Asymptotic Expansion of Risk for a Regression Model with respect to $\alpha$-Divergence with an Application to the Sample Size Problem -- Complete Version

For a regression model, we consider the risk of the maximum likelihood estimator with respect to $\alpha$-divergence, which includes the special cases of Kullback-Leibler divergence, Hellinger distance and $\chi^2$ divergence. The asymptotic expansion of the risk with respect to the sample size $n$ is given up to the order $n^{-2}$. We are interested in how the risk convergence speed (to zero) is affected by the error term distributions of the regression model and the magnitude of the joint moments of the standardized explanatory variables. Besides the general result (which is given by Mathematica program), we consider three concrete error term distributions; a normal distribution, a t-distribution and a skew-normal distribution. We use the (approximated) risk of m.l.e. as a measure of the difficulty of estimation for the regression model. Especially comparing the value of the (approximated) risk with that of a binomial distribution, we can give a certain standard for the sample size required to estimate the regression model.