*Bounty: 50*

*Bounty: 50*

Let $P_0=mathcal{N}(0,1)$ and $P_1=mathcal{N}(mu,sigma^2)$ with $sigma > 1$

I would like to show that the Neyman-Pearson test of $P_0$ vs. $P_1$ at level $alpha$ has the form

$$varphi_{*}(x)=mathbb{1} {[x notin(mu/(1-sigma^2)pmdelta{*})]}$$

for some $delta_{*}=delta_{*}(mu,sigma,alpha)>0$ and also to determine special case of $varphi_{*}$ for $mu=0$

What I have tried: Let $f_0$ be the density of probability distribution $P_0$, so $f_0=frac{1}{sqrt{2pi}}exp(-frac{x^2}{2})$ and similarly for $f_1=frac{1}{sqrt{2pi} sigma}exp(-frac{(x-mu)^2}{2sigma^2})$.

Then I found what is monotone density ratios:

$frac{f_1}{f_0}=frac{1}{sigma} frac{exp(-frac{(x-mu)^2}{2sigma^2})}{exp(-frac{x^2}{2})}=frac{1}{sigma} exp(-frac{(x-mu)^2}{2sigma^2}+frac{x^2}{2})=g(T(x))$

$g(t)=frac{1}{sigma} exp(-frac{(t-mu)^2}{2sigma^2}+frac{t^2}{2})$

and $T(x)=x$.

Then, let $H: mathbb{R} to [0,1]$ be an auxiliary function.

$$H(r):=P_{0}(T>r)$$ for any $r in mathbb{R}$.

From this, we can determine $k_{alpha}:=min {r in mathbb{R} : H(r) leq alpha}$

and $$gamma_{alpha}=frac{alpha-P_0(T>k_{alpha})}{P_0(T=k_{alpha})} in (0,1)$$

In my lecture notes I have $varphi_{*}=gamma_{alpha}mathbb{1}_{[T(x)=k_{alpha}]}+mathbb{1}_{[T(x)>k_{alpha}]}$ for (UMP right sided-test) but I have problem with $gamma_{alpha}$ which in this case seems to be 1 and to define $k_{alpha}$ and what about $mathbb{1}_{[T(x)>k_{alpha}]}$?