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Exponential regression estimator

Source: R/erm.R
shape.erm.Rd

This function implements the exponential regression estimator of the shape parameter for the case of Pareto tails with positive shape index.

Usage

shape.erm(xdat, k, method = c("bdgm", "fh"), bounds = NULL)

Arguments

xdat

vector of observations

k

vector of integer, the number of largest observations to consider

method

string; one of bdgm for the approach of Beirlant, Dierckx, Goegebeur and Matthys (1999) or fh for Feuerverger and Hall (1999)

bounds

vector of length 2 giving the bounds for rho, the second order parameter. Default to \(\rho \in [-5, -0.5]\)

Value

a data frame with columns

  • k number of exceedances

  • shape estimate of the shape parameter

  • rho estimate of the second-order regular variation index

  • scale estimate of the scale parameter

Details

The second-order parameter is difficult to pin down, and while values within \([-1,0)\) are most logical under Hall model, the model parameters become unidentifiable when \(\rho \to 0\). The default constraint restrict \(-5 <\rho < -0.5\), with the upper bound changed to \(-0.25\) for sample of sizes larger than 5000 observations. Users can set the value of the bounds for \(\rho\) via argument bounds. The optimization is initialized at the Hill estimator.

References

Feuerverger, A. and P. Hall (1999), Estimating a tail exponent by modelling departure from a Pareto distribution, The Annals of Statistics 27(2), 760-781. <doi:10.1214/aos/1018031215>

Beirlant, J., Dierckx, G., Goegebeur, Y. G. Matthys (1999). Tail Index Estimation and an Exponential Regression Model. Extremes, 2, 177–200 (1999). <doi:10.1023/A:1009975020370>