lmomco (version 2.4.14)

rreslife.lmoms: L-moments of Reversed Residual Life

Description

This function computes the L-moments of reversed residual life for a quantile function \(x(F)\) for an exceedance threshold in probabiliy of \(u\). The L-moments of residual life are thoroughly described by Nair et al. (2013, p. 211). These L-moments are define as $${}_\mathrm{r}\lambda(u)_r = \sum_{k=0}^{r-1} (-1)^k {r-1 \choose k}^2 \int_0^u \left(\frac{p}{u}\right)^{r-k-1} \left(1 - \frac{p}{u}\right)^k \frac{x(p)}{u}\,\mathrm{d}p \mbox{,}$$ where \({}_\mathrm{r}\lambda(u)_r\) is the \(r\)th L-moment at residual life probability \(u\). The L-moment ratios \({}_\mathrm{r}\tau(u)_r\) have the usual definitions. The implementation here exclusively uses the quantile function of the distribution. If \(u=0\), then the usual L-moments of the quantile function are returned because the integration domain is the entire potential lifetime range. If \(u=0\), then \({}_\mathrm{r}\lambda(1)_1 = x(0)\) is returned, which is the minimum lifetime of the distribution (the value for the lower support of the distribution), and the remaining \({}_\mathrm{r}\lambda(1)_r\) for \(r \ge 2\) are set to NA. The reversal aspect is denoted by the prepended romanscript \(\mathrm{r}\) to the \(\lambda\)'s and \(\tau\)'s. Lastly, the notation \((u)\) is neither super or subscripted to avoid confusion with L-moment order \(r\) or the TL-moments that indicate trimming level as a superscript (see TLmoms).

Usage

rreslife.lmoms(f, para, nmom=5)

Value

An R

list is returned.

lambdas

Vector of the L-moments. First element is \({}_\mathrm{r}\lambda_1\), second element is \({}_\mathrm{r}\lambda_2\), and so on.

ratios

Vector of the L-moment ratios. Second element is \({}_\mathrm{r}\tau\), third element is \({}_\mathrm{r}\tau_3\) and so on.

life.notexceeds

The value for \(x(F)\) for \(F=\) f.

life.percentile

The value \(100\times\)f.

trim

Level of symmetrical trimming used in the computation, which is NULL because no trimming theory for L-moments of residual life have been developed or researched.

leftrim

Level of left-tail trimming used in the computation, which is NULL because no trimming theory for L-moments of residual life have been developed or researched.

rightrim

Level of right-tail trimming used in the computation, which is NULL because no trimming theory for L-moments of residual life have been developed or researched.

source

An attribute identifying the computational source of the L-moments: “rreslife.lmoms”.

Arguments

f

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

para

The parameters from lmom2par or vec2par.

nmom

The number of moments to compute. Default is 5.

Author

W.H. Asquith

References

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

See Also

rmlmomco, reslife.lmoms

Examples

# 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
"afunc" <- function(p)        { return(par2qua(p,A,paracheck=FALSE)) }
"bfunc" <- function(p,u=NULL) { return((2*p - u)*par2qua(p,A,paracheck=FALSE)) }
f <- 0.35
rL1a <- integrate(afunc, lower=0, upper=f)$value      / f   # Nair et al. (2013, eq. 6.18)
rL2a <- integrate(bfunc, lower=0, upper=f, u=f)$value / f^2 # Nair et al. (2013, eq. 6.19)
rL <- rreslife.lmoms(f, A, nmom=2) # The data.frame shows equality of the two approaches.
rL1b <- rL$lambdas[1]; rL2b <- rL$lambdas[2]
print(data.frame(rL1a=rL1a, rL1b=rL1b, rL2b=rL2b, rL2b=rL2b))
if (FALSE) {
# 2nd Example, let us look at Tau3, each of the L-skews are the same.
T3    <- par2lmom(A)$ratios[3]
T3.0  <-  reslife.lmoms(0, A)$ratios[3]
rT3.1 <- rreslife.lmoms(1, A)$ratios[3]
}
if (FALSE) {
# Nair et al. (2013, p. 212), test shows rL2(u=0.77) = 12.6034
A <- vec2par(c(230, 269, 3.3), type="gpa"); F <- 0.77
"afunc" <- function(p) { return(p*rrmlmomco(p,A)) }
rL2u1 <- (F)^(-2)*integrate(afunc,0,F)$value
rL2u2 <- rreslife.lmoms(F,A)$lambdas[2]
}