lrzlmomco {lmomco}

Lorenz Curve of the Distributions

Description

This function computes the Lorenz Curve for quantile function x(F) (par2qua, qlmomco). The function is defined by Nair et al. (2013, p. 174) as

L(u) = \frac{1}{\mu}\int_0^u x(p)\; \mathrm{d}p\mbox{,}

where L(u) is the Lorenz curve for nonexceedance probability u. The Lorenz curve is related to the Bonferroni curve (B(u), bfrlmomco) by

L(u) = \mu B(u)\mbox{.}

Usage

lrzlmomco(f, para)

Arguments

f

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

para

The parameters from lmom2par or vec2par.

Value

Lorzen 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, bfrlmomco

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
f <- c(0.25, 0.75) # Both computations report: 0.02402977 and 0.51653731
Lu1 <-   lrzlmomco(f, A)
Lu2 <- f*bfrlmomco(f, A)

# The Lorenz curve is related to the Gini index (G), which is L-CV:
"afunc" <- function(u) { return(lrzlmomco(f=u, A)) }
L <- integrate(afunc, lower=0, upper=1)$value
G <- 1 - 2*L                                                    # 0.4129159
G <- 1 - expect.min.ostat(2,para=A,qua=quagov)*cmlmomco(f=0,A)  # 0.4129159
LCV <- lmomgov(A)$ratios[2]                                     # 0.41291585