ProbReg: Estimator of small tail probability in regression

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

Estimator of small tail probability in regression

Description

Estimator of small tail probability 1-F_i(q) in the regression case where \gamma is constant and the regression modelling is thus only solely placed on the scale parameter.

Usage

ProbReg(Z, A, q, plot = FALSE, add = FALSE, 
        main = "Estimates of small 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.
q	The used large quantile (we estimate P(X_i>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".
...	Additional arguments for the plot function, see plot for more details.

Details

The estimator is defined as

1-\hat{F}_i(q) = \hat{A}(i/n) (k+1)/(n+1) (q/Z_{n-k,n})^{-1/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.

Value

A list with following components:

k	Vector of the values of the tail parameter k.
P	Vector of the corresponding probability estimates.
q	The used large quantile.

Author(s)

Tom Reynkens.

References

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

See Also

QuantReg, ScaleReg, Prob

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)

ReIns documentation built on Nov. 3, 2023, 5:08 p.m.