# 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}{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

 1 lkhlmomco(f, para)

### Arguments

 f Nonexceedance probability (0 ≤ F ≤ 1). para The parameters from lmom2par or vec2par.

### Value

Leimkuhler curve value for F.

W.H. Asquith

### References

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

### Examples

 1 2 3 4 5 6 7 8 # 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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