Package RobLox. September 13, Version 0.9. Date Title Optimally robust influence curves and estimators for location and scale

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1 Package RobLox Septembe 13, 2013 Vesion 0.9 Date Title Optimally obust influence cuves and estimatos fo location and scale Functions fo the detemination of optimally obust influence cuves and estimatos in case of nomal location and/o scale Depends R(>= ), stats, distmod(>= 2.5.2), RobAStBase(>= 0.9) Impots lattice, RColoBewe, Biobase, RandVa(>= 0.9.2), dist(>= 2.5.2) Suggests MASS Autho Maintaine LazyLoad yes ByteCompile yes License LGPL-3 Encoding latin1 URL LastChangedDate {$LastChangedDate: :24: (Do, 12. Sep 2013) $} LastChangedRevision {$LastChangedRevision: 707 $} SVNRevision 696 NeedsCompilation no Repositoy CRAN Date/Publication :55:21 1

2 2 RobLox-package R topics documented: RobLox-package finitesamplecoection loptic lsoptic.al lsoptic.an lsoptic.an lsoptic.anmad lsoptic.bm lsoptic.ha lsoptic.ha lsoptic.hamad lsoptic.hu lsoptic.hu lsoptic.hu2a lsoptic.hu lsoptic.humad lsoptic.m lsoptic.mm lsoptic.tu lsoptic.tu lsoptic.tumad oblox owroblox and colroblox soptic showdown Index 38 RobLox-package Optimally obust influence cuves and estimatos fo location and scale Functions fo the detemination of optimally obust influence cuves and estimatos in case of nomal location and/o scale. Package: RobLox Vesion: 0.9 Date: Depends: R(>= 2.7.0), stats, lattice, RColoBewe, RandVa, Biobase, dist, distmod, RobAStBase Suggests: MASS LazyLoad: yes ByteCompile: yes

3 RobLox-package 3 License: LGPL-3 URL: SVNRevision: 696 Package vesions Note: The fist two numbes of package vesions do not necessaily eflect package-individual development, but athe ae chosen fo the RobAStXXX family as a whole in ode to ease updating "depends" infomation. Matthias Kohl <matthias.kohl@stamats.de> M. Kohl (2005). Numeical Contibutions to the Asymptotic Theoy of Robustness. Dissetation. Univesity of Bayeuth. Riede, H. (1994) Robust Asymptotic Statistics. New Yok: Spinge. Riede, H., Kohl, M. and Ruckdeschel, P. (2008). The Costs of not Knowing the Radius. Statistical Methods and Applications 17(1) Extended vesion: M. Kohl, P. Ruckdeschel, and H. Riede (2010). Infinitesimally Robust Estimation in Geneal Smoothly Paametized Models. Statistical Methods and Application, 19(3): RobAStBase-package libay(roblox) ind <- binom(100, size=1, pob=0.05) x <- nom(100, mean=ind*3, sd=(1-ind) + ind*9) oblox(x) es <- oblox(x, eps.lowe = 0.01, eps.uppe = 0.1, etunic = TRUE) estimate(es) confint(es) confint(es, method = symmeticbias()) pic(es) ## don t un to educe check time on CRAN ## Not un: checkic(pic(es)) Risks(pIC(es)) Infos(pIC(es))

4 4 finitesamplecoection plot(pic(es)) infoplot(pic(es)) ## End(Not un) ## ow-wise application ind <- binom(200, size=1, pob=0.05) X <- matix(nom(200, mean=ind*3, sd=(1-ind) + ind*9), now = 2) owroblox(x) finitesamplecoection Function to compute finite-sample coected adii Given some adius and some sample size the function computes the coesponding finite-sample coected adius. finitesamplecoection(, n, model = "locsc") n model asymptotic adius (non-negative numeic) sample size has to be "locsc" (fo location and scale), "loc" (fo location) o "sc" (fo scale), espectively. The finite-sample coection is based on empiical esults obtained via simulation studies. Given some adius of a shinking contamination neighbohood which leads to an asymptotically optimal obust estimato, the finite-sample empiical MSE based on contaminated samples was minimized fo this class of asymptotically optimal estimatos and the coesponding finite-sample adius detemined and saved. The computation is based on the saved esults of these Monte-Calo simulations. Finite-sample coected adius.

5 loptic 5 Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation. Riede, H. (1994) Robust Asymptotic Statistics. New Yok: Spinge. Riede, H., Kohl, M. and Ruckdeschel, P. (2008) The Costs of not Knowing the Radius. Statistical Methods and Applications 17(1) Extended vesion: oblox, owroblox, colroblox finitesamplecoection(n = 3, = 0.001, model = "locsc") finitesamplecoection(n = 10, = 0.02, model = "loc") finitesamplecoection(n = 250, = 0.15, model = "sc") loptic Computation of the optimally obust IC fo AL estimatos The function loptic computes the optimally obust IC fo AL estimatos in case of nomal location and (convex) contamination neighbohoods. The definition of these estimatos can be found in Riede (1994) o Kohl (2005), espectively. loptic(, mean = 0, sd = 1, bup = 1000, computeic = TRUE) mean sd bup computeic non-negative eal: neighbohood adius. specified mean. specified standad deviation. positive eal: the uppe end point of the inteval to be seached fo the clipping bound b. logical: should IC be computed. See details below. If computeic is FALSE only the Lagange multiplies A, a, and b contained in the optimally obust IC ae computed.

6 6 lsoptic.al If computeic is TRUE an object of class "ContIC" is etuned, othewise a list of Lagange multiplies A a b standadizing constant centeing constant; always = 0 is this symmetic setup optimal clipping bound Riede, H. (1994) Robust Asymptotic Statistics. New Yok: Spinge. Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation. ContIC-class, oblox IC1 <- loptic( = 0.1) distexoptions("eelativetoleance" = 1e-12) checkic(ic1) distexoptions("eelativetoleance" =.Machine$double.eps^0.25) # default cent(ic1) clip(ic1) stand(ic1) lsoptic.al Computation of the optimally obust IC fo AL estimatos The function lsoptic.al computes the optimally obust IC fo AL estimatos in case of nomal location with unknown scale and (convex) contamination neighbohoods. The definition of these estimatos can be found in Section 8.2 of Kohl (2005). lsoptic.al(, mean = 0, sd = 1, A.loc.stat = 1, a.sc.stat = 0, A.sc.stat = 0.5, bup = 1000, delta = 1e-6, itmax = 100, check = FALSE, computeic = TRUE)

7 lsoptic.al 7 mean sd A.loc.stat a.sc.stat A.sc.stat bup delta itmax check computeic non-negative eal: neighbohood adius. specified mean. specified standad deviation. positive eal: stating value fo the standadizing constant of the location pat. eal: stating value fo centeing constant of the scale pat. positive eal: stating value fo the standadizing constant of the scale pat. positive eal: the uppe end point of the inteval to be seached fo the clipping bound b. the desied accuacy (convegence toleance). the maximum numbe of iteations. logical: should constaints be checked. logical: should IC be computed. See details below. The Lagange multiplies contained in the expession of the optimally obust IC can be accessed via the accesso functions cent, clip and stand. If computeic is FALSE only the Lagange multiplies A, a, and b contained in the optimally obust IC ae computed. If computeic is TRUE an object of class "ContIC" is etuned, othewise a list of Lagange multiplies A a b standadizing matix centeing vecto optimal clipping bound Riede, H. (1994) Robust Asymptotic Statistics. New Yok: Spinge. Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation. ContIC-class, oblox

8 8 lsoptic.al IC1 <- lsoptic.al( = 0.1, check = TRUE) distexoptions("eelativetoleance" = 1e-12) checkic(ic1) distexoptions("eelativetoleance" =.Machine$double.eps^0.25) # default cent(ic1) clip(ic1) stand(ic1) ## don t un to educe check time on CRAN ## Not un: infoplot(ic1) ## k-step estimation ## bette use function oblox (see?oblox) ## 1. data: andom sample ind <- binom(100, size=1, pob=0.05) x <- nom(100, mean=0, sd=(1-ind) + ind*9) mean(x) sd(x) median(x) mad(x) ## 2. Kolmogoov(-Sminov) minimum distance estimato (default) ## -> we use it as initial estimate fo one-step constuction (est0 <- MDEstimato(x, PaamFamily = NomLocationScaleFamily())) ## 3.1 one-step estimation: adius known IC1 <- lsoptic.al( = 0.5, mean = estimate(est0)[1], sd = estimate(est0)[2]) (est1 <- onestepestimato(x, IC1, est0)) ## 3.2 k-step estimation: adius known ## Choose k = 3 (est2 <- kstepestimato(x, IC1, est0, steps = 3L)) ## 4.1 one-step estimation: adius unknown ## take least favoable adius = ## cf. Table 8.1 in Kohl(2005) IC2 <- lsoptic.al( = 0.579, mean = estimate(est0)[1], sd = estimate(est0)[2]) (est3 <- onestepestimato(x, IC2, est0)) ## 4.2 k-step estimation: adius unknown ## take least favoable adius = ## cf. Table 8.1 in Kohl(2005) ## choose k = 3 (est4 <- kstepestimato(x, IC2, est0, steps = 3L)) ## End(Not un)

9 lsoptic.an1 9 lsoptic.an1 Computation of the optimally obust IC fo An1 estimatos The function lsoptic.an1 computes the optimally obust IC fo An1 estimatos in case of nomal location with unknown scale and (convex) contamination neighbohoods. The definition of these estimatos can be found in Subsection of Kohl (2005). lsoptic.an1(, aup = 2.5, delta = 1e-06) non-negative eal: neighbohood adius. aup positive eal: the uppe end point of the inteval to be seached fo a. delta the desied accuacy (convegence toleance). The optimal value of the tuning constant a can be ead off fom the slot Infos of the esulting IC. Object of class "IC" Andews, D.F., Bickel, P.J., Hampel, F.R., Hube, P.J., Roges, W.H. and Tukey, J.W. (1972) Robust estimates of location. Pinceton Univesity Pess. Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation. IC-class

10 10 lsoptic.an2 IC1 <- lsoptic.an1( = 0.1) checkic(ic1) Infos(IC1) ## don t un to educe check time on CRAN ## Not un: infoplot(ic1) ## End(Not un) lsoptic.an2 Computation of the optimally obust IC fo An2 estimatos The function lsoptic.an2 computes the optimally obust IC fo An2 estimatos in case of nomal location with unknown scale and (convex) contamination neighbohoods. The definition of these estimatos can be found in Subsection of Kohl (2005). lsoptic.an2(, a.stat = 1.5, k.stat = 1.5, delta = 1e-06, MAX = 100) non-negative eal: neighbohood adius. a.stat positive eal: stating value fo a. k.stat positive eal: stating value fo k. delta MAX the desied accuacy (convegence toleance). if a o k ae beyond the admitted values, MAX is etuned. The computation of the optimally obust IC fo An2 estimatos is based on optim whee MAX is used to contol the constaints on a and k. The optimal values of the tuning constants a and k can be ead off fom the slot Infos of the esulting IC. Object of class "IC"

11 lsoptic.anmad 11 Andews, D.F., Bickel, P.J., Hampel, F.R., Hube, P.J., Roges, W.H. and Tukey, J.W. (1972) Robust estimates of location. Pinceton Univesity Pess. Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation. IC-class IC1 <- lsoptic.an2( = 0.1) checkic(ic1) Infos(IC1) infoplot(ic1) lsoptic.anmad Computation of the optimally obust IC fo AnMad estimatos The function lsoptic.anmad computes the optimally obust IC fo AnMad estimatos in case of nomal location with unknown scale and (convex) contamination neighbohoods. These estimatos wee consideed in Andews et al. (1972). A definition of these estimatos can also be found in Subsection of Kohl (2005). lsoptic.anmad(, aup = 2.5, delta = 1e-06) non-negative eal: neighbohood adius. aup positive eal: the uppe end point of the inteval to be seached fo a. delta the desied accuacy (convegence toleance). The optimal value of the tuning constant a can be ead off fom the slot Infos of the esulting IC. Object of class "IC"

12 12 lsoptic.bm Andews, D.F., Bickel, P.J., Hampel, F.R., Hube, P.J., Roges, W.H. and Tukey, J.W. (1972) Robust estimates of location. Pinceton Univesity Pess. Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation. IC-class IC1 <- lsoptic.anmad( = 0.1) checkic(ic1) Infos(IC1) infoplot(ic1) lsoptic.bm Computation of the optimally obust IC fo BM estimatos The function lsoptic.bm computes the optimally obust IC fo BM estimatos in case of nomal location with unknown scale and (convex) contamination neighbohoods. These estimatos wee poposed by Bednaski and Muelle (2001). A definition of these estimatos can also be found in Section 8.4 of Kohl (2005). lsoptic.bm(, bl.stat = 2, bs.stat = 1.5, delta = 1e-06, MAX = 100) non-negative eal: neighbohood adius. bl.stat positive eal: stating value fo b loc. bs.stat positive eal: stating value fo b sc,0. delta the desied accuacy (convegence toleance). MAX if b loc o b sc,0 ae beyond the admitted values, MAX is etuned.

13 lsoptic.ha3 13 The computation of the optimally obust IC fo BM estimatos is based on optim whee MAX is used to contol the constaints on b loc and b sc,0. The optimal values of the tuning constants b loc, b sc,0, α and γ can be ead off fom the slot Infos of the esulting IC. Object of class "IC" Bednaski, T and Muelle, C.H. (2001) Optimal bounded influence egession and scale M-estimatos in the context of expeimental design. Statistics, 35(4): Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation. IC-class IC1 <- lsoptic.bm( = 0.1) checkic(ic1) Infos(IC1) infoplot(ic1) lsoptic.ha3 Computation of the optimally obust IC fo Ha3 estimatos The function lsoptic.ha3 computes the optimally obust IC fo Ha3 estimatos in case of nomal location with unknown scale and (convex) contamination neighbohoods. The definition of these estimatos can be found in Subsection of Kohl (2005). lsoptic.ha3(, a.stat = 0.25, b.stat = 2.5, c.stat = 5, delta = 1e-06, MAX = 100)

14 14 lsoptic.ha3 non-negative eal: neighbohood adius. a.stat positive eal: stating value fo a. b.stat positive eal: stating value fo b. c.stat positive eal: stating value fo c. delta the desied accuacy (convegence toleance). MAX if a o b o c ae beyond the admitted values, MAX is etuned. The computation of the optimally obust IC fo Ha3 estimatos is based on optim whee MAX is used to contol the constaints on a, b and c. The optimal values of the tuning constants a, b and c can be ead off fom the slot Infos of the esulting IC. Object of class "IC" Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation. IC-class IC1 <- lsoptic.ha3( = 0.1) checkic(ic1) Infos(IC1) ## don t un to educe check time on CRAN ## Not un: infoplot(ic1) ## End(Not un)

15 lsoptic.ha4 15 lsoptic.ha4 Computation of the optimally obust IC fo Ha4 estimatos The function lsoptic.ha4 computes the optimally obust IC fo Ha4 estimatos in case of nomal location with unknown scale and (convex) contamination neighbohoods. The definition of these estimatos can be found in Subsection of Kohl (2005). lsoptic.ha4(, a.stat = 0.25, b.stat = 2.5, c.stat = 5, k.stat = 1, delta = 1e-06, MAX = 100) non-negative eal: neighbohood adius. a.stat positive eal: stating value fo a. b.stat positive eal: stating value fo b. c.stat positive eal: stating value fo c. k.stat positive eal: stating value fo k. delta MAX the desied accuacy (convegence toleance). if a o b o c o k ae beyond the admitted values, MAX is etuned. The computation of the optimally obust IC fo Ha4 estimatos is based on optim whee MAX is used to contol the constaints on a, b, c and k. The optimal values of the tuning constants a, b, c and k can be ead off fom the slot Infos of the esulting IC. Object of class "IC" Maazzi, A. (1993) Algoithms, outines, and S functions fo obust statistics. Wadswoth and Books / Cole. Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation.

16 16 lsoptic.hamad IC-class IC1 <- lsoptic.ha4( = 0.1) checkic(ic1) Infos(IC1) infoplot(ic1) lsoptic.hamad Computation of the optimally obust IC fo HuMad estimatos The function lsoptic.humad computes the optimally obust IC fo HuMad estimatos in case of nomal location with unknown scale and (convex) contamination neighbohoods. These estimatos wee consideed in Andews et al. (1972). A definition of these estimatos can also be found in Subsection of Kohl (2005). lsoptic.hamad(, a.stat = 0.25, b.stat = 2.5, c.stat = 5, delta = 1e-06, MAX = 100) non-negative eal: neighbohood adius. a.stat positive eal: stating value fo a. b.stat positive eal: stating value fo b. c.stat positive eal: stating value fo c. delta MAX the desied accuacy (convegence toleance). if a o b o c ae beyond the admitted values, MAX is etuned. The computation of the optimally obust IC fo HaMad estimatos is based on optim whee MAX is used to contol the constaints on a, b and c. The optimal values of the tuning constants a, b, and c can be ead off fom the slot Infos of the esulting IC. Object of class "IC"

17 lsoptic.hu1 17 Andews, D.F., Bickel, P.J., Hampel, F.R., Hube, P.J., Roges, W.H. and Tukey, J.W. (1972) Robust estimates of location. Pinceton Univesity Pess. Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation. IC-class IC1 <- lsoptic.hamad( = 0.1) checkic(ic1) Infos(IC1) infoplot(ic1) lsoptic.hu1 Computation of the optimally obust IC fo Hu1 estimatos The function lsoptic.hu1 computes the optimally obust IC fo Hu1 estimatos in case of nomal location with unknown scale and (convex) contamination neighbohoods. These estimatos wee poposed by Hube (1964), Poposal 2. A definition of these estimatos can also be found in Subsection of Kohl (2005). lsoptic.hu1(, kup = 2.5, delta = 1e-06) non-negative eal: neighbohood adius. kup positive eal: the uppe end point of the inteval to be seached fo k. delta the desied accuacy (convegence toleance). The optimal value of the tuning constant k can be ead off fom the slot Infos of the esulting IC.

18 18 lsoptic.hu2 Object of class "IC" Hube, P.J. (1964) Robust estimation of a location paamete. Ann. Math. Stat. 35: Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation. IC-class IC1 <- lsoptic.hu1( = 0.1) checkic(ic1) Infos(IC1) infoplot(ic1) lsoptic.hu2 Computation of the optimally obust IC fo Hu2 estimatos The function lsoptic.hu2 computes the optimally obust IC fo Hu2 estimatos in case of nomal location with unknown scale and (convex) contamination neighbohoods. These estimatos wee poposed in Example of Hube (1981). A definition of these estimatos can also be found in Subsection of Kohl (2005). lsoptic.hu2(, k.stat = 1.5, c.stat = 1.5, delta = 1e-06, MAX = 100) non-negative eal: neighbohood adius. k.stat positive eal: stating value fo k. c.stat positive eal: stating value fo c. delta MAX the desied accuacy (convegence toleance). if k1 o k2 ae beyond the admitted values, MAX is etuned.

19 lsoptic.hu2a 19 The computation of the optimally obust IC fo Hu2 estimatos is based on optim whee MAX is used to contol the constaints on k and c. The optimal values of the tuning constants k and c can be ead off fom the slot Infos of the esulting IC. Object of class "IC" Hube, P.J. (1981) Robust Statistics. New Yok: Wiley. Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation. IC-class IC1 <- lsoptic.hu2( = 0.1) checkic(ic1) Infos(IC1) infoplot(ic1) lsoptic.hu2a Computation of the optimally obust IC fo Hu2a estimatos The function lsoptic.hu2a computes the optimally obust IC fo Hu2a estimatos in case of nomal location with unknown scale and (convex) contamination neighbohoods. These estimatos ae a simple modification of Hube (1964), Poposal 2 whee we, in addition, admit a clipping fom below. The definition of these estimatos can be found in Subsection of Kohl (2005). lsoptic.hu2a(, k1.stat = 0.25, k2.stat = 2.5, delta = 1e-06, MAX = 100)

20 20 lsoptic.hu2a non-negative eal: neighbohood adius. k1.stat positive eal: stating value fo k1. k2.stat positive eal: stating value fo k2. delta the desied accuacy (convegence toleance). MAX if k1 o k2 ae beyond the admitted values, MAX is etuned. The computation of the optimally obust IC fo Hu2a estimatos is based on optim whee MAX is used to contol the constaints on k1 and k2. The optimal values of the tuning constants k1 and k2 can be ead off fom the slot Infos of the esulting IC. Object of class "IC" Hube, P.J. (1964) Robust estimation of a location paamete. Ann. Math. Stat. 35: Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation. IC-class IC1 <- lsoptic.hu2a( = 0.1) checkic(ic1) Infos(IC1) infoplot(ic1)

21 lsoptic.hu3 21 lsoptic.hu3 Computation of the optimally obust IC fo Hu3 estimatos The function lsoptic.hu3 computes the optimally obust IC fo Hu3 estimatos in case of nomal location with unknown scale and (convex) contamination neighbohoods. The definition of these estimatos can be found in Subsection of Kohl (2005). lsoptic.hu3(, k.stat = 1, c1.stat = 0.1, c2.stat = 0.5, delta = 1e-06, MAX = 100) non-negative eal: neighbohood adius. k.stat positive eal: stating value fo k. c1.stat positive eal: stating value fo c1. c2.stat positive eal: stating value fo c2. delta MAX the desied accuacy (convegence toleance). if k o c1 o c2 ae beyond the admitted values, MAX is etuned. The computation of the optimally obust IC fo Hu2 estimatos is based on optim whee MAX is used to contol the constaints on k, c1 and c2. The optimal values of the tuning constants k, c1 and c2 can be ead off fom the slot Infos of the esulting IC. Object of class "IC" Hube, P.J. (1981) Robust Statistics. New Yok: Wiley. Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation. IC-class

22 22 lsoptic.humad IC1 <- lsoptic.hu3( = 0.1) checkic(ic1) Infos(IC1) infoplot(ic1) lsoptic.humad Computation of the optimally obust IC fo HuMad estimatos The function lsoptic.humad computes the optimally obust IC fo HuMad estimatos in case of nomal location with unknown scale and (convex) contamination neighbohoods. These estimatos wee poposed by Andews et al. (1972), p. 12. A definition of these estimatos can also be found in Subsection of Kohl (2005). lsoptic.humad(, kup = 2.5, delta = 1e-06) non-negative eal: neighbohood adius. kup positive eal: the uppe end point of the inteval to be seached fo k. delta the desied accuacy (convegence toleance). The optimal value of the tuning constant k can be ead off fom the slot Infos of the esulting IC. Object of class "IC" Andews, D.F., Bickel, P.J., Hampel, F.R., Hube, P.J., Roges, W.H. and Tukey, J.W. (1972) Robust estimates of location. Pinceton Univesity Pess. Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation.

23 lsoptic.m 23 IC-class IC1 <- lsoptic.humad( = 0.1) checkic(ic1) Infos(IC1) infoplot(ic1) lsoptic.m Computation of the optimally obust IC fo M estimatos The function lsoptic.m computes the optimally obust IC fo M estimatos in case of nomal location with unknown scale and (convex) contamination neighbohoods. The definition of these estimatos can be found in Section 8.3 of Kohl (2005). lsoptic.m(, gglo = 0.5, ggup = 1.5, a1.stat = 0.75, a3.stat = 0.25, bup = 1000, delta = 1e-05, itmax = 100, check = FALSE) non-negative eal: neighbohood adius. gglo non-negative eal: the lowe end point of the inteval to be seached fo γ. ggup positive eal: the uppe end point of the inteval to be seached fo γ. a1.stat eal: stating value fo α 1. a3.stat eal: stating value fo α 3. bup delta itmax check positive eal: uppe bound used in the computation of the optimal clipping bound b. the desied accuacy (convegence toleance). the maximum numbe of iteations. logical. Should constaints be checked. The optimal values of the tuning constants α 1, α 3, b and γ can be ead off fom the slot Infos of the esulting IC.

24 24 lsoptic.mm2 Object of class "IC" Hube, P.J. (1981) Robust Statistics. New Yok: Wiley. Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation. IC-class IC1 <- lsoptic.m( = 0.1, check = TRUE) distexoptions("eelativetoleance" = 1e-12) checkic(ic1, NomLocationScaleFamily()) distexoptions("eelativetoleance" =.Machine$double.eps^0.25) Infos(IC1) infoplot(ic1) lsoptic.mm2 Computation of the optimally obust IC fo MM2 estimatos The function lsoptic.mm2 computes the optimally obust IC fo MM2 estimatos in case of nomal location with unknown scale and (convex) contamination neighbohoods. These estimatos ae based on a poposal of Faiman et al. (2001), p A definition of these estimatos can also be found in Section 8.6 of Kohl (2005). lsoptic.mm2(, c.stat = 1.5, d.stat = 2, delta = 1e-06, MAX = 100) non-negative eal: neighbohood adius. c.stat positive eal: stating value fo c. d.stat positive eal: stating value fo d. delta the desied accuacy (convegence toleance). MAX if a o k ae beyond the admitted values, MAX is etuned.

25 lsoptic.tu1 25 The computation of the optimally obust IC fo MM2 estimatos is based on optim whee MAX is used to contol the constaints on c and d. The optimal values of the tuning constants c and d can be ead off fom the slot Infos of the esulting IC. Object of class "IC" Faiman, R., Yohai, V.J. and Zama, R.H. (2001) Optimal obust M-estimates of location. Ann. Stat. 29(1): Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation. IC-class IC1 <- lsoptic.mm2( = 0.1) checkic(ic1) Infos(IC1) infoplot(ic1) lsoptic.tu1 Computation of the optimally obust IC fo Tu1 estimatos The function lsoptic.tu1 computes the optimally obust IC fo Tu1 estimatos in case of nomal location with unknown scale and (convex) contamination neighbohoods. The definition of these estimatos can be found in Subsection of Kohl (2005). lsoptic.tu1(, aup = 10, delta = 1e-06)

26 26 lsoptic.tu2 non-negative eal: neighbohood adius. aup positive eal: the uppe end point of the inteval to be seached fo a. delta the desied accuacy (convegence toleance). The optimal value of the tuning constant a can be ead off fom the slot Infos of the esulting IC. Object of class "IC" Beaton, A.E. and Tukey, J.W. (1974) The fitting of powe seies, meaning polynomials, illustated on band-spectoscopic data. Discussions. Technometics 16: Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation. IC-class IC1 <- lsoptic.tu1( = 0.1) checkic(ic1) Infos(IC1) infoplot(ic1) lsoptic.tu2 Computation of the optimally obust IC fo Tu2 estimatos The function lsoptic.tu2 computes the optimally obust IC fo Tu2 estimatos in case of nomal location with unknown scale and (convex) contamination neighbohoods. The definition of these estimatos can be found in Subsection of Kohl (2005).

27 lsoptic.tu2 27 lsoptic.tu2(, a.stat = 5, k.stat = 1.5, delta = 1e-06, MAX = 100) non-negative eal: neighbohood adius. a.stat positive eal: stating value fo a. k.stat positive eal: stating value fo k. delta the desied accuacy (convegence toleance). MAX if a o k ae beyond the admitted values, MAX is etuned. The computation of the optimally obust IC fo Tu2 estimatos is based on optim whee MAX is used to contol the constaints on a and k. The optimal values of the tuning constant a and k can be ead off fom the slot Infos of the esulting IC. Object of class "IC" Beaton, A.E. and Tukey, J.W. (1974) The fitting of powe seies, meaning polynomials, illustated on band-spectoscopic data. Discussions. Technometics 16: Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation. IC-class IC1 <- lsoptic.tu2( = 0.1) checkic(ic1) Infos(IC1) infoplot(ic1)

28 28 lsoptic.tumad lsoptic.tumad Computation of the optimally obust IC fo TuMad estimatos The function lsoptic.tumad computes the optimally obust IC fo TuMad estimatos in case of nomal location with unknown scale and (convex) contamination neighbohoods. The definition of these estimatos can be found in Subsection of Kohl (2005). lsoptic.tumad(, aup = 10, delta = 1e-06) non-negative eal: neighbohood adius. aup positive eal: the uppe end point of the inteval to be seached fo a. delta the desied accuacy (convegence toleance). The optimal value of the tuning constant a can be ead off fom the slot Infos of the esulting IC. Object of class "IC" Beaton, A.E. and Tukey, J.W. (1974) The fitting of powe seies, meaning polynomials, illustated on band-spectoscopic data. Discussions. Technometics 16: Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation. IC-class

29 oblox 29 IC1 <- lsoptic.tumad( = 0.1) checkic(ic1) Infos(IC1) infoplot(ic1) oblox Optimally obust estimato fo location and/o scale The function oblox computes the optimally obust estimato and coesponding IC fo nomal location und/o scale and (convex) contamination neighbohoods. The definition of these estimatos can be found in Riede (1994) o Kohl (2005), espectively. oblox(x, mean, sd, eps, eps.lowe, eps.uppe, initial.est, k = 1L, fsco = TRUE, etunic = FALSE, mad0 = 1e-4, na.m = TRUE) x mean sd eps eps.lowe eps.uppe initial.est k fsco etunic mad0 na.m vecto x of data values, may also be a matix o data.fame with one ow, espectively one column/(numeic) vaiable. specified mean. specified standad deviation which has to be positive. positive eal (0 < eps <= 0.5): amount of goss eos. See details below. positive eal (0 <= eps.lowe <= eps.uppe): lowe bound fo the amount of goss eos. See details below. positive eal (eps.lowe <= eps.uppe <= 0.5): uppe bound fo the amount of goss eos. See details below. initial estimate fo mean and/o sd. If missing median and/o MAD ae used. positive intege. k-step is used to compute the optimally obust estimato. logical: pefom finite-sample coection. See function finitesamplecoection. logical: should IC be etuned. See details below. scale estimate used if computed MAD is equal to zeo logical: if TRUE, the estimato is evaluated at complete.cases(x).

30 30 oblox Computes the optimally obust estimato fo location with scale specified, scale with location specified, o both if neithe is specified. The computation uses a k-step constuction with an appopiate initial estimate fo location o scale o location and scale, espectively. Valid candidates ae e.g. median and/o MAD (default) as well as Kolmogoov(-Sminov) o von Mises minimum distance estimatos; cf. Riede (1994) and Kohl (2005). If the amount of goss eos (contamination) is known, it can be specified by eps. The adius of the coesponding infinitesimal contamination neighbohood is obtained by multiplying eps by the squae oot of the sample size. If the amount of goss eos (contamination) is unknown, ty to find a ough estimate fo the amount of goss eos, such that it lies between eps.lowe and eps.uppe. In case eps.lowe is specified and eps.uppe is missing, eps.uppe is set to 0.5. In case eps.uppe is specified and eps.lowe is missing, eps.lowe is set to 0. If neithe eps no eps.lowe and/o eps.uppe is specified, eps.lowe and eps.uppe ae set to 0 and 0.5, espectively. If eps is missing, the adius-minimax estimato in sense of Riede et al. (2008), espectively Section 2.2 of Kohl (2005) is etuned. In case of location, espectively scale one additionally has to specify sd, espectively mean whee sd and mean have to be a single numbe. Fo sample size <= 2, median and/o MAD ae used fo estimation. If eps = 0, mean and/o sd ae computed. In this situation it s bette to use function MLEstimato. Object of class "kstepestimate". Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation. Riede, H. (1994) Robust Asymptotic Statistics. New Yok: Spinge. Riede, H., Kohl, M. and Ruckdeschel, P. (2008) The Costs of not Knowing the Radius. Statistical Methods and Applications 17(1) Extended vesion: M. Kohl, P. Ruckdeschel, and H. Riede (2010). Infinitesimally Robust Estimation in Geneal Smoothly Paametized Models. Statistical Methods and Application, 19(3): ContIC-class, loptic, soptic, lsoptic.al, kstepestimate-class, optest

31 oblox 31 ind <- binom(100, size=1, pob=0.05) x <- nom(100, mean=ind*3, sd=(1-ind) + ind*9) ## amount of goss eos known es1 <- oblox(x, eps = 0.05, etunic = TRUE) estimate(es1) ## don t un to educe check time on CRAN ## Not un: confint(es1) confint(es1, method = symmeticbias()) pic(es1) checkic(pic(es1)) Risks(pIC(es1)) Infos(pIC(es1)) plot(pic(es1)) infoplot(pic(es1)) ## End(Not un) ## amount of goss eos unknown es2 <- oblox(x, eps.lowe = 0.01, eps.uppe = 0.1, etunic = TRUE) estimate(es2) ## don t un to educe check time on CRAN ## Not un: confint(es2) confint(es2, method = symmeticbias()) pic(es2) checkic(pic(es2)) Risks(pIC(es2)) Infos(pIC(es2)) plot(pic(es2)) infoplot(pic(es2)) ## End(Not un) ## estimato compaison # classical optimal (non-obust) c(mean(x), sd(x)) # most obust c(median(x), mad(x)) # optimally obust (amount of goss eos known) estimate(es1) # optimally obust (amount of goss eos unknown) estimate(es2) # Kolmogoov(-Sminov) minimum distance estimato (obust) (ks.est <- MDEstimato(x, PaamFamily = NomLocationScaleFamily()))

32 32 owroblox and colroblox # optimally obust (amount of goss eos known) oblox(x, eps = 0.05, initial.est = estimate(ks.est)) # Came von Mises minimum distance estimato (obust) (CvM.est <- MDEstimato(x, PaamFamily = NomLocationScaleFamily(), distance = CvMDist)) # optimally obust (amount of goss eos known) oblox(x, eps = 0.05, initial.est = estimate(cvm.est)) owroblox and colroblox Optimally obust estimation fo location and/o scale The functions owroblox and colroblox compute optimally obust estimates fo nomal location und/o scale and (convex) contamination neighbohoods. The definition of these estimatos can be found in Riede (1994) o Kohl (2005), espectively. owroblox(x, mean, sd, eps, eps.lowe, eps.uppe, initial.est, k = 1L, fsco = TRUE, mad0 = 1e-4, na.m = TRUE) colroblox(x, mean, sd, eps, eps.lowe, eps.uppe, initial.est, k = 1L, fsco = TRUE, mad0 = 1e-4, na.m = TRUE) x mean sd eps eps.lowe eps.uppe initial.est k fsco mad0 na.m matix o data.fame of (numeic) data values. specified mean. See details below. specified standad deviation which has to be positive. See also details below. positive eal (0 < eps <= 0.5): amount of goss eos. See details below. positive eal (0 <= eps.lowe <= eps.uppe): lowe bound fo the amount of goss eos. See details below. positive eal (eps.lowe <= eps.uppe <= 0.5): uppe bound fo the amount of goss eos. See details below. initial estimate fo mean and/o sd. If missing median and/o MAD ae used. positive intege. k-step is used to compute the optimally obust estimato. logical: pefom finite-sample coection. See function finitesamplecoection. scale estimate used if computed MAD is equal to zeo logical: if TRUE, the estimato is evaluated at complete.cases(x).

33 owroblox and colroblox 33 Computes the optimally obust estimato fo location with scale specified, scale with location specified, o both if neithe is specified. The computation uses a k-step constuction with an appopiate initial estimate fo location o scale o location and scale, espectively. Valid candidates ae e.g. median and/o MAD (default) as well as Kolmogoov(-Sminov) o Cam\ e von Mises minimum distance estimatos; cf. Riede (1994) and Kohl (2005). In case package Biobase fom Bioconducto is installed as is suggested, median and/o MAD ae computed using function owmedians. These functions ae optimized fo the situation whee one has a matix and wants to compute the optimally obust estimato fo evey ow, espectively column of this matix. In paticula, the amount of coss eos is assumed to be constant fo all ows, espectively columns. If the amount of goss eos (contamination) is known, it can be specified by eps. The adius of the coesponding infinitesimal contamination neighbohood is obtained by multiplying eps by the squae oot of the sample size. If the amount of goss eos (contamination) is unknown, ty to find a ough estimate fo the amount of goss eos, such that it lies between eps.lowe and eps.uppe. In case eps.lowe is specified and eps.uppe is missing, eps.uppe is set to 0.5. In case eps.uppe is specified and eps.lowe is missing, eps.lowe is set to 0. If neithe eps no eps.lowe and/o eps.uppe is specified, eps.lowe and eps.uppe ae set to 0 and 0.5, espectively. If eps is missing, the adius-minimax estimato in sense of Riede et al. (2008), espectively Section 2.2 of Kohl (2005) is etuned. In case of location, espectively scale one additionally has to specify sd, espectively mean whee sd and mean can be a single numbe, i.e., identical fo all ows, espectively columns, o a vecto with length identical to the numbe of ows, espectively columns. Fo sample size <= 2, median and/o MAD ae used fo estimation. If eps = 0, mean and/o sd ae computed. Object of class "kstepestimate". Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation. Riede, H. (1994) Robust Asymptotic Statistics. New Yok: Spinge. Riede, H., Kohl, M. and Ruckdeschel, P. (2008) The Costs of not Knowing the Radius. Statistical Methods and Applications 17(1) Extended vesion: M. Kohl, P. Ruckdeschel, and H. Riede (2010). Infinitesimally Robust Estimation in Geneal Smoothly Paametized Models. Statistical Methods and Application, 19(3):

34 34 soptic oblox, kstepestimate-class ind <- binom(200, size=1, pob=0.05) X <- matix(nom(200, mean=ind*3, sd=(1-ind) + ind*9), now = 2) owroblox(x) owroblox(x, k = 3) owroblox(x, eps = 0.05) owroblox(x, eps = 0.05, k = 3) X1 <- t(x) colroblox(x1) colroblox(x1, k = 3) colroblox(x1, eps = 0.05) colroblox(x1, eps = 0.05, k = 3) X2 <- bind(nom(100, mean = -2, sd = 3), nom(100, mean = -1, sd = 4)) owroblox(x2, sd = c(3, 4)) owroblox(x2, eps = 0.03, sd = c(3, 4)) owroblox(x2, sd = c(3, 4), k = 4) owroblox(x2, eps = 0.03, sd = c(3, 4), k = 4) X3 <- cbind(nom(100, mean = -2, sd = 3), nom(100, mean = 1, sd = 2)) colroblox(x3, mean = c(-2, 1)) colroblox(x3, eps = 0.02, mean = c(-2, 1)) colroblox(x3, mean = c(-2, 1), k = 4) colroblox(x3, eps = 0.02, mean = c(-2, 1), k = 4) soptic Computation of the optimally obust IC fo AL estimatos The function soptic computes the optimally obust IC fo AL estimatos in case of nomal scale and (convex) contamination neighbohoods. The definition of these estimatos can be found in Riede (1994) o Kohl (2005), espectively. soptic(, mean = 0, sd = 1, bup = 1000, delta = 1e-06, itmax = 100, computeic = TRUE) mean sd non-negative eal: neighbohood adius. specified mean. specified standad deviation.

35 soptic 35 bup delta itmax computeic positive eal: the uppe end point of the inteval to be seached fo the clipping bound b. the desied accuacy (convegence toleance). the maximum numbe of iteations. logical: should IC be computed. See details below. If computeic is FALSE only the Lagange multiplies A, a, and b contained in the optimally obust IC ae computed. If computeic is TRUE an object of class "ContIC" is etuned, othewise a list of Lagange multiplies A a b standadizing constant centeing constant optimal clipping bound Riede, H. (1994) Robust Asymptotic Statistics. New Yok: Spinge. Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation. ContIC-class, oblox IC1 <- soptic( = 0.1) distexoptions("eelativetoleance" = 1e-12) checkic(ic1) distexoptions("eelativetoleance" =.Machine$double.eps^0.25) # default cent(ic1) clip(ic1) stand(ic1)

36 36 showdown showdown Estimato Showdown by Monte-Calo Study. The function showdown can be used to pefom Monte-Calo studies compaing a competito with mx estimatos in case of nomal location and scale. In addition, maximum likelihood (ML) estimatos (mean and sd) and median and MAD ae computed. The compaison is based on the empiical MSE. showdown(n, M, eps, contd, seed = 123, estfun, estmean, estsd, eps.lowe = 0, eps.uppe = 0.05, steps = 3L, fsco = TRUE, plot1 = FALSE, plot2 = FALSE, plot3 = FALSE) n intege; sample size, should be at least 3. M intege; Monte-Calo eplications. eps amount of contamination in [0, 0.5]. contd object of class "UnivaiateDistibution"; contaminating distibution. seed andom seed. estfun function to compute location and scale estimato; see details below. estmean function to compute location estimato; see details below. estsd function to compute scale estimato; see details below. eps.lowe used by mx estimato. eps.uppe used by mx estimato. steps intege; steps used fo estimato constuction. fsco logical; use finite-sample coection. plot1 logical; plot cdf of ideal and eal distibution. plot2 logical; plot 20 (o M if M < 20) andomly selected samples. plot3 logical; geneate boxplots of the esults. Nomal location and scale with mean = 0 and sd = 1 is used as ideal model (without estiction due to equivaiance). Since thee is no estimato which yields eliable esults if 50 pecent o moe of the obsevations ae contaminated, we use a modification whee we e-simulate all samples including at least 50 pecent contaminated data. If estfun is specified it has to compute and etun a location and scale estimate (vecto of length 2). One can also specify the location and scale estimato sepaately by using estmean and estsd whee estmean computes and etuns the location estimate and estsd the scale estimate. We use funtion owroblox fo the computation of the mx estimato.

37 showdown 37 Data.fame including empiical MSE (standadized by sample size n) and elmse with espect to the mx estimato. Kohl, M. (2005) Numeical Contibutions to the Asymptotic Theoy of Robustness. Bayeuth: Dissetation. Riede, H. (1994) Robust Asymptotic Statistics. New Yok: Spinge. Riede, H., Kohl, M. and Ruckdeschel, P. (2008) The Costs of not Knowing the Radius. Statistical Methods and Applications 17(1) Extended vesion: M. Kohl, P. Ruckdeschel, and H. Riede (2010). Infinitesimally Robust Estimation in Geneal Smoothly Paametized Models. Statistical Methods and Application, 19(3): owroblox libay(mass) ## compae with Hube s Poposal 2 showdown(n = 20, M = 100, eps = 0.02, contd = Nom(mean = 3, sd = 3), estfun = function(x){ unlist(hubes(x)) }, plot1 = TRUE, plot2 = TRUE, plot3 = TRUE) ## compae with Hube M estimato with MAD scale showdown(n = 20, M = 100, eps = 0.02, contd = Nom(mean = 3, sd = 3), estfun = function(x){ unlist(hube(x)) }, plot1 = TRUE, plot2 = TRUE, plot3 = TRUE)

38 Index Topic package RobLox-package, 2 Topic obust finitesamplecoection, 4 loptic, 5 lsoptic.al, 6 lsoptic.an1, 9 lsoptic.an2, 10 lsoptic.anmad, 11 lsoptic.bm, 12 lsoptic.ha3, 13 lsoptic.ha4, 15 lsoptic.hamad, 16 lsoptic.hu1, 17 lsoptic.hu2, 18 lsoptic.hu2a, 19 lsoptic.hu3, 21 lsoptic.humad, 22 lsoptic.m, 23 lsoptic.mm2, 24 lsoptic.tu1, 25 lsoptic.tu2, 26 lsoptic.tumad, 28 oblox, 29 owroblox and colroblox, 32 soptic, 34 showdown, 36 lsoptic.bm, 12 lsoptic.ha3, 13 lsoptic.ha4, 15 lsoptic.hamad, 16 lsoptic.hu1, 17 lsoptic.hu2, 18 lsoptic.hu2a, 19 lsoptic.hu3, 21 lsoptic.humad, 22 lsoptic.m, 23 lsoptic.mm2, 24 lsoptic.tu1, 25 lsoptic.tu2, 26 lsoptic.tumad, 28 RobLox (RobLox-package), 2 oblox, 5 7, 29, 34, 35 RobLox-package, 2 optest, 30 owroblox, 5, 36, 37 owroblox (owroblox and colroblox), 32 owroblox and colroblox, 32 soptic, 30, 34 showdown, 36 colroblox, 5 colroblox (owroblox and colroblox), 32 finitesamplecoection, 4, 29, 32 MLEstimato, 30 loptic, 5, 30 lsoptic.al, 6, 30 lsoptic.an1, 9 lsoptic.an2, 10 lsoptic.anmad, 11 38