Can the NeymanPearson lemma apply to the case when a simple null and
a simple alternative don’t belong to the same family of distributions?
From its proof, I don’t see why it can’t.For example, when the simple null is a normal distribution and the
simple alternative is a exponential distribution. Is the likelihood ratio test a good way to test a composite null against a
composite alternative when both belong to different families of
distributions?Thanks and regards!
Answer
Yes Neyman Pearson Lemma can apply to the case when simple null and simple alternative don’t belong to the same family of distributions.
Let we want to construct a Most Powerful(MP) test of H0:X∼N(0,1) against H1:X∼Exp(1) of its size.
For a particular k, our critical function by Neyman Pearson lemma is
ϕ(x)={1,f1(x)f0(x)>k0,Otherwise
is a MP test of H0 against H1 of its size.
Here r(x)=f1(x)f0(x)=e−x1√2πe−x2/2=√2πe(x22−x)
Note that r′(x)=√2πe(x22−x)(x−1){<0,x<1>0,x>1
Now if you draw the picture of r(x) [I don’t know how to construct a Picture in answer ], from graph it will be clear that r(x)>k⟹x>c.
So, for a particualr c
ϕ(x)={1,x>c0,Otherwise
is a MP test of Ho against H1 of its size.
You can test

 H0:X∼N(0,12) against H1:X∼Cauchy(0,1)
 H0:X∼N(0,1) against H1:X∼Cauchy(0,1)
 H0:X∼N(0,1) against H1:X∼Double Exponential(0,1)
By Neyman Pearson lemma.
Normally the likelihood ration test(LRT) is not a good way for composite null and composite alternative which belong to different family of distributions.The LRT is specially useful when θ is a multiparameter and we wish to test hypothesis concerning one of the parameters.
That’s all from me.
Attribution
Source : Link , Question Author : Tim , Answer Author : A.D