getInfGamma function

Generic Function for the Computation of the Optimal Clipping Bound

Generic Function for the Computation of the Optimal Clipping Bound

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.

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

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.

Details

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

Returns

The optimal clipping height is computed. More specifically, the optimal clipping height bb is determined in a zero search of a certain function gammagamma, where the respective getInf-method will return the value of gamma(b)gamma(b). The actual function gammagamma

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.

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.

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.

Author(s)

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

See Also

asGRisk-class, asMSE-class, asUnOvShoot-class, ContIC-class, TotalVarIC-class