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ISI MStat PSB 2010 Problem 10 | Uniform Modified

This is a very elegant sample problem from ISI MStat PSB 2010 Problem 10. It's mostly based on propertes of uniform, and its behaviour when modified . Try it!

Problem- ISI MStat PSB 2010 Problem 10


Let X be a random variable uniformly distributed over (0,2\theta ), \theta>0, and Y=max(X,2\theta -X).

(a) Find \mu =E(Y).

(b) Let X_1,X_2,.....X_n be a random sample from the above distribution with unknown \theta. Find two distinct unbiased estimators of \mu, as defined in (a), based on the entire sample.

Prerequisites


Uniform Distribution

Law of Total Expectation

Unbiased Estimators

Solution :

Well, this is a very straight forward problem, where we just need to be aware the way Y is defined.

As, we need E(Y) and by definition of Y , we clearly see that Y is dependent in X where X \sim Unif( 0, 2\theta).

So, using Law of Total Expectation,

E(Y)= E(X|X>2\theta-X)P(X>2\theta-X)+E(2\theta-X|X \le 2\theta-X)P(X \le 2\theta-X). <!-- /wp:paragraph -->  <!-- wp:paragraph --> Observe that, \(P(X \le \theta)=\frac{1}{2}, why ??

Also, conditional pdf of X|X>\theta is,

f_{X|X>\theta}(x)=\frac{f_X(x)}{P(X>\theta)}==\frac{1}{\theta} \&  \theta< x \le 2\theta. [where f_X is the pdf of X].

the other conditional pdf is also same due to symmetry.(Verify!!).

So, E(Y)=E(X|X\sim Unif(\theta,2\theta))\frac{1}{2}+E(X|X\sim Unif(0,\theta))\\frac{1}{2}=\frac{1}{2}(\frac{3\theta}{2}+2\theta-\frac{\theta}{2})=\frac{3\theta}{2}.

hence, \mu=\frac{3\theta}{2}.

Now, for the next part, one trivial unbiased estimator of \theta is T_n=\frac{1}{n}\sum_{i=1}^n X_i (based on the given sample). So,

\frac{3T_n}{2}=\frac{3}{2n}\sum_{i=1}^n X_i is an obvious unbiased estimator of \mu.

For another we need to change our way of looking on conventional way and look for the order statistics, since we know that X_{(n)} is sufficient for \theta.(Don't Know ?? Look for Factorization Theorem .)

So, verify that E(X_{(n)})=\frac{2n}{n+1}\theta.

Hence, \frac{n+1}{2n}X_{(n)} is another unbiased estimator of theta. So, \frac{3(n+1)}{4n}X_{(n)} is also another unbiased estimator of \mu a defined in (a).

Hence the solution concludes.


Food For Thought

Let us think about some unpopular but very beautiful relationship between discrete random variables besides the Universality of uniform. Let Xbe a discrete random variable with cdf F_X(x) and define the random variable Y=F_X(x).

Can you verify that, Y is stochastically greater that a uniform(0,1) random variable U. i.e.

P(Y>y) \ge P(U>y)=1-y for all y, 0<y<1,

P(Y>y) > P(U>y) =1-y, for some y, 0<y<1.

Hint: Draw a typical picture of a discrete cdf, and observe the jump points ! you may jump to the solution!! Think it over.


Similar Problems and Solutions



ISI MStat PSB 2008 Problem 10
Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

Subscribe to Cheenta at Youtube


This is a very elegant sample problem from ISI MStat PSB 2010 Problem 10. It's mostly based on propertes of uniform, and its behaviour when modified . Try it!

Problem- ISI MStat PSB 2010 Problem 10


Let X be a random variable uniformly distributed over (0,2\theta ), \theta>0, and Y=max(X,2\theta -X).

(a) Find \mu =E(Y).

(b) Let X_1,X_2,.....X_n be a random sample from the above distribution with unknown \theta. Find two distinct unbiased estimators of \mu, as defined in (a), based on the entire sample.

Prerequisites


Uniform Distribution

Law of Total Expectation

Unbiased Estimators

Solution :

Well, this is a very straight forward problem, where we just need to be aware the way Y is defined.

As, we need E(Y) and by definition of Y , we clearly see that Y is dependent in X where X \sim Unif( 0, 2\theta).

So, using Law of Total Expectation,

E(Y)= E(X|X>2\theta-X)P(X>2\theta-X)+E(2\theta-X|X \le 2\theta-X)P(X \le 2\theta-X). <!-- /wp:paragraph -->  <!-- wp:paragraph --> Observe that, \(P(X \le \theta)=\frac{1}{2}, why ??

Also, conditional pdf of X|X>\theta is,

f_{X|X>\theta}(x)=\frac{f_X(x)}{P(X>\theta)}==\frac{1}{\theta} \&  \theta< x \le 2\theta. [where f_X is the pdf of X].

the other conditional pdf is also same due to symmetry.(Verify!!).

So, E(Y)=E(X|X\sim Unif(\theta,2\theta))\frac{1}{2}+E(X|X\sim Unif(0,\theta))\\frac{1}{2}=\frac{1}{2}(\frac{3\theta}{2}+2\theta-\frac{\theta}{2})=\frac{3\theta}{2}.

hence, \mu=\frac{3\theta}{2}.

Now, for the next part, one trivial unbiased estimator of \theta is T_n=\frac{1}{n}\sum_{i=1}^n X_i (based on the given sample). So,

\frac{3T_n}{2}=\frac{3}{2n}\sum_{i=1}^n X_i is an obvious unbiased estimator of \mu.

For another we need to change our way of looking on conventional way and look for the order statistics, since we know that X_{(n)} is sufficient for \theta.(Don't Know ?? Look for Factorization Theorem .)

So, verify that E(X_{(n)})=\frac{2n}{n+1}\theta.

Hence, \frac{n+1}{2n}X_{(n)} is another unbiased estimator of theta. So, \frac{3(n+1)}{4n}X_{(n)} is also another unbiased estimator of \mu a defined in (a).

Hence the solution concludes.


Food For Thought

Let us think about some unpopular but very beautiful relationship between discrete random variables besides the Universality of uniform. Let Xbe a discrete random variable with cdf F_X(x) and define the random variable Y=F_X(x).

Can you verify that, Y is stochastically greater that a uniform(0,1) random variable U. i.e.

P(Y>y) \ge P(U>y)=1-y for all y, 0<y<1,

P(Y>y) > P(U>y) =1-y, for some y, 0<y<1.

Hint: Draw a typical picture of a discrete cdf, and observe the jump points ! you may jump to the solution!! Think it over.


Similar Problems and Solutions



ISI MStat PSB 2008 Problem 10
Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

Subscribe to Cheenta at Youtube


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