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>).

Package versions

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.

Author(s)

Matthias Kohl matthias.kohl@stamats.de

References

M. Kohl (2005). Numerical Contributions to the Asymptotic Theory of Robustness. Dissertation. University of Bayreuth. https://epub.uni-bayreuth.de/id/eprint/839/2/DissMKohl.pdf.

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

See Also

RobAStBase-package

Examples

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)

Function to compute finite-sample corrected radii

Description

Given some radius and some sample size the function computes the corresponding finite-sample corrected radius.

Usage

finiteSampleCorrection(r, n, model = "locsc")

Arguments

Details

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.

Value

Finite-sample corrected radius.

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

M. Kohl (2005). Numerical Contributions to the Asymptotic Theory of Robustness. Dissertation. University of Bayreuth. https://epub.uni-bayreuth.de/id/eprint/839/2/DissMKohl.pdf.

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.

See Also

roblox, rowRoblox, colRoblox

Examples

finiteSampleCorrection(n = 3, r = 0.001, model = "locsc")
finiteSampleCorrection(n = 10, r = 0.02, model = "loc")
finiteSampleCorrection(n = 250, r = 0.15, model = "sc")

Computation of the optimally robust IC for AL estimators

Description

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.

Usage

rlOptIC(r, mean = 0, sd = 1, bUp = 1000, computeIC = TRUE)

Arguments

Details

If 'computeIC' is 'FALSE' only the Lagrange multipliers 'A', 'a', and 'b' contained in the optimally robust IC are computed.

Value

If 'computeIC' is 'TRUE' an object of class "ContIC" is returned, otherwise a list of Lagrange multipliers

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

M. Kohl (2005). Numerical Contributions to the Asymptotic Theory of Robustness. Dissertation. University of Bayreuth. https://epub.uni-bayreuth.de/id/eprint/839/2/DissMKohl.pdf.

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.

See Also

ContIC-class, roblox

Examples

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)

Computation of the optimally robust IC for AL estimators

Description

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).

Usage

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)

Arguments

Details

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.

Value

If 'computeIC' is 'TRUE' an object of class "ContIC" is returned, otherwise a list of Lagrange multipliers

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

M. Kohl (2005). Numerical Contributions to the Asymptotic Theory of Robustness. Dissertation. University of Bayreuth. https://epub.uni-bayreuth.de/id/eprint/839/2/DissMKohl.pdf.

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.

See Also

ContIC-class, roblox

Examples

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)

Computation of the optimally robust IC for An1 estimators

Description

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).

Usage

rlsOptIC.An1(r, aUp = 2.5, delta = 1e-06)

Arguments

Details

The optimal value of the tuning constant a can be read off from the slot Infos of the resulting IC.

Value

Object of class "IC"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

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.

See Also

IC-class

Examples

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)

Computation of the optimally robust IC for An2 estimators

Description

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).

Usage

rlsOptIC.An2(r, a.start = 1.5, k.start = 1.5, delta = 1e-06, MAX = 100)

Arguments

Details

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.

Value

Object of class "IC"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

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.

See Also

IC-class

Examples

IC1 <- rlsOptIC.An2(r = 0.1)
checkIC(IC1)
Risks(IC1)
Infos(IC1)
plot(IC1)
infoPlot(IC1)

Computation of the optimally robust IC for AnMad estimators

Description

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).

Usage

rlsOptIC.AnMad(r, aUp = 2.5, delta = 1e-06)

Arguments

Details

The optimal value of the tuning constant a can be read off from the slot Infos of the resulting IC.

Value

Object of class "IC"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

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.

See Also

IC-class

Examples

IC1 <- rlsOptIC.AnMad(r = 0.1)
checkIC(IC1)
Risks(IC1)
Infos(IC1)
plot(IC1)
infoPlot(IC1)

Computation of the optimally robust IC for BM estimators

Description

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).

Usage

rlsOptIC.BM(r, bL.start = 2, bS.start = 1.5, delta = 1e-06, MAX = 100)

Arguments

Details

The computation of the optimally robust IC for BM estimators is based on optim where MAX is used to control the constraints on b_{\rm loc} and b_{{\rm sc},0}. The optimal values of the tuning constants b_{\rm loc}, b_{{\rm sc},0}, \alpha and \gamma can be read off from the slot Infos of the resulting IC.

Value

Object of class "IC"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

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

See Also

IC-class

Examples

IC1 <- rlsOptIC.BM(r = 0.1)
checkIC(IC1)
Risks(IC1)
Infos(IC1)
plot(IC1)
infoPlot(IC1)

Computation of the optimally robust IC for Ha3 estimators

Description

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).

Usage

rlsOptIC.Ha3(r, a.start = 0.25, b.start = 2.5, c.start = 5, 
             delta = 1e-06, MAX = 100)

Arguments

Details

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.

Value

Object of class "IC"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

M. Kohl (2005). Numerical Contributions to the Asymptotic Theory of Robustness. Dissertation. University of Bayreuth. https://epub.uni-bayreuth.de/id/eprint/839/2/DissMKohl.pdf.

See Also

IC-class

Examples

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)

Computation of the optimally robust IC for Ha4 estimators

Description

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).

Usage

rlsOptIC.Ha4(r, a.start = 0.25, b.start = 2.5, c.start = 5, 
             k.start = 1, delta = 1e-06, MAX = 100)

Arguments

Details

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.

Value

Object of class "IC"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

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.

See Also

IC-class

Examples

IC1 <- rlsOptIC.Ha4(r = 0.1)
checkIC(IC1)
Risks(IC1)
Infos(IC1)
plot(IC1)
infoPlot(IC1)

Computation of the optimally robust IC for HuMad estimators

Description

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).

Usage

rlsOptIC.HaMad(r, a.start = 0.25, b.start = 2.5, c.start = 5, 
               delta = 1e-06, MAX = 100)

Arguments

Details

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.

Value

Object of class "IC"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

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.

See Also

IC-class

Examples

IC1 <- rlsOptIC.HaMad(r = 0.1)
checkIC(IC1)
Risks(IC1)
Infos(IC1)
plot(IC1)
infoPlot(IC1)

Computation of the optimally robust IC for Hu1 estimators

Description

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).

Usage

rlsOptIC.Hu1(r, kUp = 2.5, delta = 1e-06)

Arguments

Details

The optimal value of the tuning constant k can be read off from the slot Infos of the resulting IC.

Value

Object of class "IC"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

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.

See Also

IC-class

Examples

IC1 <- rlsOptIC.Hu1(r = 0.1)
checkIC(IC1)
Risks(IC1)
Infos(IC1)
plot(IC1)
infoPlot(IC1)

Computation of the optimally robust IC for Hu2 estimators

Description

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).

Usage

rlsOptIC.Hu2(r, k.start = 1.5, c.start = 1.5, delta = 1e-06, MAX = 100)

Arguments

Details

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.

Value

Object of class "IC"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

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.

See Also

IC-class

Examples

IC1 <- rlsOptIC.Hu2(r = 0.1)
checkIC(IC1)
Risks(IC1)
Infos(IC1)
plot(IC1)
infoPlot(IC1)

Computation of the optimally robust IC for Hu2a estimators

Description

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).

Usage

rlsOptIC.Hu2a(r, k1.start = 0.25, k2.start = 2.5, delta = 1e-06, MAX = 100)

Arguments

Details

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.

Value

Object of class "IC"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

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.

See Also

IC-class

Examples

IC1 <- rlsOptIC.Hu2a(r = 0.1)
checkIC(IC1)
Risks(IC1)
Infos(IC1)
plot(IC1)
infoPlot(IC1)

Computation of the optimally robust IC for Hu3 estimators

Description

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).

Usage

rlsOptIC.Hu3(r, k.start = 1, c1.start = 0.1, c2.start = 0.5, 
             delta = 1e-06, MAX = 100)

Arguments

Details

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.

Value

Object of class "IC"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

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.

See Also

IC-class

Examples

IC1 <- rlsOptIC.Hu3(r = 0.1)
checkIC(IC1)
Risks(IC1)
Infos(IC1)
plot(IC1)
infoPlot(IC1)

Computation of the optimally robust IC for HuMad estimators

Description

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).

Usage

rlsOptIC.HuMad(r, kUp = 2.5, delta = 1e-06)

Arguments

Details

The optimal value of the tuning constant k can be read off from the slot Infos of the resulting IC.

Value

Object of class "IC"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

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.

See Also

IC-class

Examples

IC1 <- rlsOptIC.HuMad(r = 0.1)
checkIC(IC1)
Risks(IC1)
Infos(IC1)
plot(IC1)
infoPlot(IC1)

Computation of the optimally robust IC for M estimators

Description

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).

Usage

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)

Arguments

Details

The optimal values of the tuning constants \alpha_1, \alpha_3, b and \gamma can be read off from the slot Infos of the resulting IC.

Value

Object of class "IC"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

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.

See Also

IC-class

Examples

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)

Computation of the optimally robust IC for MM2 estimators

Description

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).

Usage

rlsOptIC.MM2(r, c.start = 1.5, d.start = 2, delta = 1e-06, MAX = 100)

Arguments

Details

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.

Value

Object of class "IC"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

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.

See Also

IC-class

Examples

IC1 <- rlsOptIC.MM2(r = 0.1)
checkIC(IC1)
Risks(IC1)
Infos(IC1)
plot(IC1)
infoPlot(IC1)

Computation of the optimally robust IC for Tu1 estimators

Description

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).

Usage

rlsOptIC.Tu1(r, aUp = 10, delta = 1e-06)

Arguments

Details

The optimal value of the tuning constant a can be read off from the slot Infos of the resulting IC.

Value

Object of class "IC"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

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.

See Also

IC-class

Examples

IC1 <- rlsOptIC.Tu1(r = 0.1)
checkIC(IC1)
Risks(IC1)
Infos(IC1)
plot(IC1)
infoPlot(IC1)

Computation of the optimally robust IC for Tu2 estimators

Description

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).

Usage

rlsOptIC.Tu2(r, a.start = 5, k.start = 1.5, delta = 1e-06, MAX = 100)

Arguments

Details

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.

Value

Object of class "IC"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

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.

See Also

IC-class

Examples

IC1 <- rlsOptIC.Tu2(r = 0.1)
checkIC(IC1)
Risks(IC1)
Infos(IC1)
plot(IC1)
infoPlot(IC1)

Computation of the optimally robust IC for TuMad estimators

Description

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).

Usage

rlsOptIC.TuMad(r, aUp = 10, delta = 1e-06)

Arguments

Details

The optimal value of the tuning constant a can be read off from the slot Infos of the resulting IC.

Value

Object of class "IC"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

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.

See Also

IC-class

Examples

IC1 <- rlsOptIC.TuMad(r = 0.1)
checkIC(IC1)
Risks(IC1)
Infos(IC1)
plot(IC1)
infoPlot(IC1)

Optimally robust estimator for location and/or scale

Description

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.

Usage

roblox(x, mean, sd, eps, eps.lower, eps.upper, initial.est, k = 1L, 
       fsCor = TRUE, returnIC = FALSE, mad0 = 1e-4, na.rm = TRUE)

Arguments

Details

Computes the optimally robust estimator for location with scale specified, scale with location specified, or both if neither is specified. The computation uses a k-step construction with an appropriate initial estimate for location or scale or location and scale, respectively. Valid candidates are e.g. median and/or MAD (default) as well as Kolmogorov(-Smirnov) or von Mises minimum distance estimators; cf. Rieder (1994) and Kohl (2005).

If the amount of gross errors (contamination) is known, it can be specified by eps. The radius of the corresponding infinitesimal contamination neighborhood is obtained by multiplying eps by the square root of the sample size.

If the amount of gross errors (contamination) is unknown, try to find a rough estimate for the amount of gross errors, such that it lies between eps.lower and eps.upper.

In case eps.lower is specified and eps.upper is missing, eps.upper is set to 0.5. In case eps.upper is specified and eps.lower is missing, eps.lower is set to 0.

If neither eps nor eps.lower and/or eps.upper is specified, eps.lower and eps.upper are set to 0 and 0.5, respectively.

If eps is missing, the radius-minimax estimator in sense of Rieder et al. (2008), respectively Section 2.2 of Kohl (2005) is returned.

In case of location, respectively scale one additionally has to specify sd, respectively mean where sd and mean have to be a single number.

For sample size <= 2, median and/or MAD are used for estimation.

If eps = 0, mean and/or sd are computed. In this situation it's better to use function MLEstimator.

Value

Object of class "kStepEstimate".

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

M. Kohl (2005). Numerical Contributions to the Asymptotic Theory of Robustness. Dissertation. University of Bayreuth. https://epub.uni-bayreuth.de/id/eprint/839/2/DissMKohl.pdf.

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.

See Also

ContIC-class, rlOptIC, rsOptIC, rlsOptIC.AL, kStepEstimate-class, roptest

Examples

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))

Optimally robust estimation for location and/or scale

Description

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.

Usage

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)

Arguments

Details

Computes the optimally robust estimator for location with scale specified, scale with location specified, or both if neither is specified. The computation uses a k-step construction with an appropriate initial estimate for location or scale or location and scale, respectively. Valid candidates are e.g. median and/or MAD (default) as well as Kolmogorov(-Smirnov) or 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.

Value

Object of class "kStepEstimate".

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

M. Kohl (2005). Numerical Contributions to the Asymptotic Theory of Robustness. Dissertation. University of Bayreuth. https://epub.uni-bayreuth.de/id/eprint/839/2/DissMKohl.pdf.

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.

See Also

roblox, kStepEstimate-class

Examples

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)

Computation of the optimally robust IC for AL estimators

Description

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.

Usage

rsOptIC(r, mean = 0, sd = 1, bUp = 1000, delta = 1e-06, itmax = 100, computeIC = TRUE)

Arguments

Details

If 'computeIC' is 'FALSE' only the Lagrange multipliers 'A', 'a', and 'b' contained in the optimally robust IC are computed.

Value

If 'computeIC' is 'TRUE' an object of class "ContIC" is returned, otherwise a list of Lagrange multipliers

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

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.

See Also

ContIC-class, roblox

Examples

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)

Estimator Showdown by Monte-Carlo Study.

Description

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.

Usage

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)

Arguments

Details

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.

Value

Data.frame including empirical MSE (standardized by sample size n) and relMSE with respect to the rmx estimator.

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

References

M. Kohl (2005). Numerical Contributions to the Asymptotic Theory of Robustness. Dissertation. University of Bayreuth. https://epub.uni-bayreuth.de/id/eprint/839/2/DissMKohl.pdf.

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.

See Also

rowRoblox

Examples

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)