# How to do this hypothesis test?

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#1
I am a bit stuck on the following question: We have a p.d.f for a r.v X

if (0<x<1) (theta >1) and 0 otherwise.

I want to find the most powerful test for H0: against HA: .

So I use the Neyman-Pearson lemma. To say the MP test is the likelihood ratio test.

I want to take the likelihood ratio but I don't really know how since the null hypothesis is not simple if it was simple I would just get

and then reject for small values of this i.e. () and use the p.d.f. to find what value of c we need to use here to get the correct level we desire.

However I really don't know what the ratio would be here as I don't know what the values of

and are surely the values we reject on would be depending on this yet the question makes no mention. (I.e. if then the exponent would be negative so we would look for larger values of x rather than smaller to make the ratio smaller for instance.)

I have seen sometimes where you use the supremum on the numerator (taken over a restriction of values ) and then the supremum (of all possible values) on the denominator but I don't know how to do that here? Should I be looking at MLE's? (maximum likelihood estimates) to proceed or what.

I think when I get this confusion cleared about how to deal with the composite null hypothesis rather than simple and how that effects the ratio test I will be able to complete this question so any help is appreciated.

Thanks

(hopefully I explained myself well enough for someone to understand what I mean.)
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3 years ago
#2
(Original post by BenB52058)
I am a bit stuck on the following question: We have a p.d.f for a r.v X

if (0<x<1) (theta >1) and 0 otherwise.

I want to find the most powerful test for H0: against HA: .

So I use the Neyman-Pearson lemma. To say the MP test is the likelihood ratio test.

I want to take the likelihood ratio but I don't really know how since the null hypothesis is not simple.
Why do you think that this null hypothesis is not simple?
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#3
(Original post by Gregorius)
Why do you think that this null hypothesis is not simple?
I thought because theta 0 is not known that it wouldn't be simple but I guess from your reply that it would be.

Okay so maybe I've just confused a definition but suppose that it is simple how would I proceed because it would still depend on the vales of theta 0 and 1 what the rejection region is wouldn't it I mean since 0<x<1 then if the exponent is negative we would reject for large x but if the exponent was positive we would reject for small x.

Am I supposed to break this into cases I.e theta0<theta1 and vice versa?

Thanks!
0
3 years ago
#4
(Original post by BenB52058)
I thought because theta 0 is not known that it wouldn't be simple but I guess from your reply that it would be.

Okay so maybe I've just confused a definition but suppose that it is simple how would I proceed because it would still depend on the vales of theta 0 and 1 what the rejection region is wouldn't it I mean since 0<x<1 then if the exponent is negative we would reject for large x but if the exponent was positive we would reject for small x.

Am I supposed to break this into cases I.e theta0<theta1 and vice versa?

Thanks!
A simple null hypothesis is one that specifies completely the distribution concerned.

Splitting your test according to the sign of theta 0 minus theta 1 would be fine.
1
#5
(Original post by Gregorius)
A simple null hypothesis is one that specifies completely the distribution concerned.

Splitting your test according to the sign of theta 0 minus theta 1 would be fine.
thanks!!!
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