lmomco (version 2.4.14)

lrzlmomco: 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)

Value

Lorzen 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, 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