# Proving a distribution is a member of generalised exponential family

1. Mar 29, 2012

### johnaphun

I've been asked to prove that the following distribution is a member of the generalised exponential family of distributions.

f(y;β) = (ky2β(y+k))/((β+3)(y+2k)(y+1)1/2)

I know that i have to transform the equation into the form

f(y) = exp{(yθ-bθ)/a∅ +c(y,∅)}

and that to do this i should take the exponential of the log

exp{log(f(y;β))}

I understand how to do this for the more simple distributions (poisson, binomial etc) however i always struggle with more complicated ones. Are there any tips or anything to look out for when answering this type of question.

So far i have
exp[(y+k)log(βy)-(y+2k)log(β+3)+log ky - (1/2)log(y+1)]

but i'm pretty sure thats not correct

2. Mar 29, 2012

### Dickfore

what is the empty set symbol in the general formula supposed to be?

Also, could you please demonstrate how you did it for the Poisson distribution?

3. Mar 29, 2012

### johnaphun

Sorry thats meant to be phi not an empty set. I typed this without my glasses! It's meant to represent the dispersion/scale parameter

For the poisson distribution I did the following;

f(y;θ) = λye/y!

= exp{log(λy/y!)}

= exp{ylogλ - λ - logy!}
with

θ = logλ , a(phi) = phi = 1 , b(θ) = eθ, c(y,phi) = -logy!

4. Mar 30, 2012

### chiro

Hey johnaphun and welcome to the forums.

Are you aware of transformation theorems to find distributions in terms of a distribution U and a transformed distribution f(U)?

5. Mar 30, 2012

### johnaphun

Hi chiro, thanks for the reply.

No i'm unaware of these theorems, would be able to explain for me?

6. Mar 30, 2012

### Dickfore

So, your argument in the exponential may be rewritten as:
$$\left[ \log(\beta) - \log(\beta + 3) \right] y + k \log(\beta) - 2 k \log(\beta + 3) + \log(k) + (y + k + 1) \log(y) - \frac{1}{2} \log(y + 1)$$
Can you read off your $\theta$, b, $\phi$, and $c(y, \phi)$. Although, presonally, I don't know what you're doing and what is this generalized exponential.

7. Mar 30, 2012