rmlmomco: Mean Residual Quantile Function of the Distributions

rmlmomcoR Documentation

Mean Residual Quantile Function of the Distributions


This function computes the Mean Residual Quantile Function for quantile function x(F) (par2qua, qlmomco). The function is defined by Nair et al. (2013, p. 51) as

M(u) = \frac{1}{1-u}\int_u^1 [x(p) - x(u)]\; \mathrm{d}p\mbox{,}

where M(u) is the mean residual quantile for nonexceedance probability u and x(u) is a constant for x(F = u). The variance of M(u) is provided in rmvarlmomco.

The integration instead of from 0 \rightarrow 1 for the usual quantile function is u \rightarrow 1. Note that x(u) is a constant, so

M(u) = \frac{1}{1-u}\int_u^1 x(p)\; \mathrm{d}p - x(u)\mbox{,}

is equivalent and the basis for the implementation in rmlmomco. Assuming that x(F) is a life distribution, the M(u) is interpreted (see Nair et al. [2013, p. 51]) as the average remaining life beyond the 100(1-F)\% of the distribution. Alternatively, M(u) is the mean residual life conditioned that survival to lifetime x(F) has occurred.

If u = 0, then M(0) is the expectation of the life distribution or in otherwords M(0) = λ_1 of the parent quantile function. If u = 1, then M(u) = 0 (death has occurred)—there is zero residual life remaining. The implementation intercepts an intermediate and returns 0 for u = 1.

The M(u) is referred to as a quantile function but this quantity is not to be interpreted as a type of probability distribution. The second example produces a M(u) that is not monotonic increasing with u and therefore it is immediately apparent that M(u) is not the quantile function of some probability distribution by itself. Nair et al. (2013) provide extensive details on quantile-based reliability analysis.


rmlmomco(f, para)



Nonexceedance probability (0 ≤ F ≤ 1).


The parameters from lmom2par or vec2par.


Mean residual value for F.


The Mean Residual Quantile Function is the first of many other functions and “curves” associated with lifetime/reliability analysis operations that at their root use the quantile function (QF, x(F)) of a distribution. Nair et al. (2013) (NSB) is the authoritative text on which the following functions in lmomco were based

Residual mean QF M(u) rmlmomco NSB[p.51]
Variance residual QF V(u) rmvarlmomco NSB[p.54]
α-percentile residual QF P_α(u) ralmomco NSB[p.56]
Reversed α-percentile residual QF R_α(u) rralmomco NSB[p.69--70]
Reversed residual mean QF R(u) rrmlmomco NSB[p.57]
Reversed variance residual QF D(u) rrmvarlmomco NSB[p.58]
Conditional mean QF μ(u) cmlmomco NSB[p.68]
Vitality function (see conditional mean)
Total time on test transform QF T(u) tttlmomco NSB[p.171--172, 176]
Scaled total time on test transform QF φ(u) stttlmomco NSB[p.173]
Lorenz curve L(u) lrzlmomco NSB[p.174]
Bonferroni curve B(u) bfrlmomco NSB[p.179]
Leimkuhler curve K(u) lkhlmomco NSB[p.181]
Income gap ratio curve G(u) riglmomco NSB[p.230]
Mean life: μ \equiv μ(0) \equiv λ_1(u=0) \equiv λ_1
L-moments of residual life λ_r(u) reslife.lmoms NSB[p.202]
L-moments of reversed residual life {}_\mathrm{r}λ_r(u) rreslife.lmoms NSB[p.211]


W.H. Asquith


Kupka, J., and Loo, S., 1989, The hazard and vitality measures of ageing: Journal of Applied Probability, v. 26, pp. 532–542.

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

See Also

qlmomco, cmlmomco, rmvarlmomco


# 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
qlmomco(0.5, A)  # The median lifetime = 1261 days
rmlmomco(0.5, A) # The average remaining life given survival to the median = 861 days

# 2nd example with discussion points
F <- nonexceeds(f01=TRUE)
plot(F, qlmomco(F, A), type="l", # usual quantile plot as seen throughout lmomco
lines(F, rmlmomco(F, A), col=2, lwd=3)           # mean residual life
L1 <- lmomgov(A)$lambdas[1]                      # mean lifetime at start/birth
lines(c(0,1), c(L1,L1), lty=2)                   # line "ML" (mean life)
# Notice how ML intersects M(F|F=0) and again later in "time" (about F = 1/4)  showing
# that this Govindarajulu has a peak mean residual life that is **greater** than the
# expected lifetime at start. The M(F) then tapers off to zero at infinity time (F=1).
# M(F) is non-monotonic for this example---not a proper probability distribution.

lmomco documentation built on Aug. 27, 2022, 1:06 a.m.