Feb 18, 2016

# Solution to Student Puzzle Corner 12

Editor Anirban DasGupta writes:

Well done to Lu Mao (pictured below), of the department of biostatistics at the University of North Carolina at Chapel Hill, who sent a carefully written solution.

Lu Mao

The problem was to find a minimax estimator of $m = |µ|$ under squared error loss when there are $n$ iid observations from $N (µ, 1), µ ∈ \mathcal{R}$. The claim is that the plug-in estimator $|\bar{X}|$ is unique minimax.

First, note that by the triangular inequality,

$sup_µ E_µ[(|\bar{X}| − m)^2] ≤ sup_µ E_µ [(\bar{X} − µ)^2 ] = \frac{1}{n}.$

Consider now any other estimator $δ(X_1 , ··· , X_n)$ of $m$; we may assume $δ$ to be a function of $\bar{X}$ by Rao–Blackwell. Then,

$sup_µ E_µ [(δ − m)^2] ≥ sup_{µ≥0} E_µ[(δ − µ)^2] ≥ \frac{1}{n},$

the last inequality following from a standard application of the Cramér–Rao inequality. Thus |\bar{X}| is minimax; unique minimaxity follows because the last inequality above is strict by completeness. The exact risk function of |\bar{X}| equals

$R(θ,|\bar{X}|) = \frac{1}{n}[1 + 4θ^2 (1−Φ(θ))−4θφ(θ)]$, where $θ=|\sqrt{n} µ|$.

By elementary calculus, the function $g(x) = x^2 (1 − Φ(x)) − xφ(x)$, $x ≥ 0$ has a unique minimum at $x_0$ where $x_0$ is the unique root of $\frac{x(1−Φ(x))}{φ(x)}=\frac{1}{2}$; $x_0 ≈ .61200$.

Plugging this back into the exact risk formula, one gets that the minimum risk of $|\bar{X}|$ is $\frac{1−2×0 φ(x0)}{n} = \frac{.59509}{n}$.

## Welcome!

Welcome to the IMS Bulletin website! We are developing the way we communicate news and information more effectively with members. The print Bulletin is still with us (free with IMS membership), and still available as a PDF to download, but in addition, we are placing some of the news, columns and articles on this blog site, which will allow you the opportunity to interact more. We are always keen to hear from IMS members, and encourage you to write articles and reports that other IMS members would find interesting. Contact the IMS Bulletin at bulletin@imstat.org

## What is “Open Forum”?

In the Open Forum, any IMS member can propose a topic for discussion. Email your subject and an opening paragraph (to bulletin@imstat.org) and we'll post it to start off the discussion. Other readers can join in the debate by commenting on the post. Search other Open Forum posts by using the Open Forum category link below. Start a discussion today!