icHill: Hill estimator for interval censored data

In ReIns: Functions from "Reinsurance: Actuarial and Statistical Aspects"

Hill estimator for interval censored data

Description

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.

Usage

icHill(L, U, censored, trunclower = 0, truncupper = Inf, 
       logk = FALSE, plot = TRUE, add = FALSE, main = "Hill estimates of the EVI", ...)

Arguments

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 Inf (no upper truncation).
logk	Logical indicating if the estimates are plotted as a function of \log(k) (logk=TRUE) or as a function of k. Default is FALSE.
plot	Logical indicating if the estimates of γ should be plotted as a function of k, default is FALSE.
add	Logical indicating if the estimates of γ should be added to an existing plot, default is FALSE.
main	Title for the plot, default is "Hill estimates of the EVI".
...	Additional arguments for the plot function, see plot for more details.

Details

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.

Value

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.

Author(s)

Tom Reynkens

References

Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and Statistical Aspects, Wiley, Chichester.

See Also

cHill, Hill, MeanExcess_TB, icParetoQQ, Turnbull, icfit

Examples

# 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)

ReIns documentation built on Feb. 16, 2023, 9:42 p.m.