View source: R/IntervalCensoring.R
icHill | R Documentation |
Computes the Hill estimator for positive extreme value indices, adapted for interval censoring, as a function of the tail parameter k. Optionally, these estimates are plotted as a function of k.
icHill(L, U, censored, trunclower = 0, truncupper = Inf, logk = FALSE, plot = TRUE, add = FALSE, main = "Hill estimates of the EVI", ...)
L |
Vector of length n with the lower boundaries of the intervals for interval censored data or the observed data for right censored data. |
U |
Vector of length n with the upper boundaries of the intervals. |
censored |
A logical vector of length n indicating if an observation is censored. |
trunclower |
Lower truncation point. Default is 0. |
truncupper |
Upper truncation point. Default is |
logk |
Logical indicating if the estimates are plotted as a function of \log(k) ( |
plot |
Logical indicating if the estimates of γ should be plotted as a function of k, default is |
add |
Logical indicating if the estimates of γ should be added to an existing plot, default is |
main |
Title for the plot, default is |
... |
Additional arguments for the |
This estimator is given by
H^{TB}(x)=(\int_x^{∞} (1-\hat{F}^{TB}(u))/u du)/(1-\hat{F}^{TB}(x)),
where \hat{F}^{TB} is the Turnbull estimator for the CDF. More specifically, we use the values x=\hat{Q}^{TB}(p) for p=1/(n+1), …, (n-1)/(n+1) where \hat{Q}^{TB}(p) is the empirical quantile function corresponding to the Turnbull estimator. We then denote
H^{TB}_{k,n}=H^{TB}(x_{n-k,n})
with
x_{n-k,n}=\hat{Q}^{TB}((n-k)/(n+1))=\hat{Q}^{TB}(1-(k+1)/(n+1)).
Right censored data should be entered as L=l
and U=truncupper
, and left censored data should be entered as L=trunclower
and U=u
.
If the interval package is installed, the icfit
function is used to compute the Turnbull estimator. Otherwise, survfit.formula
from survival is used.
Use Hill
for non-censored data or cHill
for right censored data.
See Section 4.3 in Albrecher et al. (2017) for more details.
A list with following components:
k |
Vector of the values of the tail parameter k. |
gamma |
Vector of the corresponding Hill estimates. |
X |
Vector of thresholds x_{n-k,n} used when estimating γ. |
Tom Reynkens
Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and Statistical Aspects, Wiley, Chichester.
cHill
, Hill
, MeanExcess_TB
, icParetoQQ
, Turnbull
, icfit
# Pareto random sample X <- rpareto(500, shape=2) # Censoring variable Y <- rpareto(500, shape=1) # Observed sample Z <- pmin(X,Y) # Censoring indicator censored <- (X>Y) # Right boundary U <- Z U[censored] <- Inf # Hill estimator adapted for interval censoring icHill(Z, U, censored, ylim=c(0,1)) # Hill estimator adapted for right censoring cHill(Z, censored, lty=2, add=TRUE) # True value of gamma abline(h=1/2, lty=3, col="blue") # Legend legend("topright", c("icHill", "cHill"), lty=1:2)
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