Leimkuhler Curve of the Distributions

Description

This function computes the Leimkuhler Curve for quantile function x(F) (par2qua, qlmomco). The function is defined by Nair et al. (2013, p. 181) as

K(u) = 1 - \frac{1}{μ}\int_0^{1-u} x(p)\; \mathrm{d}p\mbox{,}

where K(u) is Leimkuhler curve for nonexceedance probability u. The Leimkuhler curve is related to the Lorenz curve (L(u), lrzlmomco) by

K(u) = 1-L(1-u)\mbox{,}

and related to the reversed residual mean quantile function (R(u), rrmlmomco) and conditional mean (μ, cmlmomco) for u=0 by

K(u) = \frac{1}{μ} [μ - (1-u)(x(1-u) - R(1-u))] \mbox{.}

Usage

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lkhlmomco(f, para)

Arguments

f

Nonexceedance probability (0 ≤ F ≤ 1).

para

The parameters from lmom2par or vec2par.

Value

Leimkuhler curve value for F.

Author(s)

W.H. Asquith

References

Nair, N.U., Sankaran, P.G., and Balakrishnan, N., 2013, Quantile-based reliability analysis: Springer, New York.

See Also

qlmomco, lrzlmomco

Examples

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# It is easiest to think about residual life as starting at the origin, units in days.
A <- vec2par(c(0.0, 2649, 2.11), type="gov") # so set lower bounds = 0.0

"afunc" <- function(u) { return(par2qua(u,A,paracheck=FALSE)) }
f <- 0.35 # All three computations report: Ku = 0.6413727
Ku1 <- 1 - 1/cmlmomco(f=0,A) * integrate(afunc,0,1-f)$value
Ku2 <- (cmlmomco(0,A) - (1-f)*(quagov(1-f,A) - rrmlmomco(1-f,A)))/cmlmomco(0,A)
Ku3 <- lkhlmomco(f, A)

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