# Optimally robust estimator for location and/or scale The function `roblox` computes the optimally robust estimator and corresponding IC for normal location und/or scale and (convex) contamination neighborhoods. The definition of these estimators can be found in Rieder (1994) or Kohl (2005), respectively. ```r roblox(x, mean, sd, eps, eps.lower, eps.upper, initial.est, k = 1L, fsCor = TRUE, returnIC = FALSE, mad0 = 1e-4, na.rm = TRUE) ``` ## Arguments - `x`: vector `x` of data values, may also be a matrix or data.frame with one row, respectively one column/(numeric) variable. - `mean`: specified mean. - `sd`: specified standard deviation which has to be positive. - `eps`: positive real (0 < `eps` <= 0.5): amount of gross errors. See details below. - `eps.lower`: positive real (0 <= `eps.lower` <= `eps.upper`): lower bound for the amount of gross errors. See details below. - `eps.upper`: positive real (`eps.lower` <= `eps.upper` <= 0.5): upper bound for the amount of gross errors. See details below. - `initial.est`: initial estimate for `mean` and/or `sd`. If missing median and/or MAD are used. - `k`: positive integer. k-step is used to compute the optimally robust estimator. - `fsCor`: logical: perform finite-sample correction. See function `finiteSampleCorrection`. - `returnIC`: logical: should IC be returned. See details below. - `mad0`: scale estimate used if computed MAD is equal to zero - `na.rm`: logical: if `TRUE`, the estimator is evaluated at `complete.cases(x)`. ## Details Computes the optimally robust estimator for location with scale specified, scale with location specified, or both if neither is specified. The computation uses a k-step construction with an appropriate initial estimate for location or scale or location and scale, respectively. Valid candidates are e.g. median and/or MAD (default) as well as Kolmogorov(-Smirnov) or von Mises minimum distance estimators; cf. Rieder (1994) and Kohl (2005). If the amount of gross errors (contamination) is known, it can be specified by `eps`. The radius of the corresponding infinitesimal contamination neighborhood is obtained by multiplying `eps` by the square root of the sample size. If the amount of gross errors (contamination) is unknown, try to find a rough estimate for the amount of gross errors, such that it lies between `eps.lower` and `eps.upper`. In case `eps.lower` is specified and `eps.upper` is missing, `eps.upper` is set to 0.5. In case `eps.upper` is specified and `eps.lower` is missing, `eps.lower` is set to 0. If neither `eps` nor `eps.lower` and/or `eps.upper` is specified, `eps.lower` and `eps.upper` are set to 0 and 0.5, respectively. If `eps` is missing, the radius-minimax estimator in sense of Rieder et al. (2008), respectively Section 2.2 of Kohl (2005) is returned. In case of location, respectively scale one additionally has to specify `sd`, respectively `mean` where `sd` and `mean` have to be a single number. For sample size <= 2, median and/or MAD are used for estimation. If `eps = 0`, mean and/or sd are computed. In this situation it's better to use function `MLEstimator`. ## Returns Object of class `"kStepEstimate"`. ## References M. Kohl (2005). Numerical Contributions to the Asymptotic Theory of Robustness. Dissertation. University of Bayreuth. [https://epub.uni-bayreuth.de/id/eprint/839/2/DissMKohl.pdf](https://epub.uni-bayreuth.de/id/eprint/839/2/DissMKohl.pdf). H. Rieder (1994): Robust Asymptotic Statistics. Springer. tools:::Rd_expr_doi("10.1007/978-1-4684-0624-5") H. Rieder, M. Kohl, and P. Ruckdeschel (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") M. Kohl, P. Ruckdeschel, and H. Rieder (2010). Infinitesimally Robust Estimation in General Smoothly Parametrized Models. Statistical Methods and Applications **19**(3): 333-354. tools:::Rd_expr_doi("10.1007/s10260-010-0133-0") . M. Kohl and H.P. Deigner (2010). Preprocessing of gene expression data by optimally robust estimators. BMC Bioinformatics **11**, 583. tools:::Rd_expr_doi("10.1186/1471-2105-11-583") . ## Author(s) Matthias Kohl Matthias.Kohl@stamats.de ## See Also `ContIC-class`, `rlOptIC`, `rsOptIC`, `rlsOptIC.AL`, `kStepEstimate-class`, `roptest` ## Examples ```r ind <- rbinom(100, size=1, prob=0.05) x <- rnorm(100, mean=ind*3, sd=(1-ind) + ind*9) ## amount of gross errors known res1 <- roblox(x, eps = 0.05, returnIC = TRUE) estimate(res1) ## don't run to reduce check time on CRAN ## Not run: confint(res1) confint(res1, method = symmetricBias()) pIC(res1) checkIC(pIC(res1)) Risks(pIC(res1)) Infos(pIC(res1)) plot(pIC(res1)) infoPlot(pIC(res1)) ## End(Not run) ## amount of gross errors unknown res2 <- roblox(x, eps.lower = 0.01, eps.upper = 0.1, returnIC = TRUE) estimate(res2) ## don't run to reduce check time on CRAN ## Not run: confint(res2) confint(res2, method = symmetricBias()) pIC(res2) checkIC(pIC(res2)) Risks(pIC(res2)) Infos(pIC(res2)) plot(pIC(res2)) infoPlot(pIC(res2)) ## End(Not run) ## estimator comparison # classical optimal (non-robust) c(mean(x), sd(x)) # most robust c(median(x), mad(x)) # optimally robust (amount of gross errors known) estimate(res1) # optimally robust (amount of gross errors unknown) estimate(res2) # Kolmogorov(-Smirnov) minimum distance estimator (robust) (ks.est <- MDEstimator(x, ParamFamily = NormLocationScaleFamily())) # optimally robust (amount of gross errors known) roblox(x, eps = 0.05, initial.est = estimate(ks.est)) # Cramer von Mises minimum distance estimator (robust) (CvM.est <- MDEstimator(x, ParamFamily = NormLocationScaleFamily(), distance = CvMDist)) # optimally robust (amount of gross errors known) roblox(x, eps = 0.05, initial.est = estimate(CvM.est)) ```

roblox function