Example: Parametric Bootstrap estimate of mean number of calls per hour at a telephone exchange

Imagine that we want to predict the number of phone calls there will be at an exchange during a particular hour in the working day (say 2pm-3pm). Imagine that we have collected data from this period on n separate, randomly selected days. It is reasonable to assume that telephone calls will arrive at a Poisson rate since each call will be, roughly speaking, independent of every other call. Thus we could use a Poisson distribution to model the number of calls in an hour. The maximum likelihood estimate (MLE) of the mean number of calls per hour at this time of day is simply the average number of calls observed in the test periods x. Thus our Bootstrap replicate is a set of n independent Poisson(x) distributions.

To generate our uncertainty about the true mean number of phone calls per hour at this time of the day, we calculate the mean of the sum of the Bootstrap replicate, i.e. the average of n independent Poisson(x) distributions. The sum of n independent Poisson(x) distributions is simply Poisson(n.x), so the average of n Poisson(x) distributions is Poisson(n*x)/n. (n*x) is the sum of the observations. So, in general, if one has a observations from n periods, the Poisson parametric Bootstrap for the mean number of observations per period l is given by:

l = Poisson(a)/n

This is the crude classical statistics result we discuss in the classical statistics section. We don't recommend using this method, however, especially for small a, because more sophisticated approaches are available.

Example: Parametric Bootstrap estimate of mean number of calls per hour at a telephone exchange

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