cProbEPD function

Estimator of small exceedance probabilities and large return periods using censored EPD

Computes estimates of a small exceedance probability $P(X>q)$ or large return period $1/P(X>q)$ using the parameters from the EPD fit adapted for right censoring.

cProbEPD(data, censored, gamma1, kappa1, beta, q, plot = FALSE, add = FALSE,
         main = "Estimates of small exceedance probability", ...)

cReturnEPD(data, censored, gamma1, kappa1, beta, 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 cEPD.

• kappa1: Vector of $n-1$ estimates for $\kappa_1$ obtained from cEPD.

• beta: Vector of $n-1$ estimates for $\beta$ obtained from cEPD.

• 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 cProbEPD

  and "Estimates of large return period" for cReturnEPD.

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

Details

The probability is estimated as

with $F$

the CDF of the EPD with estimated parameters $\hat{\gamma}_1$, $\hat{\kappa}_1$ and $\hat{\tau}=-\hat{\beta}$

and $km$ the Kaplan-Meier estimator for the CDF evaluated in $Z_{n-k,n}$ (the $(k+1)$-th largest data point).

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 cProbEPD.

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

• q: The used large quantile.

References

Beirlant, J., Bardoutsos, A., de Wet, T. and Gijbels, I. (2016). "Bias Reduced Tail Estimation for Censored Pareto Type Distributions." Statistics & Probability Letters, 109, 78--88.

Author(s)

Tom Reynkens.

See Also

cEPD, ProbEPD, Prob, 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)

# EPD estimator adapted for right censoring
cepd <- cEPD(Z, censored=censored, plot=TRUE)

# Small exceedance probability
q <- 10
cProbEPD(Z, censored=censored, gamma1=cepd$gamma1,
        kappa1=cepd$kappa1, beta=cepd$beta, q=q, plot=TRUE)

# Return period
cReturnEPD(Z, censored=censored, gamma1=cepd$gamma1,
        kappa1=cepd$kappa1, beta=cepd$beta, q=q, plot=TRUE)