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This is a very simple and regular sample problem from ISI MStat PSB 2009 Problem 8. It It is based on testing the nature of the mean of Exponential distribution. Give it a Try it !

Let be i.i.d. observation from the density,

where is an unknown parameter.

Consider the problem of testing the hypothesis against .

(a) Show that the test with critical region , where is the th quantile of the distribution, has size .

(b) Give an expression of the power in terms of the c.d.f. of the distribution.

Likelihood Ratio Test

Exponential Distribution

Chi-squared Distribution

This problem is quite regular and simple, from the given form of the hypotheses , it is almost clear that using Neyman-Pearson can land you in trouble. So, lets go for something more general , that is Likelihood Ratio Testing.

Hence, the Likelihood function of the for the given sample is ,

, also observe that sample mean is the MLE of .

So, the Likelihood Ratio statistic is,

So, our test function is ,

.

We, reject at size , when , for some , ,

Hence, .

for some constant, .

Let , and observe that is,

decreasing function of for ,

Hence, there exists a such that ,we have . See the figure.

So, the critical region of the test is of form , for some such that,

, for some , where is the size of the test.

Now, our task is to find , and for that observe, if , then ,

Hence, in this problem, since the 's follows , hence, , we have,

,

which gives ,

Hence, the rejection region is indeed, .

Hence Proved !

(b) Now, we know that the power of the test is,

.

Hence, the power of the test is of form of a cdf of chi-squared distribution.

Can you use any other testing procedure to conduct this test ?

Think about it !!

This is a very simple and regular sample problem from ISI MStat PSB 2009 Problem 8. It It is based on testing the nature of the mean of Exponential distribution. Give it a Try it !

Let be i.i.d. observation from the density,

where is an unknown parameter.

Consider the problem of testing the hypothesis against .

(a) Show that the test with critical region , where is the th quantile of the distribution, has size .

(b) Give an expression of the power in terms of the c.d.f. of the distribution.

Likelihood Ratio Test

Exponential Distribution

Chi-squared Distribution

This problem is quite regular and simple, from the given form of the hypotheses , it is almost clear that using Neyman-Pearson can land you in trouble. So, lets go for something more general , that is Likelihood Ratio Testing.

Hence, the Likelihood function of the for the given sample is ,

, also observe that sample mean is the MLE of .

So, the Likelihood Ratio statistic is,

So, our test function is ,

.

We, reject at size , when , for some , ,

Hence, .

for some constant, .

Let , and observe that is,

decreasing function of for ,

Hence, there exists a such that ,we have . See the figure.

So, the critical region of the test is of form , for some such that,

, for some , where is the size of the test.

Now, our task is to find , and for that observe, if , then ,

Hence, in this problem, since the 's follows , hence, , we have,

,

which gives ,

Hence, the rejection region is indeed, .

Hence Proved !

(b) Now, we know that the power of the test is,

.

Hence, the power of the test is of form of a cdf of chi-squared distribution.

Can you use any other testing procedure to conduct this test ?

Think about it !!

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