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Exercise: Prove that random samples from the following distributions form exponential families: Poisson, geometric, Gamma.

What I know:

A parametric family with parameter space $\Omega$ and density $f_X(x|\theta)$ with respect to measure $\nu$ on $(\mathcal{X}, \mathcal{B})$ is called an exponential family if $$f_X(x|\theta) = c(\theta)h(x)\exp\bigg(\sum\limits_{i = 1}^k\pi_i(\theta)\tau_i(x)\bigg),$$ for some measurable functions $\pi_1, ..., \pi_k, \tau_1, ..., \tau_k$ and some integer $k$. As the function $c(\theta)$ can be written as $$c(\theta) = 1\bigg/\int h(x)\exp\bigg(\sum\limits_{i = 1}^k\pi_i(\theta)\tau_i(x)\bigg)d\nu (x),$$ we see that we might as well parametrise the family by $$\pi = (\pi_1(\theta), ..., \pi_k(\theta))\in\mathbb{R}^k.$$

What I've tried: take a random sample $X$ from the Poisson distribution. I know that $$f_X(x|\theta) = e^{-\theta}\frac{\theta^x}{x\text{ ! }}, \text{ and that }c(\theta) = 1\bigg/\int h(x)\exp\bigg(\sum\limits_{i = 1}^k\pi_i(\theta)\tau_i(x)\bigg)d\nu (x). $$ I need to find $h(x), \pi_i(\theta)$ and $\tau_i(x)$ such that $$e^{-\theta}\frac{\theta^x}{x\text{ ! }} = \bigg(h(x)\bigg/\int h(x)\exp\bigg(\sum\limits_{i = 1}^k\pi_i(\theta)\tau_i(x)\bigg)d\nu (x)\bigg)\bigg(\exp\big(\sum\limits_{i = 1}^k \pi_i(\theta)\tau_i(x)\big)\bigg). $$ I don't see how I should proceed from here unfortunately..

Question: How do I prove that a random sample of for instance the Poisson distribution forms an exponential family?

Thanks in advance!

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1 Answer 1

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$$e^{-\theta}\frac{\theta^x}{x!} = e^{-\theta} \cdot \frac1{x!} \cdot \theta^x \qquad \text{where} \qquad c(\theta) = e^{-\theta}\,,~~h(x) = \frac1{x!}\mathbb{1}_{x\in\mathbb{N}}\,,~~\text{and} \\ \theta^x=\mathrm{Exp}(x\log\theta) \quad\Longleftrightarrow \quad\pi(\theta) = \log \theta\,,~~\tau(x) = x$$ There is only one "natural parameter": the vector $\pi$ is one-dimensional and it is $\pi(\theta) = \log\theta$. The summation with index $i$ for the products of $\pi_i(\theta) \cdot\tau_i(x)$ is just a single term.

The $\mathbb{1}_{x\in \mathbb{N}}$ (where $\mathbb{N}$ is understood as $0,1,2,\ldots$) is the indicator function, and it is a totally legitimate function of $x$ (and of $x$ alone). The $h(x)$ is usually where one takes care of the "domain" (in elementary terms). Note that the uniform distribution (with arbitrary domain $[a,b]$) is not part of the exponential family exactly because the domain necessarily involves the "parameters".

Is this what you want? I apologize if I misunderstood your question.

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