Second Order Efficiency of the MLE with Respect to any Bounded Bowl-Shape Loss Function

Abstract

Let $X_1, X_2, \cdots$ be a sequence of i.i.d. random variables, each having density $f(x, \theta_0)$ where $\{f(x, \theta)\}$ is a family of densities with respect to a dominating measure $\mu$. Suppose $n^{\frac{1}{2}}(\hat{\theta} - \theta)$ and $n^{\frac{1}{2}}(T - \theta)$, where $\hat{\theta}$ is the mle and $T$ is any other efficient estimate, have Edgeworth expansions up to $o(n^{-1})$ uniformly in a compact neighbourhood of $\theta_0$. Then (under certain regularity conditions) one can choose a function $c(\theta)$ such that $\hat{\theta}' = \hat{\theta} + c(\hat{\theta})/n$ satisfies $P_{\theta_0} \{-x_1 \leqslant n^{\frac{1}{2}}(\hat{\theta}' - \theta_0)(I(\theta_0))^{\frac{1}{2}} \leqslant x_2\} \\ \geqslant P_{\theta_0}\{-x_1 \leqslant n^{\frac{1}{2}}(T - \theta_0)(I(\theta_0))^{\frac{1}{2}} \leqslant x_2\} + o(n^{-1}),$ for all $x_1, x_2 \geqslant 0$. This result implies the second order efficiency of the mle with respect to any bounded loss function $L_n(\theta, a) = h(n^{\frac{1}{2}}(a - \theta))$, which is bowl-shaped i.e., whose minimum value is zero at $a - \theta = 0$ and which increases as $|a - \theta|$ increases. This answers a question raised by C. R. Rao (Discussion on Professor Efron's paper).