ReIns (version 1.0.10)

ScaleReg: Scale estimator in regression

Description

Estimator of the scale parameter in the regression case where \(\gamma\) is constant and the regression modelling is thus placed solely on the scale parameter.

Usage

ScaleReg(s, Z, kernel = c("normal", "uniform", "triangular", "epanechnikov", "biweight"), 
         h, plot = TRUE, add = FALSE, main = "Estimates of scale parameter", ...)

Value

A list with following components:

k

Vector of the values of the tail parameter \(k\).

A

Vector of the corresponding scale estimates.

Arguments

s

Point to evaluate the scale estimator in.

Z

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

kernel

The kernel used in the estimator. One of "normal" (default), "uniform", "triangular", "epanechnikov" and "biweight".

h

The bandwidth used in the kernel function.

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 scale parameter".

...

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

Author

Tom Reynkens

Details

The scale estimator is computed as $$\hat{A}(s) = 1/(k+1) \sum_{i=1}^n 1_{Z_i>Z_{n-k,n}} K_h(s-i/n)$$ with \(K_h(x)=K(x/h)/h,\) \(K\) the kernel function and \(h\) the bandwidth. 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, QuantReg, scale, Hill

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)