Weighted asymptotic mean squared error (AMSE) estimator
Description
Estimate the scale and shape parameters of a Pareto distribution with an iterative procedure based on minimizing the weighted asymptotic mean squared error (AMSE) of the Hill estimator.
Usage
1 2 
Arguments
x 
for 
weight 
a character vector specifying the weighting scheme to be used
in the procedure. If 
kmin 
An optional integer giving the lower bound for finding the
optimal number of observations in the tail. It defaults to
[n/100], where n denotes the number of
observations in 
kmax 
An optional integer giving the upper bound for finding the optimal number of observations in the tail (see “Details”). 
mmax 
An optional integer giving the upper bound for finding the optimal number of observations for computing the nuisance parameter rho (see “Details” and the references). 
tol 
an integer giving the desired tolerance level for finding the optimal number of observations in the tail. 
maxit 
a positive integer giving the maximum number of iterations. 
... 
additional arguments to be passed to

Details
The weights used in the weighted AMSE depend on a nuisance parameter
rho. Both the optimal number of observations in the tail and the
nuisance parameter rho are estimated iteratively using nonlinear
integer minimization. This is currently done by a brute force algorithm,
hence it is stronly recommended to supply upper bounds kmax
and
mmax
.
See the references for more details on the iterative algorithm.
Value
An object of class "minAMSE"
containing the following
components:
kopt 
the optimal number of observations in the tail. 
x0 
the corresponding threshold. 
theta 
the estimated shape parameter of the Pareto distribution. 
MSEmin 
the minimal MSE. 
rho 
the estimated nuisance parameter. 
k 
the examined range for the number of observations in the tail. 
MSE 
the corresponding MSEs. 
Author(s)
Josef Holzer and Andreas Alfons
References
Beirlant, J., Vynckier, P. and Teugels, J.L. (1996) Tail index estimation, Pareto quantile plots, and regression diagnostics. Journal of the American Statistical Association, 91(436), 1659–1667.
Beirlant, J., Vynckier, P. and Teugels, J.L. (1996) Excess functions and estimation of the extremevalue index. Bernoulli, 2(4), 293–318.
Dupuis, D.J. and VictoriaFeser, M.P. (2006) A robust prediction error criterion for Pareto modelling of upper tails. The Canadian Journal of Statistics, 34(4), 639–658.
See Also
thetaHill
Examples
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