# Introductory Statistics Questions

• Aug 22nd 2010, 08:10 PM
lpd
Introductory Statistics Questions
Hi. I really need help with these statistics problem!!

Let $\displaystyle X_1, ..., X_n$ be a random sample from a uniform distribution over $\displaystyle (0, \theta)$.

a) Find the joint density $\displaystyle f(x_1, ... , x_n| \theta)$ and use the Factorization theorem to show $\displaystyle T = max(X_1, ..., X_n)$ is sufficient for $\displaystyle \theta$.

b) Find the distribution function of $\displaystyle T$ and hence show the p.d.f. of $\displaystyle T$ is $\displaystyle h(t)= n(\frac{t}{n})^{n-1} \frac{1}{\theta}$, $\displaystyle 0 < t < \theta$.

c) Find the conditional density $\displaystyle g(x_1,... , x_n|t, \theta$) and show this does not depend on $\displaystyle \theta$.

d) Show $\displaystyle E(T)= \frac{n}{n+1}\theta$ and hence find a function of the sufficient statistic that is an unbiased estimator of $\displaystyle \theta$.
• Aug 22nd 2010, 09:59 PM
matheagle
The joint density is

$\displaystyle f(x_1,\cdots , x_n)={1\over \theta ^n} I(0<X_1,\cdots ,X_n<\theta)$

where I is the indicator functions that either 0 or 1.

NOW switch to the order stats and you can factor this as....

$\displaystyle f(x_1,\cdots , x_n)={1\over \theta ^n} I(0<X_{(1)})I(X_{(n)}<\theta)$

So the only stat stuck with theta is $\displaystyle X_{(n)}$ our suff stat.