## sufficient statistics and MVUE's

Q1: Suppose that $Y_{1},...,Y_{n}$ is a random sample from a probability density in the (one-parameter) exponential family so that

$f(y|\theta)=a(\theta)b(y)e^{-[c(\theta)d(y)]}I_{[a,b]}(y)$, where $I_{[a,b]}(y)$ is the indicator function and $a$ and $b$ do not depend on $\theta$. Show that $\sum_{i=1}^{n}d(Y_{i})$ is sufficient for $\theta$.

A1:

$L(y_{1},...,y_{n}|\theta)=f(y_{1},...,y_{n}|\theta )
$
$=\Pi_{i=1}^{n}a(\theta)b(y)e^{-[c(\theta)d(y)]}I_{[a,b]}(y)
$
= $[a(\theta)b(y)]^{n}exp[-c(\theta)\sum_{i=1}^{n}d(y_{i})]\Pi_{i=1}^{n}I_{[a,b]}(y_{i})$.

If I let $h(y_{1},...,y_{n})=\Pi_{i=1}^{n}I_{[a,b]}(y_{i})$, which does not depend on $\theta$ and $g(\sum_{i=1}^{n}d(y_{i}),\theta)=[a(\theta)b(y)]^{n}exp[-c(\theta)\sum_{i=1}^{n}d(y_{i})]$, where $\theta$ interacts with the data $y_{i}$ only through $U=\sum_{i=1}^{n}d(y_{i})$. So, by the factorization criterion, $U=\sum_{i=1}^{n}d(y_{i})$ is sufficient for $\theta$.

Q2: Let $Y_{1},...,Y_{n}$ denote a random sample from the probability density function

$f(y|\theta)=\theta\\y^{\theta\\-1}$ for $0, $\theta>0$ and $0$ otherwise.

a) Show that this density function is in the (one-parameter) exponential family and that $\sum_{i=1}^{n}-ln(Y_{i})$ is sufficient for $\theta$.

b) If $W_{i}=-ln(Y_{i})$, show that $W_{i}$ has an exponential distribution with mean $\frac{1}{\theta}$.

Furthermore, in Q2, the book tells us to refrence back to Q1 as a hint.

A2:

I am really stuck on this one. I am not sure how to show something is in the one-parameter exponential family, as I have never seen such a thing before. So, some steps might be nice to see or even an explanation of the process. Any help would be greatly appreciated.