cProbGH function

Estimator of small exceedance probabilities and large return periods using censored generalised Hill

Computes estimates of a small exceedance probability $P(X>q)$ or large return period $1/P(X>q)$ using the estimates for the EVI obtained from the generalised Hill estimator adapted for right censoring.

cProbGH(data, censored, gamma1, q, plot = FALSE, add = FALSE, 
        main = "Estimates of small exceedance probability", ...)

cReturnGH(data, censored, gamma1, q, plot = FALSE, add = FALSE, 
          main = "Estimates of large return period", ...)

Arguments

• data: Vector of $n$ observations.

• censored: A logical vector of length $n$ indicating if an observation is censored.

• gamma1: Vector of $n-1$ estimates for the EVI obtained from cgenHill.

• q: The used large quantile (we estimate $P(X>q)$ or $1/P(X>q)$ for $q$ large).

• 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 "Estimates of small exceedance probability" for cProbGH

  and "Estimates of large return period" for cReturnGH.

• ...: Additional arguments for the plot function, see plot for more details.

Details

The probability is estimated as

with $Z_{i,n}$ the $i$-th order statistic of the data, $\hat{\gamma}_1$ the generalised Hill estimator adapted for right censoring and $km$ the Kaplan-Meier estimator for the CDF evaluated in $Z_{n-k,n}$. The value $a$ is defined as

with $H_{k,n}$ the ordinary Hill estimator and $\hat{p}_k$ the proportion of the $k$ largest observations that is non-censored.

Returns

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

• P: Vector of the corresponding probability estimates, only returned for cProbGH.

• R: Vector of the corresponding estimates for the return period, only returned for cReturnGH.

• q: The used large quantile.

References

Einmahl, J.H.J., Fils-Villetard, A. and Guillou, A. (2008). "Statistics of Extremes Under Random Censoring." Bernoulli, 14, 207--227.

Author(s)

Tom Reynkens

See Also

cQuantGH, cgenHill, ProbGH, cProbMOM, KaplanMeier

Examples

# Set seed
set.seed(29072016)

# 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)

# Generalised Hill estimator adapted for right censoring
cghill <- cgenHill(Z, censored=censored, plot=TRUE)

# Small exceedance probability
q <- 10
cProbGH(Z, censored=censored, gamma1=cghill$gamma1, q=q, plot=TRUE)

# Return period
cReturnGH(Z, censored=censored, gamma1=cghill$gamma1, q=q, plot=TRUE)