# rmlmomco: Mean Residual Quantile Function of the Distributions In lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

## Description

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.

## Usage

 1 rmlmomco(f, para) 

## Arguments

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

## Value

Mean residual value for F.

## Note

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

## References

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.

qlmomco, cmlmomco, rmvarlmomco

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 # 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 xlab="NONEXCEEDANCE PROBABILITY", ylab="LIFETIME, IN DAYS") lines(F, rmlmomco(F, A), col=2, lwd=3) # mean residual life L1 <- lmomgov(A)\$lambdas # 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 as a peak mean resiudal 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 dist. 

### Example output 1261.016
 860.9676


lmomco documentation built on Aug. 4, 2021, 9:06 a.m.