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Computational Order Statistics

# 2. Discrete Parent Distribution (Function Form)

mathStatica 1.5 expands its functionality to the case of a discrete parent, again for arbitrary distributions. To illustrate, let the random variable have a Poisson two-component-mix distribution with probability mass function (pmf) :

and domain of support:

Here, and are the Poisson parameters, while is the mixing weight parameter. Figure 3 illustrates this pmf when , and :

Figure 3.

Given a sample of size , the pmf of the order statistic , denoted , is:

with domain of support:

Figure 4 plots the pmf of the minimum order statistic (i.e. ) when , and and the sample size is :

Figure 4.

## Monte Carlo 'Check' of the Exact Solution We Have Just Plotted

Here is a single pseudorandom sample of size drawn from the parent two-component-mix Poisson pmf :

If we want 50,000 such samples (each of size 10), the neatest approach is to generate all drawings in one go:

and then partition this data into 50,000 samples (each of size 10). We can then find the minimum of each of the 50,000 samples by mapping the Min function across each sample:

If is very large, efficient algorithms specifically designed for pseudorandom generation of order statistics exist; see [9]. Figure 5 plots the empirical relative frequency distribution of the sample minimum data () together with the exact pmf of the sample minimum (): a good match should see the former obscure the latter nearly everywhere.

Figure 5.