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

Is my answers all correct??please reply me..thank you!!

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)

Re: maximum likelihood estimators help

why the MLE is n/sigma(k*Xk) ??

Re: maximum likelihood estimators help

isnt the sufficient stat is summation (Xk)/k ??

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...

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.

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.

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)

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?