# Statistics

• Mar 31st 2009, 07:28 PM
maurisa
Statistics
(a) Find the maximum- likelihood estimator for u, the population mean, given a sample of size n from a population with f(x) = 1/B, 0<x<B. Estimate B by the method of moments.

(b)The sample 1.3,0.6,1.7,2.2,0.3,1.1 was drawn from a population with the density f(x) = 1/B, 0<x<B. What are the maximum-likelihood estimates of the mean and variance of the population?
• Mar 31st 2009, 07:54 PM
maurisa
so far i know that

f(x) = (1/B) ^n

ln f(x) = n (1/B)

d ln f(x)/ dB = -n/B^2

any suggestions on where i went wrong?
• Apr 1st 2009, 10:48 PM
matheagle
Quote:

Originally Posted by maurisa
so far i know that

f(x) = (1/B) ^n

ln f(x) = n (1/B)

d ln f(x)/ dB = -n/B^2

any suggestions on where i went wrong?

you can't differentiate this. You need to include the indicator function, that the max is less than B.
Since the likelihood function is $B^{-n} I(\max X_i\le B)$ the max occurs when B is as small as possible
which happens when B is the largest order statistic. Since the MLE of B is the largest order stat and the mean is B/2
I guess the estimate you want is the max over 2.