Bonferroni Curve of the Distributions

Description

This function computes the Bonferroni Curve for quantile function x(F) (par2qua, qlmomco). The function is defined by Nair et al. (2013, p. 179) as

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

where B(u) is Bonferroni curve for quantile function x(F) and μ is the conditional mean for quantile u=0 (cmlmomco). The Bonferroni curve is related to the Lorenz curve (L(u), lrzlmomco) by

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

Usage

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bfrlmomco(f, para)

Arguments

f

Nonexceedance probability (0 ≤ F ≤ 1).

para

The parameters from lmom2par or vec2par.

Value

Bonferroni 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 <- 0.65 # Both computations report: 0.5517342
Bu1 <- 1/(cmlmomco(f=0,A)*f) * integrate(afunc, 0, f)$value
Bu2 <- bfrlmomco(f, A)

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