# How should you express a negative binomial distribution in an exponential family form?

When $X$ is $X_1$,...$X_n$, how do you express the following negative binomial distribution in an exponential family form?

$$f(k;r,p)\equiv\text{Pr}(X=k)=\binom{k+r-1}{k}(1-p)^{r}p^k~~~~~\text{for}~k=0,1,2,\ldots$$

• Hey, I think you should be more specific as to what it is that you're doing and what your problem is. Your question, as it currently stands, probably won't get a very good or satisfying answer, and probably will get a number of downvotes. – Eric Peterson Jun 23 '13 at 14:27
• Sorry, I am pretty new and don't know how to write an equation in LaTex. I just know how to use it for a very simple notation. I don't know how specific the question needs to be. Any suggestion? – Python_R Jun 23 '13 at 15:11
• It's not clear to me what your question is, @Python_R. Can you say more about what you want to know? – gung - Reinstate Monica Jun 23 '13 at 15:34

Warning: The negative binomial distribution has several alternative formulations for which the formulas below change.

A distribution $f(x;\theta)$ belongs to an exponential family if it can be represented in the form:

$$f(x;\theta)=h(x)\exp\left[\eta(\theta)T(x)-A(\theta)\right]$$ The value $\eta$ is called the canonical (natural) parameter of the family, $T(x)$ is a sufficient statistic for $p$, $A(\theta)$ is called the log-partition function (it's a normalization factor and sometimes called log-normalizer or cumulant generating function), and $h(x)$ is an arbitrary function called base measure or carrier measure which is 1 in many cases (e.g. exponential distribution, gamma distribution, Bernoulli distribution, ...). If the negative binomial distribution with known parameter $r$ (if $r$ is unknown, the negative binomial family is not an exponential family) has the following distribution:

$$f(k;r,p)=\binom{k+r-1}{k}(1-p)^{r}p^k~~~~~\text{for}~k=0,1,2,\ldots$$ Then it can be rewritten in exponential form as:

\begin{align} f(k;r,p) &=\binom{k+r-1}{k}\exp\left[\ln(p^{k}(1-p)^{r})\right] \\ &=\binom{k+r-1}{k}\exp\left[k\ln(p) + r\ln(1-p)\right] \\ \end{align} So the parameter $\theta$ of the distribution is $p$ (i.e. $\theta=p$) and the natural parameter is $\eta=\ln(p)$, the sufficient statistic for $p$ is $T(k)=k$ (i.e. $T(k)=\sum X_{i}$, the sample sum), the log-partition function is $A(\eta)=-r\ln(1-p)$ and $h(k)=\binom{k+r-1}{k}$. See for example the Wikipedia page for a nice overview of the theory and many common distributions. Nice references can also be found here and here.

• +1. Also, $h(x)$ is called the carrier measure because in the measure-theoretic formulation of exponential families (e.g. Shao's book), it is an actual measure. – Neil G Jun 23 '13 at 23:07
• @NeilG Thanks Neil, I didn't know that. Do you mean Jun Shao's book Mathematical statistics? If yes, could you point out where in the book? Thanks very much. – COOLSerdash Jun 24 '13 at 6:16
• @COOLSerdash: Yes, that's the one. I don't have the book with me, but I remember seeing the measure theoretic formulation there. Just doing a quick Google search, you can find a reference to "carrier measure" in "The Geometry of Exponential Families" by Bradley Efron – Neil G Jun 24 '13 at 8:18
• The carrier measure can usually be made into a standard carrier measure (by rescaling the sample space) for a continuous distribution, but is rarely equal to one for a discrete distribution. – Neil G Jun 24 '13 at 8:23