rmlmomco: Mean Residual Quantile Function of the 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 $M(0) = \lambda_1$ 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 $\infty$ 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

rmlmomco(f, para)

Value

Mean residual value for $F$.

Arguments

f: Nonexceedance probability ($0 \le F \le 1$).
para: The parameters from lmom2par or vec2par.

Author

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.

Examples

Run this code

# 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[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.

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