Quant: Weissman estimator of extreme quantiles

In TReynkens/ReIns: Functions from "Reinsurance: Actuarial and Statistical Aspects"

Weissman estimator of extreme quantiles

Description

Compute estimates of an extreme quantile Q(1-p) using the approach of Weissman (1978).

Usage

Quant(data, gamma, p, plot = FALSE, add = FALSE, 
      main = "Estimates of extreme quantile", ...)
            
Weissman.q(data, gamma, p, plot = FALSE, add = FALSE, 
           main = "Estimates of extreme quantile", ...)

Arguments

data	Vector of n observations.
gamma	Vector of n-1 estimates for the EVI, typically Hill estimates are used.
p	The exceedance probability of the quantile (we estimate Q(1-p) for p small).
plot	Logical indicating if the estimates should be plotted as a function of k, default is FALSE.
add	Logical indicating if the estimates as a function of k should be added to an existing plot, default is FALSE.
main	Title for the plot, default is "Estimates of extreme quantile".
...	Additional arguments for the plot function, see plot for more details.

Details

See Section 4.2.1 of Albrecher et al. (2017) for more details.

Weissman.q is the same function but with a different name for compatibility with the old S-Plus code.

Value

A list with following components:

k	Vector of the values of the tail parameter k.
Q	Vector of the corresponding quantile estimates.
p	The used exceedance probability.

Author(s)

Tom Reynkens based on S-Plus code from Yuri Goegebeur.

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.

Weissman, I. (1978). "Estimation of Parameters and Large Quantiles Based on the k Largest Observations." Journal of the American Statistical Association, 73, 812–815.

See Also

Prob, Quant.2oQV

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)
# Bias-reduced estimator (QV)
H_QV <- Hill.2oQV(SOAdata)

# Exceedance probability
p <- 10^(-5)
# Weissman estimator
Quant(SOAdata, gamma=H$gamma, p=p, plot=TRUE)

# Second order Weissman estimator (QV)
Quant.2oQV(SOAdata, gamma=H_QV$gamma, beta=H_QV$beta, b=H_QV$b, p=p,
           add=TRUE, lty=2)

TReynkens/ReIns documentation built on Nov. 9, 2023, 1:29 p.m.