Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 223 / 3

Write $y_i=|x_i|$. If $n$ is odd, the sample median $t=T_n$ equals one observation, with equally many $y_i$ below and above it; the corresponding terms $\mathrm{sign}(y_i/t-1)$ cancel and the median term is zero. If $n$ is even, the conventional median lies strictly between the two middle observations when the $y_i$ are distinct, so exactly half the terms are $-1$ and half are $+1$. In either caseHence $T_n$ is a Scale M-estimator.

If $X\sim N(0,{\theta }^2)$, then $Y=|X|$ has a scaled half-normal distribution withPut $c={\Phi }^{-1}(3/4)$. The population median is $t_0=\theta c$, and the asymptotic distribution of a sample median gives

Under $H_0:{\theta }^2=1$, positivity of the scale means $\theta =1$. If $z_{1-\alpha }$ is the $(1-\alpha )$-quantile of the standard normal distribution, an asymptotically level-$\alpha$ test rejects forLarger scale makes the population median $\theta c$ larger, so this is the appropriate one-sided rejection region.