Exercise 313 E

Construct three plots ofp0(m, k) (y-axis) vs. m (x-axis) as m runs from 10 to 500 for k = 10, 2, and 1. Interpret your results.

Next, we use Eqs. (3.64) and (3.65) to obtain an iterative equation relating subsequent terms, as we did for the Poisson and binomial distributions

Figure 3.7 is a comparison of the Poisson and negative binomial distributions. Here, I have set the mean equal to 10. The Poisson distribution is thus peaked around 10 and relatively symmetrical. The negative binomial distribution, with the same mean, becomes more and more skewed as the overdispersion parameter decreases from 5 to 0.5 (panels (b-d) in Figure 3.7). For k = 0.5 (Figure 3.6d), there is more than a 20% chance of 0 events, even though the mean is 10! Consequently, the probability of a large number of events (say 20-30 or even more) is considerable. Figure 3.8, in which I have plotted the cumulative distribution as if it were a continuous one, is another way of representing the idea. Notice that the cumulative values of the negative binomial are much higher than the cumulative values of the Poisson distribution for small values of the number of events and that they rise much more slowly than the Poisson for larger number of events. For example, at 20 events, the Poisson cumulative (with mean 10) is essentially 1, but the two negative binomial distributions that I have shown have nearly 20% of the probability still to be accounted.

We will close this section with an all too brief discussion of some aspects of inference involving the negative binomial. Let's begin with Eq. (3.61), for which the data would be the meanK and sample variance SK of a collection of random variables with a negative binomial distribution, for which we would want to estimate the parameters m and v. If we replace the mean and variance in Eq. (3.61) by the sample average and sample variance and then solve Eq. (3.61) for the parameters, we obtain the method of moments estimates of the parameters, which are m = K and v = (K / — KT). These estimates are simple, but not very accurate. More accurate estimates can be obtained by using maximum likelihood procedures with Eq. (3.60), but they are somewhat beyond the scope of what I want to do here. One good place to read about maximum likelihood for the negative binomial is Kendall and Stuart (1979) (which is also a generally good book). Dick (2004) discusses some modern methods for estimating the parameters.

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