rlsOptIC.AL {RobLox}

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

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.

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

A

standardizing matrix

a

centering vector

b

optimal clipping bound

Author(s)

Matthias Kohl [e-mail]

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)