For a regression model, we consider the risk of the maximum likelihood estimator with respect to α-divergence, which includes the special cases of Kullback-Leibler divergence, Hellinger distance and χ2 divergence. The asymptotic expansion of the risk with respect to the sample size nis given up to the order n-2. We observe how the risk convergence speed (to zero) is affected by the error term distributions and the magnitude of the joint moments of the standardized explanatory variables under three concrete error term distributions: a normal distribution, a t-distribution and a skew-normal distribution. We try to 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.
Asymptotic Expansion of Risk for a Regression Model with respect to α-Divergence with an Application to the Sample Size Problem
