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 - bdgmfor the approach of Beirlant, Dierckx, Goegebeur and Matthys (1999) or- fhfor 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
- knumber of exceedances
- shapeestimate of the shape parameter
- rhoestimate of the second-order regular variation index
- scaleestimate 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>