ReIns (version 1.0.13)

QuantReg: Estimator of extreme quantiles in regression

Description

Estimator of extreme quantile \(Q_i(1-p)\) in the regression case where \(\gamma\) is constant and the regression modelling is thus only solely placed on the scale parameter.

Usage

QuantReg(Z, A, p, plot = FALSE, add = FALSE, 
         main = "Estimates of extreme quantile", ...)

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.

Arguments

Z

Vector of \(n\) observations (from the response variable).

A

Vector of \(n-1\) estimates for \(A(i/n)\) obtained from ScaleReg.

p

The exceedance probability of the quantile (we estimate \(Q_i(1-p)\) 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.

Author

Tom Reynkens.

Details

The estimator is defined as $$\hat{Q}_i(1-p) = Z_{n-k,n} ((k+1)/((n+1)\times p) \hat{A}(i/n))^{H_{k,n}},$$ with \(H_{k,n}\) the Hill estimator. Here, it is assumed that we have equidistant covariates \(x_i=i/n\).

See Section 4.4.1 in Albrecher et al. (2017) for more details.

References

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

See Also

ProbReg, ScaleReg, Quant

Examples

data(norwegianfire)

Z <- norwegianfire$size[norwegianfire$year==76]

i <- 100
n <- length(Z)

# Scale estimator in i/n
A <- ScaleReg(i/n, Z, h=0.5, kernel = "epanechnikov")$A

# Small exceedance probability
q <- 10^6
ProbReg(Z, A, q, plot=TRUE)

# Large quantile
p <- 10^(-5)
QuantReg(Z, A, p, plot=TRUE)