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.
minAMSE(
x,
weight = c("Bernoulli", "JASA"),
kmin,
kmax,
mmax,
tol = 0,
maxit = 100
)# S3 method for minAMSE
print(x, ...)
An object of class "minAMSE" with the following components:
the optimal number of observations in the tail.
the corresponding threshold.
the estimated shape parameter of the Pareto distribution.
the minimal MSE.
the estimated nuisance parameter.
the examined range for the number of observations in the tail.
the corresponding MSEs.
for minAMSE, a numeric vector. The print method is
called by the generic function if an object of class "minAMSE" is
supplied.
a character vector specifying the weighting scheme to be used
in the procedure. If "Bernoulli", the weight functions as described
in the Bernoulli paper are applied. If "JASA", the weight
functions as described in the Journal of the Americal Statistical
Association are used.
An optional integer giving the lower bound for finding the
optimal number of observations in the tail. It defaults to
\([\frac{n}{100}]\), where \(n\) denotes the number of
observations in x (see the references).
An optional integer giving the upper bound for finding the optimal number of observations in the tail (see “Details”).
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).
an integer giving the desired tolerance level for finding the optimal number of observations in the tail.
a positive integer giving the maximum number of iterations.
additional arguments to be passed to
print.default.
Josef Holzer and Andreas Alfons
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.
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 extreme-value index. Bernoulli, 2(4), 293--318.
Dupuis, D.J. and Victoria-Feser, M.-P. (2006) A robust prediction error criterion for Pareto modelling of upper tails. The Canadian Journal of Statistics, 34(4), 639--658.
thetaHill
data(eusilc)
# equivalized disposable income is equal for each household
# member, therefore only one household member is taken
minAMSE(eusilc$eqIncome[!duplicated(eusilc$db030)],
kmin = 60, kmax = 150, mmax = 250)
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