ProbReg function

Estimator of small tail probability in regression

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.

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

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.

Returns

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.

References

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

Author(s)

Tom Reynkens.

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)