Q-Q Plot - Plotting Positions - Filliben's Estimate

Filliben's Estimate

The order statistic medians are the medians of the order statistics of the distribution. These can be expressed in terms of the quantile function and the order statistic medians for the continuous uniform distribution by:

$N(i) = G(U(i))$

where U(i) are the uniform order statistic medians and G is the quantile function for the desired distribution. The quantile function is the inverse of the cumulative distribution function (probability that X is less than or equal to some value). That is, given a probability, we want the corresponding quantile of the cumulative distribution function.

James J. Filliben (Filliben 1975) uses the following estimates for the uniform order statistic medians:

$m(i) = \begin{cases} 1 - m(n) & i = 1\\ \\ \dfrac{i - 0.3175}{n + 0.365} & i = 2, 3, \ldots, n-1\\ \\ 0.5^{1/n} & i = n.\end{cases}$

The reason for this estimate is that the order statistic medians do not have a simple form.

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