L-moments of Reversed Residual Life

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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}λ(u)_r = ∑_{k=0}^{r-1} (-1)^k {r-1 \choose k}^2 \int_0^u ≤ft(\frac{p}{u}\right)^{r-k-1} ≤ft(1 - \frac{p}{u}\right)^k \frac{x(p)}{u}\,\mathrm{d}p \mbox{,}

where {}_\mathrm{r}λ(u)_r is the rth L-moment at residual life probability u. The L-moment ratios {}_\mathrm{r}τ(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}λ(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}λ(1)_r for r ≥ 2 are set to NA. The reversal aspect is denoted by the prepended romanscript \mathrm{r} to the λ's and τ'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

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

Arguments

f

Nonexceedance probability (0 ≤ F ≤ 1).

para

The parameters from lmom2par or vec2par.

nmom

The number of moments to compute. Default is 5.

Value

An R list is returned.

lambdas

Vector of the L-moments. First element is {}_\mathrm{r}λ_1, second element is {}_\mathrm{r}λ_2, and so on.

ratios

Vector of the L-moment ratios. Second element is {}_\mathrm{r}τ, third element is {}_\mathrm{r}τ_3 and so on.

life.notexceeds

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

life.percentile

The value 100\timesf.

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”.

Author(s)

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

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# 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))
## Not run: 
# 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]

## End(Not run)
## Not run: 
# 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]

## End(Not run)

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