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

## Description

This function computes the Conditional Mean Residual Quantile Function for quantile function x(F) (par2qua, qlmomco). The function is defined by Nair et al. (2013, p. 68) as

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

where μ(u) is the conditional mean for nonexceedance probability u. The μ(u) is the expectation \mathrm{E}[X | X > x]. The μ(u) also is known as the vitality function. Details can be found in Nair et al. (2013, p. 68) and Kupka and Loo (1989). Mathematically, the vitality function simply is

μ(u) = M(u) + x(u)\mbox{,}

where M(u) is the mean residual quantile function (rmlmomco), x(u) is a constant for x(F = u).

## Usage

 1 cmlmomco(f, para) 

## Arguments

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

## Value

Conditional mean residual value for F or conditional mean life for F.

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, rmlmomco
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 # 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 cmlmomco(0.5, A) # The average total life given survival to the median = 2122 days # Now create with a nonzero origin A <- vec2par(c(100, 2649, 2.11), type="gov") # so set lower bounds = 0.0 qlmomco(0.5, A) # The median lifetime = 1361 days rmlmomco(0.5, A) # The average remaining life given survival to the median = 861 days cmlmomco(0.5, A) # The average total life given survival to the median = 2222 days # Mean life (mu), which shows up in several expressions listed under rmlmomco. mu1 <- cmlmomco(0,A) mu2 <- par2lmom(A)$lambdas[1] mu3 <- reslife.lmoms(0,A)$lambdas[1] # Each mu is 1289.051 days.