ReIns: icHill – R documentation

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icHill

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.

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

Functions from "Reinsurance: Actuarial and Statistical Aspects"

v1.0.10

GPL (>= 2)

Authors

Tom Reynkens [aut, cre] (<https://orcid.org/0000-0002-5516-5107>), Roel Verbelen [aut] (R code for Mixed Erlang distribution, <https://orcid.org/0000-0002-2347-9240>), Anastasios Bardoutsos [ctb] (Original R code for cEPD estimator), Dries Cornilly [ctb] (Original R code for EVT estimators for truncated data), Yuri Goegebeur [ctb] (Original S-Plus code for basic EVT estimators), Klaus Herrmann [ctb] (Original R code for GPD estimator)

Initial release

2020-05-16