# Optimally robust estimation Function to compute optimally robust estimates for L2-differentiable parametric families via k-step construction. ```r robest(x, L2Fam, fsCor = 1, risk = asMSE(), steps = 1L, verbose = NULL, OptOrIter = "iterate", nbCtrl = gennbCtrl(), startCtrl = genstartCtrl(), startICCtrl = genstartICCtrl(), kStepCtrl = genkStepCtrl(), na.rm = TRUE, ..., debug = FALSE, withTimings = FALSE, diagnostic = FALSE) ``` ## Arguments - `x`: sample - `L2Fam`: object of class `"L2ParamFamily"` - `fsCor`: positive real: factor used to correct the neighborhood radius; see details. - `risk`: object of class `"RiskType"` - `steps`: positive integer: number of steps used for k-steps construction - `verbose`: logical: if `TRUE`, some messages are printed - `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. - `nbCtrl`: a list specifying input concerning the used neighborhood; to be generated by a respective call to `gennbCtrl`. - `startCtrl`: a list specifying input concerning the used starting estimator; to be generated by a respective call to `genstartCtrl`. - `startICCtrl`: a list specifying input concerning the call to `getStartIC` which returns the starting influence curve; to be generated by a respective call to `genstartICCtrl`. - `kStepCtrl`: a list specifying input concerning the used variant of a kstepEstimator; to be generated by a respective call to `genkStepCtrl`. - `na.rm`: logical: if `TRUE`, the estimator is evaluated at `complete.cases(x)`. - ``...``: further arguments - `debug`: logical: if `TRUE`, only the respective calls within the function are generated for debugging purposes. - `withTimings`: logical: if `TRUE`, separate (and aggregate) timings for the three steps evaluating the starting value, finding the starting influence curve, and evaluating the k-step estimator is issued. - `diagnostic`: logical; if `TRUE`, diagnostic information on the performed integrations is gathered and shipped out as attributes `kStepDiagnostic` (for the kStepEstimator-step) and `diagnostic` for the remaining steps of the return value of `robest`. ## Details A new, more structured interface to the former function `roptest`. For details, see this function. In some respects this functions allows for more granular arguments, in the sense that the different steps (a) computation of the inital estimator, resp. (a') in case `initial.est` is missing computation of the initial MDE, (b) computation of the optimal IC and (c) computation of the k-step estimator each can have individial arguments `E.arglist` to be passed on to calls to expectation operator `E` within each step. These different arguments are passed through the input generating functions `genstartCtrl`, `genstartICCtrl`, and `kStepCtrl` Diagnostics on the involved integrations are available if argument `diagnostic` is `TRUE`. Then there are attributes `diagnostic` and `kStepDiagnostic` attached to the return value, which may be inspected and assessed through `showDiagnostic` and `getDiagnostic`. ## Returns Object of class `"kStepEstimate"`. In addition, it has an attribute `"timings"` where computation time is stored. ## Author(s) Matthias Kohl Matthias.Kohl@stamats.de , Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de ## See Also `roblox`, `L2ParamFamily-class` `UncondNeighborhood-class`, `RiskType-class` ## References Kohl, M. (2005) **Numerical Contributions to the Asymptotic Theory of Robustness**. Bayreuth: Dissertation. [https://epub.uni-bayreuth.de/id/eprint/839/2/DissMKohl.pdf](https://epub.uni-bayreuth.de/id/eprint/839/2/DissMKohl.pdf). Kohl, M. and Ruckdeschel, P. (2010): R package distrMod: Object-Oriented Implementation of Probability Models. J. Statist. Softw. 35 (10), 1--27. tools:::Rd_expr_doi("10.18637/jss.v035.i10") . Kohl, M. and Ruckdeschel, P., and Rieder, H. (2010): Infinitesimally Robust Estimation in General Smoothly Parametrized Models. **Stat. Methods Appl.**, 19 , 333--354. tools:::Rd_expr_doi("10.1007/s10260-010-0133-0") . Rieder, H. (1994) **Robust Asymptotic Statistics**. New York: Springer. tools:::Rd_expr_doi("10.1007/978-1-4684-0624-5") . Rieder, H., Kohl, M. and Ruckdeschel, P. (2008) The Costs of not Knowing the Radius. Statistical Methods and Applications 17 (1) 13-40. tools:::Rd_expr_doi("10.1007/s10260-007-0047-7") . Rieder, H., Kohl, M. and Ruckdeschel, P. (2001) The Costs of not Knowing the Radius. Appeared as discussion paper Nr. 81. SFB 373 (Quantification and Simulation of Economic Processes), Humboldt University, Berlin; also available under tools:::Rd_expr_doi("10.18452/3638") ## Examples ```r ## Don't test to reduce check time on CRAN ############################# ## 1. Binomial data ############################# ## generate a sample of contaminated data set.seed(123) ind <- rbinom(100, size=1, prob=0.05) x <- rbinom(100, size=25, prob=(1-ind)*0.25 + ind*0.9) ## Family BF <- BinomFamily(size = 25) ## ML-estimate MLest <- MLEstimator(x, BF) estimate(MLest) confint(MLest) ## compute optimally robust estimator (known contamination) nb <- gennbCtrl(eps=0.05) robest1 <- robest(x, BF, nbCtrl = nb, steps = 3) estimate(robest1) confint(robest1, method = symmetricBias()) ## neglecting bias confint(robest1) plot(pIC(robest1)) tmp <- qqplot(x, robest1, cex.pch=1.5, exp.cex2.pch = -.25, exp.fadcol.pch = .55, jit.fac=.9) ## compute optimally robust estimator (unknown contamination) nb2 <- gennbCtrl(eps.lower = 0, eps.upper = 0.2) robest2 <- robest(x, BF, nbCtrl = nb2, steps = 3) estimate(robest2) confint(robest2, method = symmetricBias()) plot(pIC(robest2)) ## total variation neighborhoods (known deviation) nb3 <- gennbCtrl(eps = 0.025, neighbor = TotalVarNeighborhood()) robest3 <- robest(x, BF, nbCtrl = nb3, steps = 3) estimate(robest3) confint(robest3, method = symmetricBias()) plot(pIC(robest3)) ## total variation neighborhoods (unknown deviation) nb4 <- gennbCtrl(eps.lower = 0, eps.upper = 0.1, neighbor = TotalVarNeighborhood()) robest3 <- robest(x, BF, nbCtrl = nb4, steps = 3) robest4 <- robest(x, BinomFamily(size = 25), nbCtrl = nb4, steps = 3) estimate(robest4) confint(robest4, method = symmetricBias()) plot(pIC(robest4)) ############################# ## 2. Poisson data ############################# ## Example: Rutherford-Geiger (1910); cf. Feller~(1968), Section VI.7 (a) x <- c(rep(0, 57), rep(1, 203), rep(2, 383), rep(3, 525), rep(4, 532), rep(5, 408), rep(6, 273), rep(7, 139), rep(8, 45), rep(9, 27), rep(10, 10), rep(11, 4), rep(12, 0), rep(13, 1), rep(14, 1)) ## Family PF <- PoisFamily() ## ML-estimate MLest <- MLEstimator(x, PF) estimate(MLest) confint(MLest) ## compute optimally robust estimator (unknown contamination) nb1 <- gennbCtrl(eps.upper = 0.1) robest <- robest(x, PF, nbCtrl = nb1, steps = 3) estimate(robest) confint(robest, symmetricBias()) plot(pIC(robest)) tmp <- qqplot(x, robest, cex.pch=1.5, exp.cex2.pch = -.25, exp.fadcol.pch = .55, jit.fac=.9) ## total variation neighborhoods (unknown deviation) nb2 <- gennbCtrl(eps.upper = 0.05, neighbor = TotalVarNeighborhood()) robest1 <- robest(x, PF, nbCtrl = nb2, steps = 3) estimate(robest1) confint(robest1, symmetricBias()) plot(pIC(robest1)) ############################# ## 3. Normal (Gaussian) location and scale ############################# ## this example of a two dimensional parameter ## to be estimated will need more time than ## 5 seconds to run ## you can find it in ## system.file("scripts", "examples_taking_longer.R", ## package="ROptEst") ```

robest function