This is a another beautiful sample problem from ISI MStat PSB 2014 Problem 9. It is based on testing simple hypothesis, but reveals and uses a very cute property of Geometric distribution, which I prefer calling sister to Loss of memory . Give it a try !
Let and
be independent random variables, where Geo(p) refers to Geometric distribution whose p.m.f. f is given by,
We are interested in testing the null hypothesis against the alternative
. Intuitively it is clear that we should reject if
is large, but unfortunately, we cannot compute the cut-off because the distribution of
under
depends on the unknown (common) value
and
.
(a) Let . Find the conditional distribution of
when
.
(b) Based on the result obtained in (a), derive a level 0.05 test for against
when
is large.
Geometric Distribution.
Negative binomial distribution.
Discrete Uniform distribution .
Conditional Distribution . .
Simple Hypothesis Testing.
Well, Part (a), is quite easy, but interesting and elegant, so I'm leaving it as an exercise, for you to have the fun. Hint: verify whether the required distribution is Discrete uniform or not ! If you are done, proceed .
Now, part (b), is further interesting, because here we will not use the conventional way of analyzing the distribution of and
, whereas we will be concentrating ourselves on the conditional distribution of
! But why ?
The reason behind this adaptation of strategy is required, one of the reason is already given in the question itself, but the other reason is more interesting to observe , i.e. if you are done with (a), then by now you found that , the conditional distribution of is independent of any parameter ( i.e. ithe distribution of
looses all the information about the parameter
, when conditioned by Y=y ,
is a necessary condition), and the parameter independent conditional distribution is nothing but a Discrete Uniform {0,1,....,y}, where y is the sum of
and
.
so, under , the distribution of
is independent of the both common parameter
and
. And clearly as stated in the problem itself, its intuitively understandable , large value of
exhibits evidences against
. Since large value of
is realized, means the success doesn't come very often .i.e.
is smaller.
So, there will be strong evidence against if
, where , for some constant
, where y is given the sum of
.
So, for a level 0.05 test , the test will reject for large value of k , such that,
So, we reject at level 0.05, when we observe
, where it is given that
=y . That's it!
Can you show that for this same and
,
considering , where n=0,1,.... What about the converse? Does it hold? Find out!
But avoid loosing memory, it's beauty is exclusively for Geometric ( and exponential) !!
This is a another beautiful sample problem from ISI MStat PSB 2014 Problem 9. It is based on testing simple hypothesis, but reveals and uses a very cute property of Geometric distribution, which I prefer calling sister to Loss of memory . Give it a try !
Let and
be independent random variables, where Geo(p) refers to Geometric distribution whose p.m.f. f is given by,
We are interested in testing the null hypothesis against the alternative
. Intuitively it is clear that we should reject if
is large, but unfortunately, we cannot compute the cut-off because the distribution of
under
depends on the unknown (common) value
and
.
(a) Let . Find the conditional distribution of
when
.
(b) Based on the result obtained in (a), derive a level 0.05 test for against
when
is large.
Geometric Distribution.
Negative binomial distribution.
Discrete Uniform distribution .
Conditional Distribution . .
Simple Hypothesis Testing.
Well, Part (a), is quite easy, but interesting and elegant, so I'm leaving it as an exercise, for you to have the fun. Hint: verify whether the required distribution is Discrete uniform or not ! If you are done, proceed .
Now, part (b), is further interesting, because here we will not use the conventional way of analyzing the distribution of and
, whereas we will be concentrating ourselves on the conditional distribution of
! But why ?
The reason behind this adaptation of strategy is required, one of the reason is already given in the question itself, but the other reason is more interesting to observe , i.e. if you are done with (a), then by now you found that , the conditional distribution of is independent of any parameter ( i.e. ithe distribution of
looses all the information about the parameter
, when conditioned by Y=y ,
is a necessary condition), and the parameter independent conditional distribution is nothing but a Discrete Uniform {0,1,....,y}, where y is the sum of
and
.
so, under , the distribution of
is independent of the both common parameter
and
. And clearly as stated in the problem itself, its intuitively understandable , large value of
exhibits evidences against
. Since large value of
is realized, means the success doesn't come very often .i.e.
is smaller.
So, there will be strong evidence against if
, where , for some constant
, where y is given the sum of
.
So, for a level 0.05 test , the test will reject for large value of k , such that,
So, we reject at level 0.05, when we observe
, where it is given that
=y . That's it!
Can you show that for this same and
,
considering , where n=0,1,.... What about the converse? Does it hold? Find out!
But avoid loosing memory, it's beauty is exclusively for Geometric ( and exponential) !!