| pargpaRC {lmomco} | R Documentation |
This function estimates the parameters (xi, α, and kappa) of the Generalized Pareto distribution given
the “B”-type L-moments (through the B-type Probability-Weighted Moments) of the data under right censoring conditions (see pwmRC). The “B”-type L-moments in terms of the parameters are
λ^B_1 = xi + α m_1 mbox{,}
λ^B_2 = α (m_1 - m_2) mbox{,}
λ^B_3 = α (m_1 - 3m_2 + 2m_3)mbox{,}
λ^B_4 = α (m_1 - 6m_2 + 10m_3 - 5m_4)mbox{, and}
λ^B_5 = α (m_1 - 10m_2 + 30m_3 - 35m_4 + 14m_5)mbox{,}
noindent where m_r = lbrace 1-(1-zeta)^{r+kappa} rbrace/(r+kappa) and zeta is the right-tail censor fraction or the probability mathrm{Pr}lbrace rbrace that x is less than the quantile at zeta nonexceedance probability: (mathrm{Pr}lbrace x < X(zeta) rbrace). Finally, the RC in the function name is to denote Right-tail Censoring.
pargpaRC(lmom,zeta=1,lower=-1,upper=20,checklmom=TRUE)
lmom |
A B-type L-moment object created by a function such as pwm2lmom from B-type Probability-Weighted Moments from pwmRC. |
zeta |
The censoring fraction. The number of samples observed (noncensored) divided by the total number of samples. |
lower |
The lower value for kappa for a call to the optimize function. For the L-moments of the distribution to be valid kappa > -1. |
upper |
The upper value for kappa for a call to the optimize function. Hopefully, a large enough default is chosen for real-world data sets. |
checklmom |
Should the lmom be checked for validity using the are.lmom.valid function. Normally this should be left as the default and it is very unlikely that the L-moments will not be viable (particularly in the tau_4 and tau_3 inequality). However, for some circumstances or large simulation exercises then one might want to bypass this check. |
The optimize function is used to numerically solve for the shape parameter kappa. No test or evaluation is made on the quality of the minimization. Users should concults the contents of the optim portion of the returned list. Finally, this function should return the same parameters if zeta=1 as the pargpa function.
An R list is returned.
type |
The type of distribution: gpa. |
para |
The parameters of the distribution. |
zeta |
The right-tail censoring fraction. |
source |
The source of the parameters: “pargpaRC”. |
optim |
The list returned by the optimize function. |
W.H. Asquith
Hosking, J.R.M., 1990, L-moments—Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, vol. 52, p. 105–124.
Hosking, J.R.M., 1995, The use of L-moments in the analysis of censored data, in Recent Advances in Life-Testing and Reliability, edited by N. Balakrishnan, chapter 29, CRC Press, Boca Raton, Fla., pp. 546–560.
lmomgpa, lmomgpaRC, pargpa,
cdfgpa, quagpa
n <- 60 # samplesize
para <- vec2par(c(1500,160,.3),type="gpa") # build a GPA parameter set
fakedata <- quagpa(runif(n),para) # generate n simulated values
threshold <- 1700 # a threshold to apply the simulated censoring
fakedata <- sapply(fakedata,function(x) { if(x > threshold)
return(threshold) else return(x)
})
lmr <- lmoms(fakedata) # Ordinary L-moments without considering
# that the data is censored
estpara <- pargpa(lmr) # Estimated parameters of parent
pwm2 <- pwmRC(fakedata,threshold=threshold) # compute censored PWMs
typeBpwm <- pwm2$Bbetas # the B-type PWMs
zeta <- pwm2$zeta # the censoring fraction
cenpara <- pargpaRC(pwm2lmom(typeBpwm),zeta=zeta) # Estimated parameters
F <- nonexceeds() # nonexceedance probabilities for plotting purposes
# Visualize some data
plot( F,quagpa(F,para),type='l', lwd=3) # The true distribution
lines( F,quagpa(F,estpara),col=3) # Green estimated in the ordinary fashion
lines(F,quagpa(F,cenpara),col=2) # Red, consider that the data is censored
# now add in what the drawn sample looks like.
PP <- pp(fakedata) # plotting positions of the data
points(PP,sort(fakedata)) # sorting is needed!
# Interpretation. You should see that the red line more closely matches
# the heavy black line. The green line should be deflected to the right
# and pass through the values equal to the threshold, which reflects the
# much smaller L-skew of the ordinary L-moments compared to the type-B
# L-moments.
# Assertion, given some PWMs or L-moments, if zeta=1 then the parameter
# estimates must be identifical. The following provides a demonstration.
para1 <- pargpaRC(pwm2lmom(typeBpwm),zeta=1)
para2 <- pargpa(pwm2lmom(typeBpwm))
str(para1)
str(para2)