Chebyshev's Inequality - Finite Samples

Finite Samples

Saw et al extended Chebyshev's inequality to cases where the population mean and variance are not known but are instead replaced by their sample estimates.

where N is the sample size, m is the sample mean, k is a constant and s is the sample standard deviation. g(x) is defined as follows:

Let x ≥ 1, Q = N + 1, and R be the greatest integer less than Q / x. Let

Now

This inequality holds when the population moments do not exist and when the sample is weakly exchangeably distributed.

Kabán gives a somewhat less complex version of this inequality.

If the standard deviation is a multiple of the mean then a further inequality can be derived,

A table of values for the Saw–Yang–Mo inequality for finite sample sizes (n < 100) has been determined by Konijn.

For fixed N and large m the Saw–Yang–Mo inequality is approximately

Beasley et al have suggested a modification of this inequality

In empirical testing this modification is conservative but appears to have low statistical power. Its theoretical basis currently remains unexplored.

Read more about this topic:  Chebyshev's Inequality

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