Casella& Berger problem 8.5 2.c

Hi all,

I am trying to solve the problem 8.5 part c of Casella & Berger part c. I found the solution online but I couldn't understand why the distribution of log(X_{i})-min_{i}(log(X_{i}) is an exponential distribution where the X_{i}'s follow a pareto distribution.

Thanks for your help.

Re: Casella& Berger problem 8.5 2.c

Quote:

Originally Posted by

**camerbabe** Hi all,

I am trying to solve the problem 8.5 part c of Casella & Berger part c. I found the solution online but I couldn't understand why the distribution of log(X_{i})-min_{i}(log(X_{i}) is an exponential distribution where the X_{i}'s follow a pareto distribution.

Thanks for your help.

this might help.Pareto distribution - Wikipedia, the free encyclopedia

Re: Casella& Berger problem 8.5 2.c

Quote:

Originally Posted by

**harish21**

In this wikipedia page, xm is a value but in my case, Xm is a ramdom variable representing the minimum of the Xi's. Thanks.

Re: Casella& Berger problem 8.5 2.c

Quote:

Originally Posted by

**camerbabe** In this wikipedia page, xm is a value but in my case, Xm is a ramdom variable representing the minimum of the Xi's. Thanks.

xm in that page is also the minimum value of X, and is also a random variable(order statistic). look at the definition section of the same page.

Re: Casella& Berger problem 8.5 2.c

Quote:

Originally Posted by

**harish21** xm in that page is also the minimum value of X, and is also a random variable(order statistic). look at the definition section of the same page.

I'm sorry if you find me annoying harish21 but the case I have in hand is about a distribution where the minimum value of the Xi's is v and the the index theta. I am trying to find the distribution of the log(Xi/minXi) for all i. Thanks for your help.

Re: Casella& Berger problem 8.5 2.c

Quote:

Originally Posted by

**camerbabe** I'm sorry if you find me annoying harish21 but the case I have in hand is about a distribution where the minimum value of the Xi's is v and the the index theta. I am trying to find the distribution of the log(Xi/minXi) for all i. Thanks for your help.

$\displaystyle X_1, X_2, \cdots, X_n$ are identically distributed

So for $\displaystyle Y_i\;=\;\log\dfrac{X_i}{min X_i}$, find $\displaystyle P(Y_1 \leq y), P(Y_2 \leq y) .. $ and so on..

Each $\displaystyle Y_i$ has an exponential(1) distribution.