lmomco (version 2.3.7)

theopwms: The Theoretical Probability-Weighted Moments using Integration of the Quantile Function

Description

Compute the theoretrical probability-weighted moments (PWMs) for a distribution. A theoretrical PWM in integral form is $$ \beta_r = \int^1_0 x(F)\,F^r\,\mathrm{d}F \mbox{,}$$ where \(x(F)\) is the quantile function of the random variable \(X\) for nonexceedance probability \(F\) and \(r\) represents the order of the PWM. This function loops across the above equation for each nmom set in the argument list. The function \(x(F)\) is computed through the par2qua function. The distribution type is determined using the type attribute of the para argument, which is a parameter object of lmomco (see vec2par).

Usage

theopwms(para, nmom=5, verbose=FALSE)

Value

An R

list is returned.

betas

The PWMs. Note that convention is the have a \(\beta_0\), but this is placed in the first index i=1 of the betas vector.

source

An attribute identifying the computational source of the probability-weighted moments: “theopwms”.

Arguments

para

A distribution parameter object such as that by lmom2par or vec2par.

nmom

The number of moments to compute. Default is 5.

verbose

Toggle verbose output. Because the R function integrate is used to perform the numerical integration, it might be useful to see selected messages regarding the numerical integration.

Author

W.H. Asquith

References

Hosking, J.R.M., 1990, L-moments--Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, v. 52, p. 105--124.

See Also

theoLmoms, pwm, pwm2lmom

Examples

para     <- vec2par(c(0,1),type='nor') # standard normal
the.pwms <- theopwms(para) # compute PWMs
str(the.pwms)