A few days ago I wrote about U-statistics, statistics which can be expressed as the average of a symmetric function over all combinations of elements of a set. V-statistics can be written as an average of over all products of elements of a set.

Let S be a statistical sample of size n and let h be a symmetric function of r elements. The average of h over all subsets of S with r elements is a U-statistic. The average of h over the Cartesian product of S with itself r times

underbrace{S times S times cdots times S}_{n text{ times}}

is a V-statistic.

As in the previous post, let h(x, y) = (x − y)²/2. We can illustrate the V-statistic associated with h with Python code as before.

    import numpy as np
    from itertools import product

    def var(xs):
        n = len(xs)
        h = lambda c: (c[0] - c[1])**2/2
        return sum(h(c) for c in product(xs, repeat=2)) / n**2

    xs = np.array([2, 3, 5, 7, 11])
    print(np.var(xs))
    print(var(xs))

This time, however, we iterate over product rather than over combinations. Note also that at the bottom of the code we print

   np.var(xs)

rather than

   np.var(xs, ddof=1)

This means our code here is computing the population variance, not the sample variance. We could make this more explicit by supplying the default value of ddof.

   (np.var(xs, ddof=0)

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The post V-statistics first appeared on John D. Cook.

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