getInfGamma: Generic Function for the Computation of the Optimal Clipping Bound

Description

Generic function for the computation of the optimal clipping bound. This function is rarely called directly. It is called by getInfClip to compute optimally robust ICs.

Usage

getInfGamma(L2deriv, risk, neighbor, biastype, ...)
# S4 method for UnivariateDistribution,asGRisk,ContNeighborhood,BiasType
getInfGamma(L2deriv, 
     risk, neighbor, biastype, cent, clip)
# S4 method for UnivariateDistribution,asGRisk,TotalVarNeighborhood,BiasType
getInfGamma(L2deriv, 
     risk, neighbor, biastype, cent, clip)
# S4 method for RealRandVariable,asMSE,ContNeighborhood,BiasType
getInfGamma(L2deriv, 
     risk, neighbor, biastype, Distr, stand, cent, clip, power = 1L, ...)
# S4 method for RealRandVariable,asMSE,TotalVarNeighborhood,BiasType
getInfGamma(L2deriv,
     risk, neighbor, biastype, Distr, stand, cent, clip, power = 1L, ...)
# S4 method for UnivariateDistribution,asUnOvShoot,ContNeighborhood,BiasType
getInfGamma(L2deriv,
     risk, neighbor, biastype, cent, clip)
# S4 method for UnivariateDistribution,asMSE,ContNeighborhood,onesidedBias
getInfGamma(L2deriv, 
     risk, neighbor, biastype, cent, clip)
# S4 method for UnivariateDistribution,asMSE,ContNeighborhood,asymmetricBias
getInfGamma(L2deriv, 
    risk, neighbor, biastype, cent, clip)

Value

The optimal clipping height is computed. More specifically, the optimal clipping height \(b\) is determined in a zero search of a certain function

\(\gamma\), where the respective getInf-method will return the value of \(\gamma(b)\). The actual function \(\gamma\)

varies according to whether the parameter is one dimensional or higher dimensional, according to the risk, according to the neighborhood, and according to the bias type, which leads to the different methods.

Arguments

L2deriv: L2-derivative of some L2-differentiable family of probability measures.
risk: object of class "RiskType".
neighbor: object of class "Neighborhood".
biastype: object of class "BiasType".
...: additional parameters, in particular for expectation E.
cent: optimal centering constant.
clip: optimal clipping bound.
stand: standardizing matrix.
Distr: object of class "Distribution".
power: exponent for the integrand; by default 1, but may also be 2, for optimization in getLagrangeMultByOptim.

Methods

L2deriv = "UnivariateDistribution", risk = "asGRisk", neighbor = "ContNeighborhood", biastype = "BiasType": used by getInfClip for symmetric bias.
L2deriv = "UnivariateDistribution", risk = "asGRisk", neighbor = "TotalVarNeighborhood", biastype = "BiasType": used by getInfClip for symmetric bias.
L2deriv = "RealRandVariable", risk = "asMSE", neighbor = "ContNeighborhood", biastype = "BiasType": used by getInfClip for symmetric bias.
L2deriv = "RealRandVariable", risk = "asMSE", neighbor = "TotalVarNeighborhood", biastype = "BiasType": used by getInfClip for symmetric bias.
L2deriv = "UnivariateDistribution", risk = "asUnOvShoot", neighbor = "ContNeighborhood", biastype = "BiasType": used by getInfClip for symmetric bias.
L2deriv = "UnivariateDistribution", risk = "asMSE", neighbor = "ContNeighborhood", biastype = "onesidedBias": used by getInfClip for onesided bias.
L2deriv = "UnivariateDistribution", risk = "asMSE", neighbor = "ContNeighborhood", biastype = "asymmetricBias": used by getInfClip for asymmetric bias.

Author

Matthias Kohl Matthias.Kohl@stamats.de, Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de

Details

The function is used in case of asymptotic G-risks; confer Ruckdeschel and Rieder (2004).

References

Rieder, H. (1980) Estimates derived from robust tests. Ann. Stats. 8: 106--115.

Rieder, H. (1994) Robust Asymptotic Statistics. New York: Springer.

Ruckdeschel, P. and Rieder, H. (2004) Optimal Influence Curves for General Loss Functions. Statistics & Decisions 22, 201-223.

Ruckdeschel, P. (2005) Optimally One-Sided Bounded Influence Curves. Mathematical Methods in Statistics 14(1), 105-131.

Kohl, M. (2005) Numerical Contributions to the Asymptotic Theory of Robustness. Bayreuth: Dissertation.