# Estimate the Parameters of the Generalized Pareto Distribution with Right-Tail Censoring This function estimates the parameters ($\xi$, $\alpha$, 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 relations between distribution parameters and L-moments are seen under `lmomgpaRC`. ```r pargpaRC(lmom, zeta=1, xi=NULL, lower=-1, upper=20, checklmom=TRUE, ...) ``` ## Arguments - `lmom`: A B-type L-moment object created by a function such as `pwm2lmom` from B-type probability-weighted moments from `pwmRC`. - `zeta`: The compliment of the right-tail censoring fraction. The number of samples observed (noncensored) divided by the total number of samples. - `xi`: The lower limit of the distribution. If $\xi$ is known, then alternative algorithms are used. - `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. - `...`: Other arguments to pass. ## Returns An `list` is returned. - **type**: The type of distribution: `gpa`. - **para**: The parameters of the distribution. - **zeta**: The compliment of the right-tail censoring fraction. - **source**: The source of the parameters: pargpaRC . - **optim**: The `list` returned by the function `optimize`. ## Details 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 consult 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. ## References 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, v. 52, pp. 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. ## Author(s) W.H. Asquith ## See Also `lmomgpa`, `lmomgpaRC`, `pargpa`, `cdfgpa`, `pdfgpa`, `quagpa` ## Examples ```r 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 identical. The following provides a demonstration. para1 <- pargpaRC(pwm2lmom(typeBpwm),zeta=1) para2 <- pargpa(pwm2lmom(typeBpwm)) str(para1); str(para2) # Assertion as previous assertion, let us trigger different optimizer # algorithms with a non-NULL xi parameter and see if the two parameter # lists are the same. para1 <- pargpaRC(pwm2lmom(typeBpwm), zeta=zeta) para2 <- pargpaRC(pwm2lmom(typeBpwm), xi=para1$para[1], zeta=zeta) str(para1); str(para2) ```

pargpaRC function