maximum likelihood estimators help

• Aug 16th 2011, 10:11 PM
phinejose5
maximum likelihood estimators help
X1, X2, .... ,Xn are independet exponential random variables with E(Xk)=k times lambda where k = 1,2,...,n. What is the sufficient statistics and maximum likelihood estimators of lambda??? and is it unbiased?

(a) sufficient statistics i got summation of Xk where k is from 1 to n

(b) MLE i got (1/n) times summation (Xk)/k where k is from 1 to n

(c) it is unbiased

• Aug 16th 2011, 10:30 PM
Moo
Re: maximum likelihood estimators help
Hello,

The sufficient statistics would rather be $\displaystyle \sum kX_k$. You can't isolate $\displaystyle \sum X_k$ itself.

Please do the caculations again, the MLE is something like $\displaystyle \frac{n}{\sum kX_k}$

And it will indeed be unbiased since the expectation of the texponential distribution is 1/(k*lambda)
• Aug 16th 2011, 11:12 PM
phinejose5
Re: maximum likelihood estimators help
why the MLE is n/sigma(k*Xk) ??
• Aug 16th 2011, 11:23 PM
phinejose5
Re: maximum likelihood estimators help
isnt the sufficient stat is summation (Xk)/k ??
• Aug 17th 2011, 12:51 AM
Moo
Re: maximum likelihood estimators help
Awww yes, I was using the pdf $\displaystyle \lambda e^{-\lambda x}$, but it seems like you're using the $\displaystyle \frac 1\lambda e^{-x/\lambda}$ one...

For the MLE it is then correct. So your answers are correct.

As a sidenote, it would have been much better if you had shown your working...
• Aug 17th 2011, 02:04 AM
phinejose5
Re: maximum likelihood estimators help
yea, thank you. i have my working but it is in handwriting, a bit difficult to type here. i will scan it later.
• Aug 17th 2011, 02:19 AM
phinejose5
Re: maximum likelihood estimators help
I have another problem. Let X1,X2,...,Xn be a random sample of size n from a distribution with density function f(x;theta) = c2^[-(x-theta)^2]. if theta is known, the MLE of c is infinity is it?? i formed the likelihood function and take the derivative of the ln of the function then let it equal to zero.
• Aug 17th 2011, 03:51 AM
Moo
Re: maximum likelihood estimators help
Are you sure you have to consider the MLE of c ??? Because c is just a constant to make the function a pdf (integral equal to 1)
• Aug 18th 2011, 01:41 AM
phinejose5
Re: maximum likelihood estimators help
oh..but the question is like that..i think i just stick to my answer which is infinity. And i wanna ask if i want to find the 95% confidence interval of theta, after i found the estimator of theta, how to find the distribution of it?