MHB Binomial Distribution in the Exponential Family of Distributions

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The discussion centers on the binomial distribution's relationship with the exponential family of distributions. It is established that when the number of trials, n, is constant, the binomial distribution can be expressed in the exponential family form. However, when n is treated as a variable, the binomial distribution fails to meet the criteria for the exponential family due to the dependence of the possible values of x on n. The challenge remains in proving that the logarithm of the binomial coefficient cannot be expressed in the required summation form, which is crucial for establishing its non-membership in the exponential family. Ultimately, the binomial distribution's dependence on its parameters prevents it from fitting the exponential family framework.
Rashad9607
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A pdf is of the exponential family if it can be written $ f(x|\theta)=h(x)c(\theta)exp(\sum_{i=1}^{k}{w_{i}(\theta)t_{i}(x))}$ with $\theta$ a finite parameter vector, $c(\theta)>0$, all functions are over the reals, and only $h(x)$ is possibly constant.

I would like to show the binomial distribution with parameters $\theta=(p,n)$ is not in the exponential family.

Actually, if we consider $n$ to be constant, it is an exponential member:

$f(x|\theta)=p^{x}(1-p)^{n-x}\binom{n}{x}=\binom{n}{x}(1-p)^{n}(\frac{p}{1-p})^{x}=\binom{n}{x}(1-p)^{n}exp(x*log(\frac{p}{1-p}))$

Because $n$ is given, $\binom{n}{x}$ is a function of $x$ and will be $h(x)$.

$c(\theta)=(1-p)^{n}$.

$w_{1}(\theta)=log(\frac{p}{1-p})$ and $t_{1}(x)=x$.

If we instead want to consider the full parameter space where $n$ is not given, the binomial distribution is not a member of the exponential family.

Say we wanted to try and fit it into the exponential family model. The $\binom{n}{x}$ term would need to be split into a product of separate functions of $x$ and $n$ to be incorporated into $h(x)c(\theta)$, or split into a sum of products of separate functions to be incorporated into the summation term.

I was able to show that $\binom{n}{x}$ cannot be expressed as a product $u(n)v(x)$, so what is left is showing that it won't work in the summation term either. This means showing that $log(\binom{n}{x})$ is inexpressible as $\sum_{i=1}^{k}{w_{i}(n)t_{i}(x)}$, with $w_{i}$ and $t_{i}$ nonconstant, which I haven't been able to do. Any thoughts are appreciated.
 
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So I have read the next section in my text and learned that a characteristic of exponential family distributions is that the values $x$ can take must be the same over the entire parameter space. If we take $\theta=(p,n)$, then x=0,1,2,...,n, which depends on $\theta$, so it cannot be an exponential distribution.

But I'd still like to prove the log(nCr) thing.
 
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