ProbGH function

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

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

ProbGH(data, gamma, q, plot = FALSE, add = FALSE, 
       main = "Estimates of small exceedance probability", ...)

ReturnGH(data, gamma, q, plot = FALSE, add = FALSE, 
         main = "Estimates of large return period", ...)

Arguments

• data: Vector of $n$ observations.
• gamma: Vector of $n-2$ estimates for the EVI obtained from genHill.
• 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 ProbGH and "Estimates of large return period" for ReturnGH.
• ...: Additional arguments for the plot function, see plot for more details.

Details

See Section 4.2.2 of Albrecher et al. (2017) for more details.

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 ProbGH.

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

• q: The used large quantile.

References

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

Beirlant J., Goegebeur Y., Segers, J. and Teugels, J. (2004). Statistics of Extremes: Theory and Applications, Wiley Series in Probability, Wiley, Chichester.

Beirlant, J., Vynckier, P. and Teugels, J.L. (1996). "Excess Function and Estimation of the Extreme-value Index". Bernoulli, 2, 293--318.

Author(s)

Tom Reynkens.

See Also

QuantGH, genHill, ProbMOM, Prob

Examples

data(soa)

# Look at last 500 observations of SOA data
SOAdata <- sort(soa$size)[length(soa$size)-(0:499)]

# Hill estimator
H <- Hill(SOAdata)
# Generalised Hill estimator
gH <- genHill(SOAdata, H$gamma)

# Exceedance probability
q <- 10^7
ProbGH(SOAdata, gamma=gH$gamma, q=q, plot=TRUE)

# Return period
q <- 10^7
ReturnGH(SOAdata, gamma=gH$gamma, q=q, plot=TRUE)