crHill function

Hill-type estimator for the conditional EVI

Hill-type estimator for the conditional Extreme Value Index (EVI) adapted for censored data.

crHill(x, Xtilde, Ytilde, censored, h, 
       kernel = c("biweight", "normal", "uniform", "triangular", "epanechnikov"), 
       logk = FALSE, plot = FALSE, add = FALSE, main = "", ...)

Arguments

• x: Value of the conditioning variable $X$ to estimate the EVI at.
• Xtilde: Vector of length $n$ containing the censored sample of the conditioning variable $X$.
• Ytilde: Vector of length $n$ containing the censored sample of the variable $Y$.
• censored: A logical vector of length $n$ indicating if an observation is censored.
• h: Bandwidth of the non-parametric estimator.
• kernel: Kernel of the non-parametric estimator. One of "biweight" (default), "normal", "uniform", "triangular" and "epanechnikov".
• logk: Logical indicating if the Hill-type 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 should be plotted as a function of $k$, default is FALSE.
• add: Logical indicating if the estimates should be added to an existing plot, default is FALSE.
• main: Title for the plot, default is "" (no title).
• ...: Additional arguments for the plot function, see plot for more details.

Details

This is a Hill-type estimator of the EVI of $Y$ given $X=x$. The estimator uses the censored sample $(\tilde{X}_i, \tilde{Y}_i)$, for $i=1,\ldots,n$, where $X$ and $Y$ are censored at the same time. We assume that $Y$ and the censoring variable are conditionally independent given $X$.

See Section 4.4.3 in Albrecher et al. (2017) for more details.

Returns

A list with following components: - k: Vector of the values of the tail parameter $k$.

• gamma: Vector of the corresponding Hill-type estimates.

References

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

Author(s)

Tom Reynkens

See Also

crParetoQQ, crSurv, cHill

Examples

# Set seed
set.seed(29072016)

# Pareto random sample
Y <- rpareto(200, shape=2)

# Censoring variable
C <- rpareto(200, shape=1)

# Observed (censored) sample of variable Y
Ytilde <- pmin(Y, C)

# Censoring indicator
censored <- (Y>C)

# Conditioning variable
X <- seq(1, 10, length.out=length(Y))

# Observed (censored) sample of conditioning variable
Xtilde <- X
Xtilde[censored] <- X[censored] - runif(sum(censored), 0, 1)

# Conditional Pareto QQ-plot
crParetoQQ(x=1, Xtilde=Xtilde, Ytilde=Ytilde, censored=censored, h=2)

# Plot Hill-type estimates
crHill(x=1, Xtilde, Ytilde, censored, h=2, plot=TRUE)