# Compound negative binomial distribution

In this post, we discuss the compound negative binomial distribution and its relationship with the compound Poisson distribution.

A compound distribution is a model for a random sum $Y=X_1+X_2+ \cdots +X_N$ where the number of terms $N$ is uncertain. To make the compund distribution more tractable, we assume that the variables $X_i$ are independent and identically distributed and that each $X_i$ is independent of $N$. The random sum $Y$ can be interpreted the sum of all the measurements that are associated with certain events that occur during a fixed period of time. For example, we may be interested in the total amount of rainfall in a 24-hour period, during which the occurences of a number of events are observed and each of the events provides a measurement of an amount of rainfall. Another interpretation of compound distribution is the random variable of the aggregate claims generated by an insurance policy or a group of insurance policies during a fixed policy period. In this setting, $N$ is the number of claims generated by the portfolio of insurance policies and $X_1$ is the amount of the first claim and $X_2$ is the amount of the second claim and so on. When $N$ follows the Poisson distribution, the random sum $Y$ is said to have a compound Poisson distribution. Even though the compound Poisson distribution has many attractive properties, it is not a good model when the variance of the number of claims is greater than the mean of the number of claims. In such situations, the compound negative binomial distribution may be a better fit. See this post (Compound Poisson distribution) for a basic discussion. See the links at the end of this post for more articles on compound distributons that I posted on this blog.

Compound Negative Binomial Distribution
The random variable $N$ is said to have a negative binomial distribution if its probability function is given by the following:

$\displaystyle P[N=n]=\binom{\alpha + n-1}{\alpha-1} \thinspace \biggl(\frac{\beta}{\beta+1}\biggr)^{\alpha}\biggl(\frac{1}{\beta+1}\biggr)^{n} \ \ \ \ \ \ \ \ \ \ \ \ (1)$

where $n=0,1,2,3, \cdots$, $\beta >0$ and $\alpha$ is a positive integer.

Our formulation of negative binomial distribution is the number of failures that occur before the $\alpha^{th}$ success in a sequence of independent Bernoulli trials. But this interpretation is not important to our task at hand. Let $Y=X_1+X_2+ \cdots +X_N$ be the random sum as described in the above introductory paragraph such that $N$ follows a negative binomial distribution. We present the basic properties discussed in the post An introduction to compound distributions by plugging the negative binomial distribution into $N$.

Distribution Function
$\displaystyle F_Y(y)=\sum \limits_{n=0}^{\infty} F^{*n}(y) \thinspace P[N=n]$

where $F$ is the common distribution function for $X_i$ and $F^{*n}$ is the $n^{th}$ convolution of $F$. Of course, $P[N=n]$ is the negative binomial probability function indicated above.

Mean and Variance
$\displaystyle E[Y]=E[N] \thinspace E[X]=\frac{\alpha}{\beta} E[X]$

$\displaystyle Var[Y]=E[N] \thinspace Var[X]+Var[N] \thinspace E[X]^2$

$\displaystyle =\frac{\alpha}{\beta} Var[X]+\frac{\alpha (\beta+1)}{\beta^2} E[X]^2$

Moment Generating Function
$\displaystyle M_Y(t)=M_N[ln M_X(t)]=\biggl(\frac{p}{1-(1-p) M_X(t)}\biggr)^{\alpha}$

$\displaystyle M_Y(t)=\biggl(\frac{\beta}{\beta+1- M_X(t)}\biggr)^{\alpha}$

where $\displaystyle p=\frac{\beta}{\beta+1}$, $\displaystyle M_N(t)=\biggl(\frac{p}{1-(1-p) e^{t}}\biggr)^{\alpha}$

Cumulant Generating Function
$\displaystyle \Psi_Y(t)=\alpha \thinspace ln \biggl(\frac{\beta}{\beta+1- M_X(t)}\biggr)$

Skewness
$\displaystyle E[(Y-\mu_Y)^3]=\Psi_Y^{(3)}(0)$

$\displaystyle =\frac{2}{\alpha^2} E[N]^3 E[X]^3 +\frac{3}{\alpha} E[N]^2 E[X] E[X^2]+E[N] E[X^3]$

Measure of skewness: $\displaystyle \gamma_Y=\frac{E[(Y-\mu_Y)^3]}{(Var[Y])^{\frac{3}{2}}}$

Compound Mixed Poisson Distribution
In a previous post (Basic properties of mixtures), we showed that the negative binomial distribution is a mixture of a family of Poisson distributions with gamma mixing weights. Specifically, if $N \sim \text{Poisson}(\Lambda)$ and $\Lambda \sim \text{Gamma}(\alpha,\beta)$, then the unconditional distribution of $N$ is a negative binomial distribution and the probability function is of the form (1) given above.

Thus the negative binomial distribution is a special example of a compound mixed Poisson distribution. When an aggregate claims variable $Y=X_1+X_2+ \cdots +Y_N$ has a compound mixed Poisson distribution, the number of claims $N$ follows a Poisson distribution, but the Poisson parameter $\Lambda$ is uncertain. The uncertainty could be due to an heterogeneity of risks across the insureds in the insurance portfolio (or across various rating classes). If the information of the risk parameter $\Lambda$ can be captured in a gamma distribution, then the unconditional number of claims in a given fixed period has a negative binomial distribution.

Previous Posts on Compound Distributions
An introduction to compound distributions
Some examples of compound distributions
Compound Poisson distribution
Compound Poisson distribution-discrete example