| Title: | Optimally Robust Influence Curves and Estimators for Location and Scale |
|---|---|
| Description: | Functions for the determination of optimally robust influence curves and estimators in case of normal location and/or scale (see Chapter 8 in Kohl (2005) <https://epub.uni-bayreuth.de/839/2/DissMKohl.pdf>). |
| Authors: | Matthias Kohl [cre, cph], Peter Ruckdeschel [aut, cph] |
| Maintainer: | Matthias Kohl <[email protected]> |
| License: | LGPL-3 |
| Version: | 1.2.2 |
| Built: | 2024-09-04 19:18:17 UTC |
| Source: | https://github.com/r-forge/robast |
Functions for the determination of optimally robust influence curves and estimators in case of normal location and/or scale (see Chapter 8 in Kohl (2005) <https://epub.uni-bayreuth.de/839/2/DissMKohl.pdf>).
Note: The first two numbers of package versions do not necessarily reflect package-individual development, but rather are chosen for the RobAStXXX family as a whole in order to ease updating "depends" information.
Matthias Kohl [email protected]
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.
H. Rieder (1994): Robust Asymptotic Statistics. Springer. 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. 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. 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. doi:10.1186/1471-2105-11-583.
M. Kohl (2012). Bounded influence estimation for regression and scale. Statistics, 46(4): 437-488. doi:10.1080/02331888.2010.540668
library(RobLox) ind <- rbinom(100, size=1, prob=0.05) x <- rnorm(100, mean=ind*3, sd=(1-ind) + ind*9) roblox(x) res <- roblox(x, eps.lower = 0.01, eps.upper = 0.1, returnIC = TRUE) estimate(res) confint(res) confint(res, method = symmetricBias()) pIC(res) ## don't run to reduce check time on CRAN ## Not run: checkIC(pIC(res)) Risks(pIC(res)) Infos(pIC(res)) plot(pIC(res)) infoPlot(pIC(res)) ## End(Not run) ## row-wise application ind <- rbinom(200, size=1, prob=0.05) X <- matrix(rnorm(200, mean=ind*3, sd=(1-ind) + ind*9), nrow = 2) rowRoblox(X)library(RobLox) ind <- rbinom(100, size=1, prob=0.05) x <- rnorm(100, mean=ind*3, sd=(1-ind) + ind*9) roblox(x) res <- roblox(x, eps.lower = 0.01, eps.upper = 0.1, returnIC = TRUE) estimate(res) confint(res) confint(res, method = symmetricBias()) pIC(res) ## don't run to reduce check time on CRAN ## Not run: checkIC(pIC(res)) Risks(pIC(res)) Infos(pIC(res)) plot(pIC(res)) infoPlot(pIC(res)) ## End(Not run) ## row-wise application ind <- rbinom(200, size=1, prob=0.05) X <- matrix(rnorm(200, mean=ind*3, sd=(1-ind) + ind*9), nrow = 2) rowRoblox(X)
Given some radius and some sample size the function computes the corresponding finite-sample corrected radius.
finiteSampleCorrection(r, n, model = "locsc")finiteSampleCorrection(r, n, model = "locsc")
r |
asymptotic radius (non-negative numeric) |
n |
sample size |
model |
has to be |
The finite-sample correction is based on empirical results obtained via simulation studies.
Given some radius of a shrinking contamination neighborhood which leads to an asymptotically optimal robust estimator, the finite-sample empirical MSE based on contaminated samples was minimized for this class of asymptotically optimal estimators and the corresponding finite-sample radius determined and saved.
The computation is based on the saved results of these Monte-Carlo simulations.
Finite-sample corrected radius.
Matthias Kohl [email protected]
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.
H. Rieder (1994): Robust Asymptotic Statistics. Springer. doi:10.1007/978-1-4684-0624-5
M. Kohl and H.P. Deigner (2010). Preprocessing of gene expression data by optimally robust estimators. BMC Bioinformatics 11, 583. doi:10.1186/1471-2105-11-583.
finiteSampleCorrection(n = 3, r = 0.001, model = "locsc") finiteSampleCorrection(n = 10, r = 0.02, model = "loc") finiteSampleCorrection(n = 250, r = 0.15, model = "sc")finiteSampleCorrection(n = 3, r = 0.001, model = "locsc") finiteSampleCorrection(n = 10, r = 0.02, model = "loc") finiteSampleCorrection(n = 250, r = 0.15, model = "sc")
The function rlOptIC computes the optimally robust IC for
AL estimators in case of normal location and (convex) contamination
neighborhoods. The definition of these estimators can be found
in Rieder (1994) or Kohl (2005), respectively.
rlOptIC(r, mean = 0, sd = 1, bUp = 1000, computeIC = TRUE)rlOptIC(r, mean = 0, sd = 1, bUp = 1000, computeIC = TRUE)
r |
non-negative real: neighborhood radius. |
mean |
specified mean. |
sd |
specified standard deviation. |
bUp |
positive real: the upper end point of the interval to be searched for the clipping bound b. |
computeIC |
logical: should IC be computed. See details below. |
If 'computeIC' is 'FALSE' only the Lagrange multipliers 'A', 'a', and 'b' contained in the optimally robust IC are computed.
If 'computeIC' is 'TRUE' an object of class "ContIC" is returned,
otherwise a list of Lagrange multipliers
A |
standardizing constant |
a |
centering constant; always '= 0' is this symmetric setup |
b |
optimal clipping bound |
Matthias Kohl [email protected]
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.
H. Rieder (1994): Robust Asymptotic Statistics. Springer. doi:10.1007/978-1-4684-0624-5
M. Kohl, P. Ruckdeschel, and H. Rieder (2010). Infinitesimally Robust Estimation in General Smoothly Parametrized Models. Statistical Methods and Applications 19(3): 333-354. doi:10.1007/s10260-010-0133-0.
IC1 <- rlOptIC(r = 0.1) distrExOptions("ErelativeTolerance" = 1e-12) checkIC(IC1) distrExOptions("ErelativeTolerance" = .Machine$double.eps^0.25) # default Risks(IC1) cent(IC1) clip(IC1) stand(IC1) plot(IC1)IC1 <- rlOptIC(r = 0.1) distrExOptions("ErelativeTolerance" = 1e-12) checkIC(IC1) distrExOptions("ErelativeTolerance" = .Machine$double.eps^0.25) # default Risks(IC1) cent(IC1) clip(IC1) stand(IC1) plot(IC1)
The function rlsOptIC.AL computes the optimally robust IC for
AL estimators in case of normal location with unknown scale and
(convex) contamination neighborhoods. The definition of
these estimators can be found in Section 8.2 of Kohl (2005).
rlsOptIC.AL(r, mean = 0, sd = 1, A.loc.start = 1, a.sc.start = 0, A.sc.start = 0.5, bUp = 1000, delta = 1e-6, itmax = 100, check = FALSE, computeIC = TRUE)rlsOptIC.AL(r, mean = 0, sd = 1, A.loc.start = 1, a.sc.start = 0, A.sc.start = 0.5, bUp = 1000, delta = 1e-6, itmax = 100, check = FALSE, computeIC = TRUE)
r |
non-negative real: neighborhood radius. |
mean |
specified mean. |
sd |
specified standard deviation. |
A.loc.start |
positive real: starting value for the standardizing constant of the location part. |
a.sc.start |
real: starting value for centering constant of the scale part. |
A.sc.start |
positive real: starting value for the standardizing constant of the scale part. |
bUp |
positive real: the upper end point of the interval to be searched for the clipping bound b. |
delta |
the desired accuracy (convergence tolerance). |
itmax |
the maximum number of iterations. |
check |
logical: should constraints be checked. |
computeIC |
logical: should IC be computed. See details below. |
The Lagrange multipliers contained in the expression
of the optimally robust IC can be accessed via the
accessor functions cent, clip and stand.
If 'computeIC' is 'FALSE' only the Lagrange multipliers 'A', 'a',
and 'b' contained in the optimally robust IC are computed.
If 'computeIC' is 'TRUE' an object of class "ContIC" is returned,
otherwise a list of Lagrange multipliers
A |
standardizing matrix |
a |
centering vector |
b |
optimal clipping bound |
Matthias Kohl [email protected]
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.
H. Rieder (1994): Robust Asymptotic Statistics. Springer. doi:10.1007/978-1-4684-0624-5
M. Kohl, P. Ruckdeschel, and H. Rieder (2010). Infinitesimally Robust Estimation in General Smoothly Parametrized Models. Statistical Methods and Applications 19(3): 333-354. doi:10.1007/s10260-010-0133-0.
IC1 <- rlsOptIC.AL(r = 0.1, check = TRUE) distrExOptions("ErelativeTolerance" = 1e-12) checkIC(IC1) distrExOptions("ErelativeTolerance" = .Machine$double.eps^0.25) # default Risks(IC1) cent(IC1) clip(IC1) stand(IC1) ## don't run to reduce check time on CRAN ## Not run: plot(IC1) infoPlot(IC1) ## k-step estimation ## better use function roblox (see ?roblox) ## 1. data: random sample ind <- rbinom(100, size=1, prob=0.05) x <- rnorm(100, mean=0, sd=(1-ind) + ind*9) mean(x) sd(x) median(x) mad(x) ## 2. Kolmogorov(-Smirnov) minimum distance estimator (default) ## -> we use it as initial estimate for one-step construction (est0 <- MDEstimator(x, ParamFamily = NormLocationScaleFamily())) ## 3.1 one-step estimation: radius known IC1 <- rlsOptIC.AL(r = 0.5, mean = estimate(est0)[1], sd = estimate(est0)[2]) (est1 <- oneStepEstimator(x, IC1, est0)) ## 3.2 k-step estimation: radius known ## Choose k = 3 (est2 <- kStepEstimator(x, IC1, est0, steps = 3L)) ## 4.1 one-step estimation: radius unknown ## take least favorable radius r = 0.579 ## cf. Table 8.1 in Kohl(2005) IC2 <- rlsOptIC.AL(r = 0.579, mean = estimate(est0)[1], sd = estimate(est0)[2]) (est3 <- oneStepEstimator(x, IC2, est0)) ## 4.2 k-step estimation: radius unknown ## take least favorable radius r = 0.579 ## cf. Table 8.1 in Kohl(2005) ## choose k = 3 (est4 <- kStepEstimator(x, IC2, est0, steps = 3L)) ## End(Not run)IC1 <- rlsOptIC.AL(r = 0.1, check = TRUE) distrExOptions("ErelativeTolerance" = 1e-12) checkIC(IC1) distrExOptions("ErelativeTolerance" = .Machine$double.eps^0.25) # default Risks(IC1) cent(IC1) clip(IC1) stand(IC1) ## don't run to reduce check time on CRAN ## Not run: plot(IC1) infoPlot(IC1) ## k-step estimation ## better use function roblox (see ?roblox) ## 1. data: random sample ind <- rbinom(100, size=1, prob=0.05) x <- rnorm(100, mean=0, sd=(1-ind) + ind*9) mean(x) sd(x) median(x) mad(x) ## 2. Kolmogorov(-Smirnov) minimum distance estimator (default) ## -> we use it as initial estimate for one-step construction (est0 <- MDEstimator(x, ParamFamily = NormLocationScaleFamily())) ## 3.1 one-step estimation: radius known IC1 <- rlsOptIC.AL(r = 0.5, mean = estimate(est0)[1], sd = estimate(est0)[2]) (est1 <- oneStepEstimator(x, IC1, est0)) ## 3.2 k-step estimation: radius known ## Choose k = 3 (est2 <- kStepEstimator(x, IC1, est0, steps = 3L)) ## 4.1 one-step estimation: radius unknown ## take least favorable radius r = 0.579 ## cf. Table 8.1 in Kohl(2005) IC2 <- rlsOptIC.AL(r = 0.579, mean = estimate(est0)[1], sd = estimate(est0)[2]) (est3 <- oneStepEstimator(x, IC2, est0)) ## 4.2 k-step estimation: radius unknown ## take least favorable radius r = 0.579 ## cf. Table 8.1 in Kohl(2005) ## choose k = 3 (est4 <- kStepEstimator(x, IC2, est0, steps = 3L)) ## End(Not run)
The function rlsOptIC.An1 computes the optimally robust IC for
An1 estimators in case of normal location with unknown scale and
(convex) contamination neighborhoods. The definition of
these estimators can be found in Subsection 8.5.3 of Kohl (2005).
rlsOptIC.An1(r, aUp = 2.5, delta = 1e-06)rlsOptIC.An1(r, aUp = 2.5, delta = 1e-06)
r |
non-negative real: neighborhood radius. |
aUp |
positive real: the upper end point of the interval to be searched for a. |
delta |
the desired accuracy (convergence tolerance). |
The optimal value of the tuning constant a can be read off
from the slot Infos of the resulting IC.
Object of class "IC"
Matthias Kohl [email protected]
Andrews, D.F., Bickel, P.J., Hampel, F.R., Huber, P.J., Rogers, W.H. and Tukey, J.W. (1972) Robust estimates of location. Princeton University Press.
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.
IC1 <- rlsOptIC.An1(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) ## don't run to reduce check time on CRAN ## Not run: plot(IC1) infoPlot(IC1) ## End(Not run)IC1 <- rlsOptIC.An1(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) ## don't run to reduce check time on CRAN ## Not run: plot(IC1) infoPlot(IC1) ## End(Not run)
The function rlsOptIC.An2 computes the optimally robust IC for
An2 estimators in case of normal location with unknown scale and
(convex) contamination neighborhoods. The definition of
these estimators can be found in Subsection 8.5.3 of Kohl (2005).
rlsOptIC.An2(r, a.start = 1.5, k.start = 1.5, delta = 1e-06, MAX = 100)rlsOptIC.An2(r, a.start = 1.5, k.start = 1.5, delta = 1e-06, MAX = 100)
r |
non-negative real: neighborhood radius. |
a.start |
positive real: starting value for a. |
k.start |
positive real: starting value for k. |
delta |
the desired accuracy (convergence tolerance). |
MAX |
if a or k are beyond the admitted values,
|
The computation of the optimally robust IC for An2 estimators
is based on optim where MAX is used to
control the constraints on a and k. The optimal values of the
tuning constants a and k can be read off from the slot
Infos of the resulting IC.
Object of class "IC"
Matthias Kohl [email protected]
Andrews, D.F., Bickel, P.J., Hampel, F.R., Huber, P.J., Rogers, W.H. and Tukey, J.W. (1972) Robust estimates of location. Princeton University Press.
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.
IC1 <- rlsOptIC.An2(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)IC1 <- rlsOptIC.An2(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)
The function rlsOptIC.AnMad computes the optimally robust IC for
AnMad estimators in case of normal location with unknown scale and
(convex) contamination neighborhoods. These estimators were
considered in Andrews et al. (1972). A definition of these estimators
can also be found in Subsection 8.5.3 of Kohl (2005).
rlsOptIC.AnMad(r, aUp = 2.5, delta = 1e-06)rlsOptIC.AnMad(r, aUp = 2.5, delta = 1e-06)
r |
non-negative real: neighborhood radius. |
aUp |
positive real: the upper end point of the interval to be searched for a. |
delta |
the desired accuracy (convergence tolerance). |
The optimal value of the tuning constant a can be read off
from the slot Infos of the resulting IC.
Object of class "IC"
Matthias Kohl [email protected]
Andrews, D.F., Bickel, P.J., Hampel, F.R., Huber, P.J., Rogers, W.H. and Tukey, J.W. (1972) Robust estimates of location. Princeton University Press.
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.
IC1 <- rlsOptIC.AnMad(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)IC1 <- rlsOptIC.AnMad(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)
The function rlsOptIC.BM computes the optimally robust IC for
BM estimators in case of normal location with unknown scale and
(convex) contamination neighborhoods. These estimators were proposed
by Bednarski and Mueller (2001). A definition of these
estimators can also be found in Section 8.4 of Kohl (2005).
rlsOptIC.BM(r, bL.start = 2, bS.start = 1.5, delta = 1e-06, MAX = 100)rlsOptIC.BM(r, bL.start = 2, bS.start = 1.5, delta = 1e-06, MAX = 100)
r |
non-negative real: neighborhood radius. |
bL.start |
positive real: starting value for |
bS.start |
positive real: starting value for |
delta |
the desired accuracy (convergence tolerance). |
MAX |
if |
The computation of the optimally robust IC for BM estimators
is based on optim where MAX is used to
control the constraints on
and . The optimal values of the
tuning constants , ,
and can be read off
from the slot Infos of the resulting IC.
Object of class "IC"
Matthias Kohl [email protected]
Bednarski, T and Mueller, C.H. (2001) Optimal bounded influence regression and scale M-estimators in the context of experimental design. Statistics, 35(4): 349-369.
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.
M. Kohl (2012). Bounded influence estimation for regression and scale. Statistics, 46(4): 437-488. doi:10.1080/02331888.2010.540668
IC1 <- rlsOptIC.BM(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)IC1 <- rlsOptIC.BM(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)
The function rlsOptIC.Ha3 computes the optimally robust IC for
Ha3 estimators in case of normal location with unknown scale and
(convex) contamination neighborhoods. The definition of
these estimators can be found in Subsection 8.5.2 of Kohl (2005).
rlsOptIC.Ha3(r, a.start = 0.25, b.start = 2.5, c.start = 5, delta = 1e-06, MAX = 100)rlsOptIC.Ha3(r, a.start = 0.25, b.start = 2.5, c.start = 5, delta = 1e-06, MAX = 100)
r |
non-negative real: neighborhood radius. |
a.start |
positive real: starting value for a. |
b.start |
positive real: starting value for b. |
c.start |
positive real: starting value for c. |
delta |
the desired accuracy (convergence tolerance). |
MAX |
if a or b or c are beyond the admitted values,
|
The computation of the optimally robust IC for Ha3 estimators
is based on optim where MAX is used to
control the constraints on a, b and c. The optimal values of
the tuning constants a, b and c can be read off
from the slot Infos of the resulting IC.
Object of class "IC"
Matthias Kohl [email protected]
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.
IC1 <- rlsOptIC.Ha3(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) ## don't run to reduce check time on CRAN ## Not run: plot(IC1) infoPlot(IC1) ## End(Not run)IC1 <- rlsOptIC.Ha3(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) ## don't run to reduce check time on CRAN ## Not run: plot(IC1) infoPlot(IC1) ## End(Not run)
The function rlsOptIC.Ha4 computes the optimally robust IC for
Ha4 estimators in case of normal location with unknown scale and
(convex) contamination neighborhoods. The definition of
these estimators can be found in Subsection 8.5.2 of Kohl (2005).
rlsOptIC.Ha4(r, a.start = 0.25, b.start = 2.5, c.start = 5, k.start = 1, delta = 1e-06, MAX = 100)rlsOptIC.Ha4(r, a.start = 0.25, b.start = 2.5, c.start = 5, k.start = 1, delta = 1e-06, MAX = 100)
r |
non-negative real: neighborhood radius. |
a.start |
positive real: starting value for a. |
b.start |
positive real: starting value for b. |
c.start |
positive real: starting value for c. |
k.start |
positive real: starting value for k. |
delta |
the desired accuracy (convergence tolerance). |
MAX |
if a or b or c or k are beyond the admitted values,
|
The computation of the optimally robust IC for Ha4 estimators
is based on optim where MAX is used to
control the constraints on a, b, c and k. The optimal values of
the tuning constants a, b, c and k can be read off
from the slot Infos of the resulting IC.
Object of class "IC"
Matthias Kohl [email protected]
Marazzi, A. (1993) Algorithms, routines, and S functions for robust statistics. Wadsworth and Brooks / Cole.
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.
IC1 <- rlsOptIC.Ha4(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)IC1 <- rlsOptIC.Ha4(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)
The function rlsOptIC.HuMad computes the optimally robust IC for
HuMad estimators in case of normal location with unknown scale and
(convex) contamination neighborhoods. These estimators were
considered in Andrews et al. (1972). A definition of these estimators
can also be found in Subsection 8.5.2 of Kohl (2005).
rlsOptIC.HaMad(r, a.start = 0.25, b.start = 2.5, c.start = 5, delta = 1e-06, MAX = 100)rlsOptIC.HaMad(r, a.start = 0.25, b.start = 2.5, c.start = 5, delta = 1e-06, MAX = 100)
r |
non-negative real: neighborhood radius. |
a.start |
positive real: starting value for a. |
b.start |
positive real: starting value for b. |
c.start |
positive real: starting value for c. |
delta |
the desired accuracy (convergence tolerance). |
MAX |
if a or b or c are beyond the admitted values,
|
The computation of the optimally robust IC for HaMad estimators
is based on optim where MAX is used to
control the constraints on a, b and c. The optimal values of
the tuning constants a, b, and c can be read off
from the slot Infos of the resulting IC.
Object of class "IC"
Matthias Kohl [email protected]
Andrews, D.F., Bickel, P.J., Hampel, F.R., Huber, P.J., Rogers, W.H. and Tukey, J.W. (1972) Robust estimates of location. Princeton University Press.
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.
IC1 <- rlsOptIC.HaMad(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)IC1 <- rlsOptIC.HaMad(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)
The function rlsOptIC.Hu1 computes the optimally robust IC for
Hu1 estimators in case of normal location with unknown scale and
(convex) contamination neighborhoods. These estimators were
proposed by Huber (1964), Proposal 2. A definition of these
estimators can also be found in Subsection 8.5.1 of Kohl (2005).
rlsOptIC.Hu1(r, kUp = 2.5, delta = 1e-06)rlsOptIC.Hu1(r, kUp = 2.5, delta = 1e-06)
r |
non-negative real: neighborhood radius. |
kUp |
positive real: the upper end point of the interval to be searched for k. |
delta |
the desired accuracy (convergence tolerance). |
The optimal value of the tuning constant k can be read off
from the slot Infos of the resulting IC.
Object of class "IC"
Matthias Kohl [email protected]
Huber, P.J. (1964) Robust estimation of a location parameter. Ann. Math. Stat. 35: 73–101.
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.
IC1 <- rlsOptIC.Hu1(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)IC1 <- rlsOptIC.Hu1(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)
The function rlsOptIC.Hu2 computes the optimally robust IC for
Hu2 estimators in case of normal location with unknown scale and
(convex) contamination neighborhoods. These estimators were
proposed in Example 6.4.1 of Huber (1981). A definition of these
estimators can also be found in Subsection 8.5.1 of Kohl (2005).
rlsOptIC.Hu2(r, k.start = 1.5, c.start = 1.5, delta = 1e-06, MAX = 100)rlsOptIC.Hu2(r, k.start = 1.5, c.start = 1.5, delta = 1e-06, MAX = 100)
r |
non-negative real: neighborhood radius. |
k.start |
positive real: starting value for k. |
c.start |
positive real: starting value for c. |
delta |
the desired accuracy (convergence tolerance). |
MAX |
if k1 or k2 are beyond the admitted values,
|
The computation of the optimally robust IC for Hu2 estimators
is based on optim where MAX is used to
control the constraints on k and c. The optimal values of
the tuning constants k and c can be read off
from the slot Infos of the resulting IC.
Object of class "IC"
Matthias Kohl [email protected]
Huber, P.J. (1981) Robust Statistics. New York: Wiley.
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.
IC1 <- rlsOptIC.Hu2(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)IC1 <- rlsOptIC.Hu2(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)
The function rlsOptIC.Hu2a computes the optimally robust IC for
Hu2a estimators in case of normal location with unknown scale and
(convex) contamination neighborhoods. These estimators are a
simple modification of Huber (1964), Proposal 2 where we, in addition,
admit a clipping from below. The definition of
these estimators can be found in Subsection 8.5.1 of Kohl (2005).
rlsOptIC.Hu2a(r, k1.start = 0.25, k2.start = 2.5, delta = 1e-06, MAX = 100)rlsOptIC.Hu2a(r, k1.start = 0.25, k2.start = 2.5, delta = 1e-06, MAX = 100)
r |
non-negative real: neighborhood radius. |
k1.start |
positive real: starting value for k1. |
k2.start |
positive real: starting value for k2. |
delta |
the desired accuracy (convergence tolerance). |
MAX |
if k1 or k2 are beyond the admitted values,
|
The computation of the optimally robust IC for Hu2a estimators
is based on optim where MAX is used to
control the constraints on k1 and k2. The optimal values of
the tuning constants k1 and k2 can be read off
from the slot Infos of the resulting IC.
Object of class "IC"
Matthias Kohl [email protected]
Huber, P.J. (1964) Robust estimation of a location parameter. Ann. Math. Stat. 35: 73–101.
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.
IC1 <- rlsOptIC.Hu2a(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)IC1 <- rlsOptIC.Hu2a(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)
The function rlsOptIC.Hu3 computes the optimally robust IC for
Hu3 estimators in case of normal location with unknown scale and
(convex) contamination neighborhoods. The definition of
these estimators can be found in Subsection 8.5.1 of Kohl (2005).
rlsOptIC.Hu3(r, k.start = 1, c1.start = 0.1, c2.start = 0.5, delta = 1e-06, MAX = 100)rlsOptIC.Hu3(r, k.start = 1, c1.start = 0.1, c2.start = 0.5, delta = 1e-06, MAX = 100)
r |
non-negative real: neighborhood radius. |
k.start |
positive real: starting value for k. |
c1.start |
positive real: starting value for c1. |
c2.start |
positive real: starting value for c2. |
delta |
the desired accuracy (convergence tolerance). |
MAX |
if k or c1 or c2 are beyond the admitted values,
|
The computation of the optimally robust IC for Hu2 estimators
is based on optim where MAX is used to
control the constraints on k, c1 and c2. The optimal values of
the tuning constants k, c1 and c2 can be read off
from the slot Infos of the resulting IC.
Object of class "IC"
Matthias Kohl [email protected]
Huber, P.J. (1981) Robust Statistics. New York: Wiley.
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.
IC1 <- rlsOptIC.Hu3(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)IC1 <- rlsOptIC.Hu3(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)
The function rlsOptIC.HuMad computes the optimally robust IC for
HuMad estimators in case of normal location with unknown scale and
(convex) contamination neighborhoods. These estimators were
proposed by Andrews et al. (1972), p. 12. A definition of these
estimators can also be found in Subsection 8.5.1 of Kohl (2005).
rlsOptIC.HuMad(r, kUp = 2.5, delta = 1e-06)rlsOptIC.HuMad(r, kUp = 2.5, delta = 1e-06)
r |
non-negative real: neighborhood radius. |
kUp |
positive real: the upper end point of the interval to be searched for k. |
delta |
the desired accuracy (convergence tolerance). |
The optimal value of the tuning constant k can be read off
from the slot Infos of the resulting IC.
Object of class "IC"
Matthias Kohl [email protected]
Andrews, D.F., Bickel, P.J., Hampel, F.R., Huber, P.J., Rogers, W.H. and Tukey, J.W. (1972) Robust estimates of location. Princeton University Press.
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.
IC1 <- rlsOptIC.HuMad(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)IC1 <- rlsOptIC.HuMad(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)
The function rlsOptIC.M computes the optimally robust IC for
M estimators in case of normal location with unknown scale and
(convex) contamination neighborhoods. The definition of
these estimators can be found in Section 8.3 of Kohl (2005).
rlsOptIC.M(r, ggLo = 0.5, ggUp = 1.5, a1.start = 0.75, a3.start = 0.25, bUp = 1000, delta = 1e-05, itmax = 100, check = FALSE)rlsOptIC.M(r, ggLo = 0.5, ggUp = 1.5, a1.start = 0.75, a3.start = 0.25, bUp = 1000, delta = 1e-05, itmax = 100, check = FALSE)
r |
non-negative real: neighborhood radius. |
ggLo |
non-negative real: the lower end point of the interval to be searched
for |
ggUp |
positive real: the upper end point of the interval to be searched
for |
a1.start |
real: starting value for |
a3.start |
real: starting value for |
bUp |
positive real: upper bound used in the computation of the optimal clipping bound b. |
delta |
the desired accuracy (convergence tolerance). |
itmax |
the maximum number of iterations. |
check |
logical. Should constraints be checked. |
The optimal values of the tuning constants ,
, b and can be read off
from the slot Infos of the resulting IC.
Object of class "IC"
Matthias Kohl [email protected]
Huber, P.J. (1981) Robust Statistics. New York: Wiley.
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.
IC1 <- rlsOptIC.M(r = 0.1, check = TRUE) distrExOptions("ErelativeTolerance" = 1e-12) checkIC(IC1, NormLocationScaleFamily()) distrExOptions("ErelativeTolerance" = .Machine$double.eps^0.25) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)IC1 <- rlsOptIC.M(r = 0.1, check = TRUE) distrExOptions("ErelativeTolerance" = 1e-12) checkIC(IC1, NormLocationScaleFamily()) distrExOptions("ErelativeTolerance" = .Machine$double.eps^0.25) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)
The function rlsOptIC.MM2 computes the optimally robust IC for
MM2 estimators in case of normal location with unknown scale and
(convex) contamination neighborhoods. These estimators are based
on a proposal of Fraiman et al. (2001), p. 206. A definition of
these estimators can also be found in Section 8.6 of Kohl (2005).
rlsOptIC.MM2(r, c.start = 1.5, d.start = 2, delta = 1e-06, MAX = 100)rlsOptIC.MM2(r, c.start = 1.5, d.start = 2, delta = 1e-06, MAX = 100)
r |
non-negative real: neighborhood radius. |
c.start |
positive real: starting value for c. |
d.start |
positive real: starting value for d. |
delta |
the desired accuracy (convergence tolerance). |
MAX |
if a or k are beyond the admitted values,
|
The computation of the optimally robust IC for MM2 estimators
is based on optim where MAX is used to
control the constraints on c and d. The optimal values of
the tuning constants c and d can be read off from the slot
Infos of the resulting IC.
Object of class "IC"
Matthias Kohl [email protected]
Fraiman, R., Yohai, V.J. and Zamar, R.H. (2001) Optimal robust M-estimates of location. Ann. Stat. 29(1): 194-223.
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.
IC1 <- rlsOptIC.MM2(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)IC1 <- rlsOptIC.MM2(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)
The function rlsOptIC.Tu1 computes the optimally robust IC for
Tu1 estimators in case of normal location with unknown scale and
(convex) contamination neighborhoods. The definition of
these estimators can be found in Subsection 8.5.4 of Kohl (2005).
rlsOptIC.Tu1(r, aUp = 10, delta = 1e-06)rlsOptIC.Tu1(r, aUp = 10, delta = 1e-06)
r |
non-negative real: neighborhood radius. |
aUp |
positive real: the upper end point of the interval to be searched for a. |
delta |
the desired accuracy (convergence tolerance). |
The optimal value of the tuning constant a can be read off
from the slot Infos of the resulting IC.
Object of class "IC"
Matthias Kohl [email protected]
Beaton, A.E. and Tukey, J.W. (1974) The fitting of power series, meaning polynomials, illustrated on band-spectroscopic data. Discussions. Technometrics 16: 147–185.
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.
IC1 <- rlsOptIC.Tu1(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)IC1 <- rlsOptIC.Tu1(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)
The function rlsOptIC.Tu2 computes the optimally robust IC for
Tu2 estimators in case of normal location with unknown scale and
(convex) contamination neighborhoods. The definition of
these estimators can be found in Subsection 8.5.4 of Kohl (2005).
rlsOptIC.Tu2(r, a.start = 5, k.start = 1.5, delta = 1e-06, MAX = 100)rlsOptIC.Tu2(r, a.start = 5, k.start = 1.5, delta = 1e-06, MAX = 100)
r |
non-negative real: neighborhood radius. |
a.start |
positive real: starting value for a. |
k.start |
positive real: starting value for k. |
delta |
the desired accuracy (convergence tolerance). |
MAX |
if a or k are beyond the admitted values,
|
The computation of the optimally robust IC for Tu2 estimators
is based on optim where MAX is used to
control the constraints on a and k. The optimal values of
the tuning constant a and k can be read off from the slot
Infos of the resulting IC.
Object of class "IC"
Matthias Kohl [email protected]
Beaton, A.E. and Tukey, J.W. (1974) The fitting of power series, meaning polynomials, illustrated on band-spectroscopic data. Discussions. Technometrics 16: 147–185.
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.
IC1 <- rlsOptIC.Tu2(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)IC1 <- rlsOptIC.Tu2(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)
The function rlsOptIC.TuMad computes the optimally robust IC for
TuMad estimators in case of normal location with unknown scale and
(convex) contamination neighborhoods. The definition of
these estimators can be found in Subsection 8.5.4 of Kohl (2005).
rlsOptIC.TuMad(r, aUp = 10, delta = 1e-06)rlsOptIC.TuMad(r, aUp = 10, delta = 1e-06)
r |
non-negative real: neighborhood radius. |
aUp |
positive real: the upper end point of the interval to be searched for a. |
delta |
the desired accuracy (convergence tolerance). |
The optimal value of the tuning constant a can be read off
from the slot Infos of the resulting IC.
Object of class "IC"
Matthias Kohl [email protected]
Beaton, A.E. and Tukey, J.W. (1974) The fitting of power series, meaning polynomials, illustrated on band-spectroscopic data. Discussions. Technometrics 16: 147–185.
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.
IC1 <- rlsOptIC.TuMad(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)IC1 <- rlsOptIC.TuMad(r = 0.1) checkIC(IC1) Risks(IC1) Infos(IC1) plot(IC1) infoPlot(IC1)
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.
roblox(x, mean, sd, eps, eps.lower, eps.upper, initial.est, k = 1L, fsCor = TRUE, returnIC = FALSE, mad0 = 1e-4, na.rm = TRUE)roblox(x, mean, sd, eps, eps.lower, eps.upper, initial.est, k = 1L, fsCor = TRUE, returnIC = FALSE, mad0 = 1e-4, na.rm = TRUE)
x |
vector |
mean |
specified mean. |
sd |
specified standard deviation which has to be positive. |
eps |
positive real (0 < |
eps.lower |
positive real (0 <= |
eps.upper |
positive real ( |
initial.est |
initial estimate for |
k |
positive integer. k-step is used to compute the optimally robust estimator. |
fsCor |
logical: perform finite-sample correction. See function |
returnIC |
logical: should IC be returned. See details below. |
mad0 |
scale estimate used if computed MAD is equal to zero |
na.rm |
logical: if |
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.
Object of class "kStepEstimate".
Matthias Kohl [email protected]
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.
H. Rieder (1994): Robust Asymptotic Statistics. Springer. 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. 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. 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. doi:10.1186/1471-2105-11-583.
ContIC-class, rlOptIC,
rsOptIC, rlsOptIC.AL,
kStepEstimate-class,
roptest
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))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))
The functions rowRoblox and colRoblox compute
optimally robust estimates 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.
rowRoblox(x, mean, sd, eps, eps.lower, eps.upper, initial.est, k = 1L, fsCor = TRUE, mad0 = 1e-4, na.rm = TRUE) colRoblox(x, mean, sd, eps, eps.lower, eps.upper, initial.est, k = 1L, fsCor = TRUE, mad0 = 1e-4, na.rm = TRUE)rowRoblox(x, mean, sd, eps, eps.lower, eps.upper, initial.est, k = 1L, fsCor = TRUE, mad0 = 1e-4, na.rm = TRUE) colRoblox(x, mean, sd, eps, eps.lower, eps.upper, initial.est, k = 1L, fsCor = TRUE, mad0 = 1e-4, na.rm = TRUE)
x |
matrix or data.frame of (numeric) data values. |
mean |
specified mean. See details below. |
sd |
specified standard deviation which has to be positive. See also details below. |
eps |
positive real (0 < |
eps.lower |
positive real (0 <= |
eps.upper |
positive real ( |
initial.est |
initial estimate for |
k |
positive integer. k-step is used to compute the optimally robust estimator. |
fsCor |
logical: perform finite-sample correction. See function |
mad0 |
scale estimate used if computed MAD is equal to zero |
na.rm |
logical: if |
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 Cram\'er von
Mises minimum distance estimators; cf. Rieder (1994) and Kohl (2005). In case
package Biobase from Bioconductor is installed as is suggested,
median and/or MAD are computed using function rowMedians.
These functions are optimized for the situation where one has a matrix and wants to compute the optimally robust estimator for every row, respectively column of this matrix. In particular, the amount of cross errors is assumed to be constant for all rows, respectively columns.
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 can
be a single number, i.e., identical for all rows, respectively columns,
or a vector with length identical to the number of rows, respectively
columns.
For sample size <= 2, median and/or MAD are used for estimation.
If eps = 0, mean and/or sd are computed.
Object of class "kStepEstimate".
Matthias Kohl [email protected]
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.
H. Rieder (1994): Robust Asymptotic Statistics. Springer. 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. 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. 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. doi:10.1186/1471-2105-11-583.
ind <- rbinom(200, size=1, prob=0.05) X <- matrix(rnorm(200, mean=ind*3, sd=(1-ind) + ind*9), nrow = 2) rowRoblox(X) rowRoblox(X, k = 3) rowRoblox(X, eps = 0.05) rowRoblox(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 <- rbind(rnorm(100, mean = -2, sd = 3), rnorm(100, mean = -1, sd = 4)) rowRoblox(X2, sd = c(3, 4)) rowRoblox(X2, eps = 0.03, sd = c(3, 4)) rowRoblox(X2, sd = c(3, 4), k = 4) rowRoblox(X2, eps = 0.03, sd = c(3, 4), k = 4) X3 <- cbind(rnorm(100, mean = -2, sd = 3), rnorm(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)ind <- rbinom(200, size=1, prob=0.05) X <- matrix(rnorm(200, mean=ind*3, sd=(1-ind) + ind*9), nrow = 2) rowRoblox(X) rowRoblox(X, k = 3) rowRoblox(X, eps = 0.05) rowRoblox(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 <- rbind(rnorm(100, mean = -2, sd = 3), rnorm(100, mean = -1, sd = 4)) rowRoblox(X2, sd = c(3, 4)) rowRoblox(X2, eps = 0.03, sd = c(3, 4)) rowRoblox(X2, sd = c(3, 4), k = 4) rowRoblox(X2, eps = 0.03, sd = c(3, 4), k = 4) X3 <- cbind(rnorm(100, mean = -2, sd = 3), rnorm(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)
The function rsOptIC computes the optimally robust IC for
AL estimators in case of normal scale and (convex) contamination
neighborhoods. The definition of these estimators can be found
in Rieder (1994) or Kohl (2005), respectively.
rsOptIC(r, mean = 0, sd = 1, bUp = 1000, delta = 1e-06, itmax = 100, computeIC = TRUE)rsOptIC(r, mean = 0, sd = 1, bUp = 1000, delta = 1e-06, itmax = 100, computeIC = TRUE)
r |
non-negative real: neighborhood radius. |
mean |
specified mean. |
sd |
specified standard deviation. |
bUp |
positive real: the upper end point of the interval to be searched for the clipping bound b. |
delta |
the desired accuracy (convergence tolerance). |
itmax |
the maximum number of iterations. |
computeIC |
logical: should IC be computed. See details below. |
If 'computeIC' is 'FALSE' only the Lagrange multipliers 'A', 'a', and 'b' contained in the optimally robust IC are computed.
If 'computeIC' is 'TRUE' an object of class "ContIC" is returned,
otherwise a list of Lagrange multipliers
A |
standardizing constant |
a |
centering constant |
b |
optimal clipping bound |
Matthias Kohl [email protected]
Rieder, H. (1994) Robust Asymptotic Statistics. New York: Springer.
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.
IC1 <- rsOptIC(r = 0.1) distrExOptions("ErelativeTolerance" = 1e-12) checkIC(IC1) distrExOptions("ErelativeTolerance" = .Machine$double.eps^0.25) # default Risks(IC1) cent(IC1) clip(IC1) stand(IC1) plot(IC1)IC1 <- rsOptIC(r = 0.1) distrExOptions("ErelativeTolerance" = 1e-12) checkIC(IC1) distrExOptions("ErelativeTolerance" = .Machine$double.eps^0.25) # default Risks(IC1) cent(IC1) clip(IC1) stand(IC1) plot(IC1)
The function showdown can be used to perform Monte-Carlo studies
comparing a competitor with rmx estimators in case of normal location and scale.
In addition, maximum likelihood (ML) estimators (mean and sd) and median and
MAD are computed. The comparison is based on the empirical MSE.
showdown(n, M, eps, contD, seed = 123, estfun, estMean, estSd, eps.lower = 0, eps.upper = 0.05, steps = 3L, fsCor = TRUE, plot1 = FALSE, plot2 = FALSE, plot3 = FALSE)showdown(n, M, eps, contD, seed = 123, estfun, estMean, estSd, eps.lower = 0, eps.upper = 0.05, steps = 3L, fsCor = TRUE, plot1 = FALSE, plot2 = FALSE, plot3 = FALSE)
n |
integer; sample size, should be at least 3. |
M |
integer; Monte-Carlo replications. |
eps |
amount of contamination in [0, 0.5]. |
contD |
object of class |
seed |
random seed. |
estfun |
function to compute location and scale estimator; see details below. |
estMean |
function to compute location estimator; see details below. |
estSd |
function to compute scale estimator; see details below. |
eps.lower |
used by rmx estimator. |
eps.upper |
used by rmx estimator. |
steps |
integer; steps used for estimator construction. |
fsCor |
logical; use finite-sample correction. |
plot1 |
logical; plot cdf of ideal and real distribution. |
plot2 |
logical; plot 20 (or M if M < 20) randomly selected samples. |
plot3 |
logical; generate boxplots of the results. |
Normal location and scale with mean = 0 and sd = 1 is used as ideal model (without restriction due to equivariance).
Since there is no estimator which yields reliable results if 50 percent or more of the observations are contaminated, we use a modification where we re-simulate all samples including at least 50 percent contaminated data.
If estfun is specified it has to compute and return a location and scale estimate
(vector of length 2). One can also specify the location and scale estimator separately
by using estMean and estSd where estMean computes and returns
the location estimate and estSd the scale estimate.
We use funtion rowRoblox for the computation of the rmx estimator.
Data.frame including empirical MSE (standardized by sample size n) and relMSE with respect to the rmx estimator.
Matthias Kohl [email protected]
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.
H. Rieder (1994): Robust Asymptotic Statistics. Springer. 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. 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. 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. doi:10.1186/1471-2105-11-583.
library(MASS) ## compare with Huber's Proposal 2 showdown(n = 20, M = 100, eps = 0.02, contD = Norm(mean = 3, sd = 3), estfun = function(x){ unlist(hubers(x)) }, plot1 = TRUE, plot2 = TRUE, plot3 = TRUE) ## compare with Huber M estimator with MAD scale showdown(n = 20, M = 100, eps = 0.02, contD = Norm(mean = 3, sd = 3), estfun = function(x){ unlist(huber(x)) }, plot1 = TRUE, plot2 = TRUE, plot3 = TRUE)library(MASS) ## compare with Huber's Proposal 2 showdown(n = 20, M = 100, eps = 0.02, contD = Norm(mean = 3, sd = 3), estfun = function(x){ unlist(hubers(x)) }, plot1 = TRUE, plot2 = TRUE, plot3 = TRUE) ## compare with Huber M estimator with MAD scale showdown(n = 20, M = 100, eps = 0.02, contD = Norm(mean = 3, sd = 3), estfun = function(x){ unlist(huber(x)) }, plot1 = TRUE, plot2 = TRUE, plot3 = TRUE)