# getInfRobIC: Generic Function for the Computation of Optimally Robust ICs In ROptEst: Optimally Robust Estimation

## Description

Generic function for the computation of 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 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75``` ```getInfRobIC(L2deriv, risk, neighbor, ...) ## S4 method for signature 'UnivariateDistribution,asCov,ContNeighborhood' getInfRobIC(L2deriv, risk, neighbor, Finfo, trafo, verbose = NULL) ## S4 method for signature 'UnivariateDistribution,asCov,TotalVarNeighborhood' getInfRobIC(L2deriv, risk, neighbor, Finfo, trafo, verbose = NULL) ## S4 method for signature 'RealRandVariable,asCov,UncondNeighborhood' getInfRobIC(L2deriv, risk, neighbor, Distr, Finfo, trafo, QuadForm = diag(nrow(trafo)), verbose = NULL) ## S4 method for signature 'UnivariateDistribution,asBias,UncondNeighborhood' getInfRobIC(L2deriv, risk, neighbor, symm, trafo, maxiter, tol, warn, Finfo, verbose = NULL, ...) ## S4 method for signature 'RealRandVariable,asBias,UncondNeighborhood' getInfRobIC(L2deriv, risk, neighbor, Distr, DistrSymm, L2derivSymm, L2derivDistrSymm, z.start, A.start, Finfo, trafo, maxiter, tol, warn, verbose = NULL, ...) ## S4 method for signature 'UnivariateDistribution,asHampel,UncondNeighborhood' getInfRobIC(L2deriv, risk, neighbor, symm, Finfo, trafo, upper = NULL, lower=NULL, maxiter, tol, warn, noLow = FALSE, verbose = NULL, checkBounds = TRUE, ...) ## S4 method for signature 'RealRandVariable,asHampel,UncondNeighborhood' getInfRobIC(L2deriv, risk, neighbor, Distr, DistrSymm, L2derivSymm, L2derivDistrSymm, Finfo, trafo, onesetLM = FALSE, z.start, A.start, upper = NULL, lower=NULL, OptOrIter = "iterate", maxiter, tol, warn, verbose = NULL, checkBounds = TRUE, ..., .withEvalAsVar = TRUE) ## S4 method for signature ## 'UnivariateDistribution,asAnscombe,UncondNeighborhood' getInfRobIC( L2deriv, risk, neighbor, symm, Finfo, trafo, upper = NULL, lower=NULL, maxiter, tol, warn, noLow = FALSE, verbose = NULL, checkBounds = TRUE, ...) ## S4 method for signature 'RealRandVariable,asAnscombe,UncondNeighborhood' getInfRobIC(L2deriv, risk, neighbor, Distr, DistrSymm, L2derivSymm, L2derivDistrSymm, Finfo, trafo, onesetLM = FALSE, z.start, A.start, upper = NULL, lower=NULL, OptOrIter = "iterate", maxiter, tol, warn, verbose = NULL, checkBounds = TRUE, ...) ## S4 method for signature 'UnivariateDistribution,asGRisk,UncondNeighborhood' getInfRobIC(L2deriv, risk, neighbor, symm, Finfo, trafo, upper = NULL, lower = NULL, maxiter, tol, warn, noLow = FALSE, verbose = NULL, ...) ## S4 method for signature 'RealRandVariable,asGRisk,UncondNeighborhood' getInfRobIC(L2deriv, risk, neighbor, Distr, DistrSymm, L2derivSymm, L2derivDistrSymm, Finfo, trafo, onesetLM = FALSE, z.start, A.start, upper = NULL, lower = NULL, OptOrIter = "iterate", maxiter, tol, warn, verbose = NULL, withPICcheck = TRUE, ..., .withEvalAsVar = TRUE) ## S4 method for signature ## 'UnivariateDistribution,asUnOvShoot,UncondNeighborhood' getInfRobIC( L2deriv, risk, neighbor, symm, Finfo, trafo, upper, lower, maxiter, tol, warn, verbose = NULL, ...) ```

## Arguments

 `L2deriv` L2-derivative of some L2-differentiable family of probability measures. `risk` object of class `"RiskType"`. `neighbor` object of class `"Neighborhood"`. `...` additional parameters (mainly for `optim`). `Distr` object of class `"Distribution"`. `symm` logical: indicating symmetry of `L2deriv`. `DistrSymm` object of class `"DistributionSymmetry"`. `L2derivSymm` object of class `"FunSymmList"`. `L2derivDistrSymm` object of class `"DistrSymmList"`. `Finfo` Fisher information matrix. `z.start` initial value for the centering constant. `A.start` initial value for the standardizing matrix. `trafo` matrix: transformation of the parameter. `upper` upper bound for the optimal clipping bound. `lower` lower bound for the optimal clipping bound. `OptOrIter` character; which method to be used for determining Lagrange multipliers `A` and `a`: if (partially) matched to `"optimize"`, `getLagrangeMultByOptim` is used; otherwise: by default, or if matched to `"iterate"` or to `"doubleiterate"`, `getLagrangeMultByIter` is used. More specifically, when using `getLagrangeMultByIter`, and if argument `risk` is of class `"asGRisk"`, by default and if matched to `"iterate"` we use only one (inner) iteration, if matched to `"doubleiterate"` we use up to `Maxiter` (inner) iterations. `maxiter` the maximum number of iterations. `tol` the desired accuracy (convergence tolerance). `warn` logical: print warnings. `noLow` logical: is lower case to be computed? `onesetLM` logical: use one set of Lagrange multipliers? `QuadForm` matrix of (or which may coerced to) class `PosSemDefSymmMatrix` for use of different (standardizing) norm `verbose` logical: if `TRUE`, some messages are printed `checkBounds` logical: if `TRUE`, minimal and maximal clipping bound are computed to check if a valid bound was specified. `withPICcheck` logical: at the end of the algorithm, shall we check how accurately this is a pIC; this will only be done if `withPICcheck && verbose`. `.withEvalAsVar` logical (of length 1): if `TRUE`, risks based on covariances are to be evaluated (default), otherwise just a call is returned.

## Value

The optimally robust IC is computed.

## Methods

L2deriv = "UnivariateDistribution", risk = "asCov", neighbor = "ContNeighborhood"

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

L2deriv = "UnivariateDistribution", risk = "asCov", neighbor = "TotalVarNeighborhood"

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

L2deriv = "RealRandVariable", risk = "asCov", neighbor = "UncondNeighborhood"

computes the classical optimal influence curve for L2 differentiable parametric families with unknown k-dimensional parameter (k > 1) where the underlying distribution is univariate; for total variation neighborhoods only is implemented for the case where there is a 1 x k transformation `trafo` matrix.

L2deriv = "UnivariateDistribution", risk = "asBias", neighbor = "UncondNeighborhood"

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

L2deriv = "RealRandVariable", risk = "asBias", neighbor = "UncondNeighborhood"

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

L2deriv = "UnivariateDistribution", risk = "asHampel", neighbor = "UncondNeighborhood"

computes the optimally robust influence curve for L2 differentiable parametric families with unknown one-dimensional parameter.

L2deriv = "RealRandVariable", risk = "asHampel", neighbor = "UncondNeighborhood"

computes the optimally robust influence curve for L2 differentiable parametric families with unknown k-dimensional parameter (k > 1) where the underlying distribution is univariate; for total variation neighborhoods only is implemented for the case where there is a 1 x k transformation `trafo` matrix.

L2deriv = "UnivariateDistribution", risk = "asAnscombe", neighbor = "UncondNeighborhood"

computes the optimally bias-robust influence curve to given ARE in the ideal model for L2 differentiable parametric families with unknown one-dimensional parameter.

L2deriv = "RealRandVariable", risk = "asAnscombe", neighbor = "UncondNeighborhood"

computes the optimally bias-robust influence curve to given ARE in the ideal modelfor L2 differentiable parametric families with unknown k-dimensional parameter (k > 1) where the underlying distribution is univariate; for total variation neighborhoods only is implemented for the case where there is a 1 x k transformation `trafo` matrix.

L2deriv = "UnivariateDistribution", risk = "asGRisk", neighbor = "UncondNeighborhood"

computes the optimally robust influence curve for L2 differentiable parametric families with unknown one-dimensional parameter.

L2deriv = "RealRandVariable", risk = "asGRisk", neighbor = "UncondNeighborhood"

computes the optimally robust influence curve for L2 differentiable parametric families with unknown k-dimensional parameter (k > 1) where the underlying distribution is univariate; for total variation neighborhoods only is implemented for the case where there is a 1 x k transformation `trafo` matrix.

L2deriv = "UnivariateDistribution", risk = "asUnOvShoot", neighbor = "UncondNeighborhood"

computes the optimally robust influence curve for one-dimensional L2 differentiable parametric families and asymptotic under-/overshoot risk.

## 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. 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.

`InfRobModel-class`