rralmomco {lmomco}

Reversed Alpha-Percentile Residual Quantile Function of the Distributions

Description

This function computes the Reversed \alpha-Percentile Residual Quantile Function for quantile function x(F) (par2qua, qlmomco). The function is defined by Nair and Vineshkumar (2011, p. 87) and Midhu et al. (2013, p. 13) as

R_\alpha(u) = x(u) - x(u[1-\alpha])\mbox{,}

where R_\alpha(u) is the reversed \alpha-percentile residual quantile for nonexceedance probability u and percentile \alpha and x(u[1-\alpha]) is a constant for x(F = u[1-\alpha]). The nonreversed \alpha-percentile residual quantile is available under ralmomco.

Usage

rralmomco(f, para, alpha=0)

Arguments

f

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

para

The parameters from lmom2par or vec2par.

alpha

The \alpha percentile, which is divided by 100 inside the function ahead of calling the quantile function of the distribution.

Value

Reversed \alpha-percentile residual quantile value for F.

Note

Technically it seems that Nair et al. (2013) do not explictly define the reversed \alpha-percentile residual quantile but their index points to pp. 69–70 for a derivation involving the Generalized Lambda distribution (GLD) but that derivation (top of p. 70) has incorrect algebra. A possibilty is that Nair et al. (2013) forgot to include R_\alpha(u) as an explicit definition in juxtaposition to P_\alpha(u) (ralmomco) and then apparently made an easy-to-see algebra error in trying to collect terms for the GLD.

Author(s)

W.H. Asquith

References

Nair, N.U., and Vineshkumar, B., 2011, Reversed percentile residual life and related concepts: Journal of the Korean Statistical Society, v. 40, no. 1, pp. 85–92.

Midhu, N.N., Sankaran, P.G., and Nair, N.U., 2013, A class of distributions with linear mean residual quantile function and it's generalizations: Statistical Methodology, v. 15, pp. 1–24.

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

See Also

qlmomco, ralmomco

Examples

# It is easiest to think about residual life as starting at the origin, units in days.
A <- vec2par(c(145, 2649, 2.11), type="gov") # so set lower bounds = 0.0
rralmomco(0.78, A, alpha=50)
## Not run: 
F <- nonexceeds(f01=TRUE); r <- range(rralmomco(F,A, alpha=50), ralmomco(F,A, alpha=50))
plot(F, rralmomco(F,A, alpha=50), type="l", xlab="NONEXCEEDANCE PROBABILITY",
                  ylim=r, ylab="MEDIAN RESIDUAL OR REVERSED LIFETIME, IN DAYS")
lines(F, ralmomco(F, A, alpha=50), col=2) # notice the lack of symmetry

## End(Not run)