icHill function

Hill estimator for interval censored data

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", ...)

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 $\gamma$ should be plotted as a function of $k$, default is FALSE.
• add: Logical indicating if the estimates of $\gamma$ 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

$$H^{TB}(x)=(\int_x^{\infty} (1-\hat{F}^{TB}(u))/u du)/(1-\hat{F}^{TB}(x)),$$

where $\hat{F}^{TB}$ is the Turnbull estimator for the CDF. More specifically, we use the values $x=\hat{Q}^{TB}(p)$ for $p=1/(n+1), \ldots, (n-1)/(n+1)$ 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.

Returns

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 $\gamma$.

References

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

Author(s)

Tom Reynkens

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)