RATES OF CONVERGENCE IN EMPIRICAL BAYESIAN ESTIMATION OF COEFFICIENT AND ERROR VARIANCE IN LINEAR REGRESSION

In this paper, we consider linear regression $Y=X \beta+8,8 \sim N\left(0, \sigma^2 I_k\right)$ where $\beta$ and $\sigma^2$ are unknown parameters. With general quadratic loss function, we first construct Bayesian estimators of the parameters of multiparameter exponential families. Using multiple kernel function, we construct the $E B$ estimators of the parameters. We obtain their rates of convergence $2(\sigma r-1) /(2 r+p+1)$ closing to 1.