Description Usage Arguments Value Author(s) References See Also Examples
This function computes the Lmoments of reversed residual life for a quantile function x(F) for an exceedance threshold in probabiliy of u. The Lmoments of residual life are thoroughly described by Nair et al. (2013, p. 211). These Lmoments are define as
{}_\mathrm{r}λ(u)_r = ∑_{k=0}^{r1} (1)^k {r1 \choose k}^2 \int_0^u ≤ft(\frac{p}{u}\right)^{rk1} ≤ft(1  \frac{p}{u}\right)^k \frac{x(p)}{u}\,\mathrm{d}p \mbox{,}
where {}_\mathrm{r}λ(u)_r is the rth Lmoment at residual life probability u. The Lmoment 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 Lmoments 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 Lmoment order r or the TLmoments that indicate trimming level as a superscript (see TLmoms
).
1  rreslife.lmoms(f, para, nmom=5)

f 
Nonexceedance probability (0 ≤ F ≤ 1). 
para 
The parameters from 
nmom 
The number of moments to compute. Default is 5. 
An R list
is returned.
lambdas 
Vector of the Lmoments. First element is {}_\mathrm{r}λ_1, second element is {}_\mathrm{r}λ_2, and so on. 
ratios 
Vector of the Lmoment 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= 
life.percentile 
The value 100\times 
trim 
Level of symmetrical trimming used in the computation, which is 
leftrim 
Level of lefttail trimming used in the computation, which is 
rightrim 
Level of righttail trimming used in the computation, which is 
source 
An attribute identifying the computational source of the Lmoments: “rreslife.lmoms”. 
W.H. Asquith
Nair, N.U., Sankaran, P.G., and Balakrishnan, N., 2013, Quantilebased reliability analysis: Springer, New York.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25  # 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 Lskews 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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