We observe [a single observation!] \(X \sim \mathcal{N} (\xi, \sigma^2)\), with both \(\xi\) and \(\sigma\) unknown. Construct a confidence set for \(\xi\), i.e.
\[ \mathbb{P}_{\xi , \sigma} \{ \xi \in (\underline{\xi}(X) , \bar \xi (X) \} \geq 1 - \alpha, \]
for all \((\xi , \sigma)\), with \(\bar \xi (X) = c |X|\), and \(\underline{\xi}(X) = -c |X|\), for \(c > 0\).
Let \(X = \xi + \sigma Y\), then
\[ \begin{align*} \mathbb{P} \{ |\xi| < c|X| \} &= \mathbb{P} \{ |\xi| < c|\xi + \sigma Y| \}\\ &= \mathbb{P} \{ |\xi|/\sigma < c|\xi/\sigma + Y| \} \end{align*} \]
so by symmetry,
\[ \begin{align*} \inf_{\lambda \geq 0} \mathbb{P} \{ |\xi| < c|X| \} &= \inf_{\lambda \geq 0} \mathbb{P} \{ \lambda < c|\lambda + Y| \}\\ &= \inf_{\lambda \geq 0} [\mathbb{P} \{ Y > - \lambda, \lambda < c \lambda + Y \} + \mathbb{P} \{ Y < - \lambda, \lambda < -c (\lambda + Y) \}]\\ &= \inf_{\lambda \geq 0} [\mathbb{P} \{ Y > - \lambda, Y < \lambda/c - \lambda \} + \mathbb{P} \{ Y < - \lambda, Y < - \frac{1+c}{c} \lambda \}]\\ &= \inf_{\lambda \geq 0} [\mathbb{P} \{ Y > \lambda/c - \lambda \} + \mathbb{P} \{ Y < - \lambda - \lambda/c \}] \end{align*} \]
This is the probability of the complement of the interval of length \(2\lambda/c\), centered at \(-\lambda\). [Now optimize, over \(\lambda\) for each \(c\), and then find \(c\) to achieve level \(\alpha\). Using the R function Findc below one obtains \(c_{0.05} \approx 9.68\).]
Findc <- function(alpha){
U <- function(c, a) {
coverage <- function(z,c) 1 - pnorm(-z + z/c) + pnorm(-z - z/c)
optimize(coverage, c(0,5), c = c)$obj - (1 - a)
}
uniroot(U, c(1, 20), a = alpha)$root
}Version: Nov 06 2015. Transcript of a fragment of class notes taken by Stephen Portnoy from a course given, circa 1967, at Stanford by Charles Stein.}↩