We find the distribution that has maximum entropy conditional on having
specified values of its first r L-moments. This condition is equivalent to specifying
the expected values of the order statistics of a sample of size r. We show that
the maximum-entropy distribution has a density-quantile function, the reciprocal of
the derivative of the quantile function, that is a polynomial of degree r; the quantile
function of the distribution can then be found by integration. This class of maximum-entropy distributions includes the uniform, exponential and logistic, and two new generalizations of the logistic distribution that may be useful for modeling data.
We also derive maximum-entropy distributions subject to constraints on expected
values of linear combinations of order statistics.
By: J. R. Hosking
Published in: RC22508 in 2002
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