# bfrlmomco: 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

 1 bfrlmomco(f, para) 

### Arguments

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

### Value

Bonferroni 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.

qlmomco, lrzlmomco

### Examples

 1 2 3 4 5 6 7 # 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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