Get inspired by the success stories of our students in IIT JAM MS, ISI MStat, CMI MSc DS. Learn More

This is a really beautiful sample problem from ISI MStat PSB 2008 Problem 10. It is based on testing simple hypothesis. This problem teaches me how observation, makes life simple. Go for it!

Consider a population with three kinds of individuals labelled 1,2 and 3. Suppose the proportion of individuals of the three types are given by , k=1,2,3 where 0< <1.

Let be a random sample from this population. Find the most powerful test for testing versus . ().

Binomial Distribution.

Neyman-Pearson Lemma.

Test function and power function.

Hypothesis Testing.

This is a quite beautiful problem, only when you observe it closely. Here the distribution of X may seem non-trivial ( non-theoretical), but if one observes the distribution of Y=X-1 (say), instead of X , one will find that .

so, now let, p= 1- , so, 0<p<1, and let, and .

and since , , and our hypotheses, reduces to,

versus .

so, under , our joint pmf ( of Y=X-1), is ; where

and under , our joint pmf is, ; where

So, now we can use, widely used Neyman-Pearson Lemma , and end up with,

=== .

now we define a test function, . for some positive constant k.

Where

so, our test rule is, we reject if , and we choose k such that the for a give level ,

, for a given ,

with a power function , . Can you find the more subtle condition when, ? Try It!

Suppose, , can you verify, that there for any constant c, . Can you generalize the situation, what kind distribution must X follow ?? Think over it, until we meet again !

This is a really beautiful sample problem from ISI MStat PSB 2008 Problem 10. It is based on testing simple hypothesis. This problem teaches me how observation, makes life simple. Go for it!

Consider a population with three kinds of individuals labelled 1,2 and 3. Suppose the proportion of individuals of the three types are given by , k=1,2,3 where 0< <1.

Let be a random sample from this population. Find the most powerful test for testing versus . ().

Binomial Distribution.

Neyman-Pearson Lemma.

Test function and power function.

Hypothesis Testing.

This is a quite beautiful problem, only when you observe it closely. Here the distribution of X may seem non-trivial ( non-theoretical), but if one observes the distribution of Y=X-1 (say), instead of X , one will find that .

so, now let, p= 1- , so, 0<p<1, and let, and .

and since , , and our hypotheses, reduces to,

versus .

so, under , our joint pmf ( of Y=X-1), is ; where

and under , our joint pmf is, ; where

So, now we can use, widely used Neyman-Pearson Lemma , and end up with,

=== .

now we define a test function, . for some positive constant k.

Where

so, our test rule is, we reject if , and we choose k such that the for a give level ,

, for a given ,

with a power function , . Can you find the more subtle condition when, ? Try It!

Suppose, , can you verify, that there for any constant c, . Can you generalize the situation, what kind distribution must X follow ?? Think over it, until we meet again !

Cheenta is a knowledge partner of Aditya Birla Education Academy

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

JOIN TRIALAcademic Programs

Free Resources

Why Cheenta?