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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.

Let $m=F^{-1}(1/2)$ be the unique median. The influence function of the sample median isaway from $m$. Its second moment, equivalently the asymptotic distribution of a sample median, givesNo symmetry assumption is used: the density is evaluated at the actual median of $F$.

Now both the number of groups and their size equal $\sqrt{n}$. A group mean is approximately $N(\mu ,{\sigma }^2/\sqrt{n})$, whose density at $\mu$ is approximately $n^{1/4}/(\sigma \sqrt{2\pi })$. The asymptotic distribution of a sample median based on $\sqrt{n}$ such values therefore has varianceThis suggests the conjectureprovided a sufficiently uniform central and local limit approximation controls the triangular array.

First establish the limiting law needed by the delta method. For fixed $t$, put $u_n=1/2+t/\sqrt{n}$ and let $N_n$ count how many of the $U_i$ are at most $u_n$. Then $N_n$ has a binomial distribution with expectation $n/2+t\sqrt{n}$ and variance $n/4-t^2$. The event $M_U\le u_n$ is exactly $N_n\ge (n+1)/2$. Its standardized threshold tends to $-2t$, so the central limit theorem for these binomial counts givesHere $\Phi$ is the cumulative distribution function of the standard normal distribution. Equivalently,Use the usual local regularity for the delta method: $f$ is continuous at the population median $\theta$ and $f(\theta )>0$. The inverse function theorem, applied at $F(\theta )=1/2$, givesThereforeThe asymptotic normality gives the requested leading variance approximationHere the variance parameter for the rescaled limiting law is $1/(4f(\theta {)}^2)$, while the source states the corresponding $1/n$ approximation for the unscaled estimator. Strict increase of $F$ alone does not force $f(\theta )>0$; that local condition is needed for this form of the asymptotic distribution of a sample median.