Results 1 to 2 of 2

Thread: exponential family help

  1. #1
    Senior Member Danneedshelp's Avatar
    Joined
    Apr 2009
    Posts
    303

    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor harish21's Avatar
    Joined
    Feb 2010
    From
    Dirty South
    Posts
    1,036
    Thanks
    10
    Quote Originally Posted by Danneedshelp View Post
    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:

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

    so your pdf is :
    $\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 $
    Last edited by harish21; Mar 24th 2010 at 12:49 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Inverse Gaussian Exponential Family
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: Mar 15th 2012, 01:17 PM
  2. Multiparameter exponential family
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: May 20th 2010, 07:54 AM
  3. Exponential Family
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: Feb 4th 2010, 06:32 PM
  4. Bayesian - Exponential Family
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: Nov 7th 2009, 07:04 AM
  5. Exponential Family & Sufficient Statistic.
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: Aug 19th 2009, 03:45 PM

Search Tags


/mathhelpforum @mathhelpforum