smoothm {smoothmest}

Smoothed and unsmoothed 1-d location M-estimators

Description

smoothm is an interface for all the smoothed M-estimators introduced in Hampel, Hennig and Ronchetti (2011) for one-dimensional location, the Huber- and Bisquare-M-estimator and the ML-estimator of the Cauchy distribution, calling all the other functions documented on this page.

Usage

   smoothm(y, method="smhuber",
     k=0.862, sn=sqrt(2.046/length(y)),
     tol=1e-06,  s=mad(y), init="median")

   sehuber(y, k = 0.862, tol = 1e-06, s=mad(y), init="median")

   smhuber(y, k = 0.862, sn=sqrt(2.046/length(y)), tol = 1e-06, s=mad(y),
     smmed=FALSE, init="median")

   mbisquare(y, k=4.685, tol = 1e-06, s=mad(y), init="median")

   smbisquare(y, k=4.685, tol = 1e-06, sn=sqrt(1.0526/length(y)),
     s=mad(y), init="median")

   mlcauchy(y, tol = 1e-06, s=mad(y))

   smcauchy(y, tol = 1e-06, sn=sqrt(2/length(y)), s=mad(y))

Arguments

y

numeric vector. Data set.

method

one of "huber", "smhuber", "bisquare", "smbisquare", "cauchy", "smcauchy", "smmed". See details.

k

numeric. Tuning constant. This is used for method one of "huber", "smhuber", "bisquare", "smbisquare" in smoothm and the corresponding functions. Tuning constants are defined for the Huber- and Bisquare M-estimator as in Maronna et al. (2006). The default values refer to a Huber M-estimator which is optimal under 20% contamination (0.862) and to a Bisquare M-estimator with 95% efficiency under the Gaussian model (4.685).

sn

numeric. This is used for method one of "smhuber", "smbisquare", "smcauchy", "smmed". This is the smoothing standard error \sigma_n in Hampel et al. (2011) depending on the sample size and the asymptotic variance of the base estimator. The default value of smoothm and smhuber is based on a Huber estimator with k=0.862 under Huber's least favourable distribution for which it is ML. The default value of smbisquare is based on the Bisquare estimator with k=4.685 under the Gaussian distribution. The default value of smcauchy is based on the Cauchy ML estimator under the Cauchy distribution. A value that can be used for the smoothed median is sqrt(1.056/length(y)), which is based on the median under the double exponential (Laplace) distribution with 1.4826 MAD=1. Note that the distributional "assumptions" for these choices are by no means critical; they should work well under many other distributions as well.

tol

numeric. Stopping criterion for algorithms (absolute difference between two successive values).

s

numeric. Estimated or assumed scale/standard deviation.

init

"median" or "mean". Initial estimator for iteration. Ignored if method=="cauchy" or "smcauchy".

smmed

logical. If FALSE, the smoothed Huber estimator is computed, otherwise the smoothed median by smhuber.

Details

The following estimators can be computed (some computational details are given in Hampel et al. 2011):

Huber estimator.

method="huber" and function sehuber compute the standard Huber estimator (Huber and Ronchetti 2009). The only differences from huber are that s and init can be specified and that the default k is different.

Smoothed Huber estimator.

method="smhuber" and function smhuber compute the smoothed Huber estimator (Hampel et al. 2011).

Bisquare estimator.

method="bisquare" and function bisquare compute the bisquare M-estimator (Maronna et al. 2006). This uses psi.bisquare.

Smoothed bisquare estimator.

method="smbisquare" and function smbisquare compute the smoothed bisquare M-estimator (Hampel et al. 2011). This uses psi.bisquare

ML estimator for Cauchy distribution.

method="cauchy" and function mlcauchy compute the ML-estimator for the Cauchy distribution.

Smoothed ML estimator for Cauchy distribution.

method="smcauchy" and function smcauchy compute the smoothed ML-estimator for the Cauchy distribution (Hampel et al. 2011).

Smoothed median.

method="smmed" and function smhuber with median=TRUE compute the smoothed median (Hampel et al. 2011).

Value

A list with components

mu

the location estimator.

method

see above.

k

see above.

sn

see above.

tol

see above.

s

see above.

Author(s)

Christian Hennig [e-mail] http://www.homepages.ucl.ac.uk/~ucakche/

References

Hampel, F., Hennig, C. and Ronchetti, E. (2011) A smoothing principle for the Huber and other location M-estimators. Computational Statistics and Data Analysis 55, 324-337.

Huber, P. J. and Ronchetti, E. (2009) Robust Statistics (2nd ed.). Wiley, New York.

Maronna, A.R., Martin, D.R., Yohai, V.J. (2006). Robust Statistics: Theory and Methods. Wiley, New York

See Also

pitman, huber, rlm

Examples

  library(MASS)
  set.seed(10001)
  y <- rdoublex(7)
  median(y)
  huber(y)$mu
  smoothm(y)$mu
  smoothm(y,method="huber")$mu
  smoothm(y,method="bisquare",k=4.685)$mu
  smoothm(y,method="smbisquare",k=4.685,sn=sqrt(1.0526/7))$mu
  smoothm(y,method="cauchy")$mu
  smoothm(y,method="smcauchy",sn=sqrt(2/7))$mu
  smoothm(y,method="smmed",sn=sqrt(1.0526/7))$mu
  smoothm(y,method="smmed",sn=sqrt(1.0526/7),init="mean")$mu