# minmaxBias: Generic Function for the Computation of Bias-Optimally Robust... In ROptEst: Optimally Robust Estimation

## Description

Generic function for the computation of bias-optimally robust ICs in case of infinitesimal robust models. This function is rarely called directly.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30``` ```minmaxBias(L2deriv, neighbor, biastype, ...) ## S4 method for signature 'UnivariateDistribution,ContNeighborhood,BiasType' minmaxBias(L2deriv, neighbor, biastype, symm, trafo, maxiter, tol, warn, Finfo, verbose = NULL) ## S4 method for signature ## 'UnivariateDistribution,ContNeighborhood,asymmetricBias' minmaxBias( L2deriv, neighbor, biastype, symm, trafo, maxiter, tol, warn, Finfo, verbose = NULL) ## S4 method for signature ## 'UnivariateDistribution,ContNeighborhood,onesidedBias' minmaxBias( L2deriv, neighbor, biastype, symm, trafo, maxiter, tol, warn, Finfo, verbose = NULL) ## S4 method for signature ## 'UnivariateDistribution,TotalVarNeighborhood,BiasType' minmaxBias( L2deriv, neighbor, biastype, symm, trafo, maxiter, tol, warn, Finfo, verbose = NULL) ## S4 method for signature 'RealRandVariable,ContNeighborhood,BiasType' minmaxBias(L2deriv, neighbor, biastype, normtype, Distr, z.start, A.start, z.comp, A.comp, Finfo, trafo, maxiter, tol, verbose = NULL, ...) ## S4 method for signature 'RealRandVariable,TotalVarNeighborhood,BiasType' minmaxBias(L2deriv, neighbor, biastype, normtype, Distr, z.start, A.start, z.comp, A.comp, Finfo, trafo, maxiter, tol, verbose = NULL, ...) ```

## Arguments

 `L2deriv` L2-derivative of some L2-differentiable family of probability measures. `neighbor` object of class `"Neighborhood"`. `biastype` object of class `"BiasType"`. `normtype` object of class `"NormType"`. `...` additional arguments to be passed to `E` `Distr` object of class `"Distribution"`. `symm` logical: indicating symmetry of `L2deriv`. `z.start` initial value for the centering constant. `A.start` initial value for the standardizing matrix. `z.comp` `logical` indicator which indices need to be computed and which are 0 due to symmetry. `A.comp` `matrix` of `logical` indicator which indices need to be computed and which are 0 due to symmetry. `trafo` matrix: transformation of the parameter. `maxiter` the maximum number of iterations. `tol` the desired accuracy (convergence tolerance). `warn` logical: print warnings. `Finfo` Fisher information matrix. `verbose` logical: if `TRUE`, some messages are printed

## Value

The bias-optimally robust IC is computed.

## Methods

L2deriv = "UnivariateDistribution", neighbor = "ContNeighborhood", biastype = "BiasType"

computes the bias optimal influence curve for symmetric bias for L2 differentiable parametric families with unknown one-dimensional parameter.

L2deriv = "UnivariateDistribution", neighbor = "ContNeighborhood", biastype = "asymmetricBias"

computes the bias optimal influence curve for asymmetric bias for L2 differentiable parametric families with unknown one-dimensional parameter.

L2deriv = "UnivariateDistribution", neighbor = "TotalVarNeighborhood", biastype = "BiasType"

computes the bias optimal influence curve for symmetric bias for L2 differentiable parametric families with unknown one-dimensional parameter.

L2deriv = "RealRandVariable", neighbor = "ContNeighborhood", biastype = "BiasType"

computes the bias optimal influence curve for symmetric bias for L2 differentiable parametric families with unknown k-dimensional parameter (k > 1) where the underlying distribution is univariate.

L2deriv = "RealRandVariable", neighbor = "TotalNeighborhood", biastype = "BiasType"

computes the bias optimal influence curve for symmetric bias for L2 differentiable parametric families in a setting where we are interested in a p=1 dimensional aspect of an unknown k-dimensional parameter (k > 1) where the underlying distribution is univariate.

## Author(s)

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

## 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. (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.

`InfRobModel-class`