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 !!