Weighted asymptotic mean squared error (AMSE) estimator

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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

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minAMSE(x, weight = c("Bernoulli", "JASA"), kmin, kmax, mmax, tol = 0,
  maxit = 100)

Arguments

x

for minAMSE, a numeric vector. The print method is called by the generic function if an object of class "minAMSE" is supplied.

weight

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.

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 x (see the references).

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 print.default.

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 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.

See Also

thetaHill

Examples

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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 = 50, kmax = 150, mmax = 250)

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