Question

Verify whether the following belong to the exponential family and find a sufficient statistic for the unknown parameters?

- Gamma distribution with α or β or both α and β unknown

Answer 1

K-parameter exponential family: If a K-parameter family wth pdf/pmf can be expressed as-

then it belongs to K-parameter exponential family.

where

• The support   is independent of
• The parameter space is an open region in containing a k-dimensional rectangle.
• 

Verification of Gamma distribution:

The density of a random variable with distribution  , can be written as

Clearly,

  

Hence belongs to two parameter exponential family

• To find sufficient statistics:

Sufficient Statistics:A statistic is said to be sufficient for if the conditional distribution of given T = t does not depend on θ for any value of t.

Theorem: If   is a random sample from a k-parameter exponential family:

Then are complete sufficient statistics of

parameter .

• For gamma distribution we have already proved it belongs to 2-parameter exponential family with:
•