along with model selection consistency, we also investigate the performance of the proposed prior when the true model does not belong to the model space considered. We find conditions under which the proposed prior is consistent in appropriate sense when normal linear models are considered. Further, we consider the case with non-normal errors in the regression model and study the performance of the model selection procedure. We also compare the performance of the proposed prior with that of several other mixtures available in the literature, both theoretically and using simulated data sets.