Quant2o function

Second order refined Weissman estimator of extreme quantiles (QV)

Compute second order refined Weissman estimator of extreme quantiles $Q(1-p)$ using the quantile view.

Quant.2oQV(data, gamma, b, beta, p, plot = FALSE, add = FALSE, 
           main = "Estimates of extreme quantile", ...)
                
Weissman.q.2oQV(data, gamma, b, beta, 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 obtained from Hill.2oQV.
• b: Vector of $n-1$ estimates for $b$ obtained from Hill.2oQV.
• beta: Vector of $n-1$ estimates for $\beta$ obtained from Hill.2oQV.
• 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 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.2oQV is the same function but with a different name for compatibility with the old S-Plus code.

Returns

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.

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.

Author(s)

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

See Also

Quant, Hill.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)