1. ## exponential family help

Q: Let $\displaystyle Y_{1},...,Y_{n}$ denote a random sample from the probability density function
$\displaystyle f(y|\theta)=\theta\\y^{\theta\\-1}$ for $\displaystyle 0<y<1$, $\displaystyle \theta>0$ and 0 otherwise.
a) Show that this density function is in the (one-parameter) exponential family and that $\displaystyle \sum_{i=1}^{n}-ln(Y_{i})$ is sufficient for $\displaystyle \theta$ (see previous exercise).
b) If $\displaystyle W_{i}=-ln(Y_{i})$, show that $\displaystyle W_{i}$ has an exponential distribution with mean $\displaystyle \frac{1}{\theta}$.

Previews exercise mentioned in (a):

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

$\displaystyle f(y|\theta)=a(\theta)b(y)e^{-[c(\theta)d(y)]}I_{[a,b]}(y)$, where $\displaystyle I_{[a,b]}(y)$ is the indicator function and a and b do not depend on $\displaystyle \theta$.

I think I am supposed to define each function in the density directly above for the orginal question; however, I am having a hard time doing so. Some direction would be great. Thanks.

2. Originally Posted by Danneedshelp
Q: Let $\displaystyle Y_{1},...,Y_{n}$ denote a random sample from the probability density function
$\displaystyle f(y|\theta)=\theta\\y^{\theta\\-1}$ for $\displaystyle 0<y<1$, $\displaystyle \theta>0$ and 0 otherwise.
a) Show that this density function is in the (one-parameter) exponential family and that $\displaystyle \sum_{i=1}^{n}-ln(Y_{i})$ is sufficient for $\displaystyle \theta$ (see previous exercise).
b) If $\displaystyle W_{i}=-ln(Y_{i})$, show that $\displaystyle W_{i}$ has an exponential distribution with mean $\displaystyle \frac{1}{\theta}$.

Previews exercise mentioned in (a):

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

$\displaystyle f(y|\theta)=a(\theta)b(y)e^{-[c(\theta)d(y)]}I_{[a,b]}(y)$, where $\displaystyle I_{[a,b]}(y)$ is the indicator function and a and b do not depend on $\displaystyle \theta$.

I think I am supposed to define each function in the density directly above for the orginal question; however, I am having a hard time doing so. Some direction would be great. Thanks.
You can say that the distribution $\displaystyle f(y,\theta)$ belongs to a one parameter exponential family if you can express its pdf in this form:

$\displaystyle f(y,\theta) = a(\theta) \times g(y) \times exp(b(\theta) \times R(y))$

$\displaystyle \theta\\y^{\theta\\-1}$

= $\displaystyle \theta \frac{y^{\theta}}{y}$

= $\displaystyle \theta \times \frac{1}{y} \times {y^{\theta}}$

=$\displaystyle \theta \times \frac{1}{y} \times exp(\theta \times log(y))$

compare the above system with $\displaystyle f(y,\theta) = a(\theta) \times g(y) \times exp(b(\theta) \times R(y))$, you can say that:

$\displaystyle a(\theta) = \theta$

$\displaystyle g(y) = \frac{1}{y}$

$\displaystyle b(\theta) = \theta$

and $\displaystyle R(y) = log(y)$

and

$\displaystyle \sum_{i=1}^{n} R(y)$ is a complete sufficient statistic for $\displaystyle \theta$