Do order statistic densities satisfy the monotone likelihood ratio property?
Suppose we have $n$ independent random variables, $X_1, X_2, \ldots, X_n$, where each $X_p$ is distributed $\mathcal{N}(\mu_p,\sigma^2)$. Let $\mathbf{X}$ denote the corresponding random vector. For this population of random variables, let $g_{(i)}(y;\boldsymbol{\mu})$ denote the pdf of the $i^{th}$ order statistic when $\mathbb{E}[\mathbf{X}]=\boldsymbol{\mu}$. Thus we have a family of distributions indexed by $\boldsymbol{\mu}$.
Does $g_{(i)}(y;\boldsymbol{\mu})$ satisfy the monotone likelihood ratio property? I.e., is the case that for $\boldsymbol{\mu}_1>\boldsymbol{\mu}_0$ and $y_1 > y_0$: $$
\frac{g_{(i)}(y_1;\boldsymbol{\mu}_1)}{g_{(i)}(y_1;\boldsymbol{\mu}_0)}
\geq
\frac{g_{(i)}(y_0;\boldsymbol{\mu}_1)}{g_{(i)}(y_0;\boldsymbol{\mu}_0)}
$$
I think this is true, but how can I prove this? Thanks!
