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Let X1, X2, X3, X4 denote 4 independent observations from a distribution with density f(x;theta)=(1+theta)x^theta, if...

Let X1, X2, X3, X4 denote 4 independent observations from a distribution with density f(x;theta)=(1+theta)x^theta, if 0<=x<=1; 0 Otherwise.. What is the form of the LRT critical regoon for testing H0:theta =2 versus H1:theta=5

Expert Solution

here,

Now, we are interested in testing the null hypothesis against the alternative hypothesis

Now, to find the likelihood ratio, as defined above, we first need to find . Well, when the null hypothesis is true, can take on only one value, namely, . Therefore:

We also need to find in order to define the likelihood ratio. Well, when the alternate hypothesis is true, can take on only one value, namely, . Therefore:

Now, putting it all together to form the likelihood ratio, we get:

which simplifies to:

Now, the likelihood ratio test tells us to reject the null hypothesis when the likelihood ratio λ is small, that is, when:

where k is chosen to ensure that, in this case, α = 0.05. Well, by taking the natural log of both sides of the inequality, we can show that λ ≤ k is equivalent to:

Now, as are independent observations, thus,

This distribution can be used to obtained the value of C at a given level of significance.

milcah answered 7 months ago

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