lmomco (version 2.4.14)

lkhlmomco: 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 $$K(u) = 1 - \frac{1}{\mu}\int_0^{1-u} x(p)\; \mathrm{d}p\mbox{,}$$ 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 (\(\mu\), cmlmomco) for \(u=0\) by $$K(u) = \frac{1}{\mu} [\mu - (1-u)(x(1-u) - R(1-u))] \mbox{.}$$

Usage

lkhlmomco(f, para)

Value

Leimkuhler curve value for \(F\).

Arguments

f

Nonexceedance probability (\(0 \le F \le 1\)).

para

The parameters from lmom2par or vec2par.

Author

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

# 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)