## The Annals of Statistics

### 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).

#### Article information

Source
Ann. Statist., Volume 8, Number 3 (1980), 506-521.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176345005

Digital Object Identifier
doi:10.1214/aos/1176345005

Mathematical Reviews number (MathSciNet)
MR568717

Zentralblatt MATH identifier
0436.62031

JSTOR
links.jstor.org

Subjects
Primary: 62B10: Information-theoretic topics [See also 94A17]
Secondary: 62F20

#### Citation

Ghosh, J. K.; Sinha, B. K.; Wieand, H. S. Second Order Efficiency of the MLE with Respect to any Bounded Bowl-Shape Loss Function. Ann. Statist. 8 (1980), no. 3, 506--521. doi:10.1214/aos/1176345005. https://projecteuclid.org/euclid.aos/1176345005