Probability Weighted Moments Algorithm

A Probability Weighted Moments Algorithm is a parameter estimation algorithm for cumulative distribution functions whose inverse forms are explicitly defined.

• AKA: PWM.
• Context:
  • It can be defined as: [math]\displaystyle{ M(i,j,k) = E[X^iF^j(1 - F)^k]=\int_0^1[xF]^iF^j(1-F)^k DF }[/math] where X is the random variable, [math]\displaystyle{ F }[/math] is the cumulative distribution function (i. e.[math]\displaystyle{ F\equiv F(x)=P(X\leq x) }[/math]) and [math]\displaystyle{ i,j,k }[/math] are real numbers.
  • It can be used to derive L-Moments.
• Example(s)
  • (Hosking & Wallis, 2005).
  • …
• Counter-Example(s):
  • Method of Moments Algorithm.
• See: Moment, Discrete Random Variable, Statistic, Cumulative Distribution Function, Gumbel Distribution.

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References

2015

• (Wikipedia, 2015) ⇒ http://wikipedia.org/wiki/L-moment#Related_quantities
  • QUOTE: L-moments are statistical quantities that are derived from probability weighted moments (PWM) which were defined earlier (1979). PWM are used to efficiently estimate the parameters of distributions expressable in inverse form such as the Gumbel, the Tukey, and the Wakeby distributions.

2005

• (Hosking & Wallis, 2005) ⇒ Jonathan Richard Morley Hosking, and James R. Wallis. (2005). “Regional Frequency Analysis: an approach based on L-moments." Cambridge University Press.

1979

• (Greenwood et al., 1979) ⇒ J. Arthur Greenwood, J. Maciunas Landwehr, N. C. Matalas and J. R. Wallis (1979). “Probability Weighted Moments: Definition and Relation to Parameters of Several Distributions Expressable in Inverse Form", Vol. 15, No. 5, Water Resources Research
  • QUOTE: A distribution function [math]\displaystyle{ F \equiv F(x) = P(X \leq x) }[/math]may be characterized by probability weighted moments, which are defined as

[math]\displaystyle{ M(l,j,k) = E[X^lF^j(1 - F)^k]=\int_0^1[xF]^lF^j(1-F)^k DF }[/math]

where [math]\displaystyle{ l, j }[/math], and [math]\displaystyle{ k }[/math] are real numbers. If [math]\displaystyle{ j = k = 0 }[/math] and [math]\displaystyle{ l }[/math] is a nonnegative integer, then [math]\displaystyle{ M_{l,0,0} }[/math] represents the conventional moment about the origin of order 1.

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