R: Smoothed and unsmoothed 1-d location M-estimators

smoothm {smoothmest} | R Documentation

## 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 σ_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 chrish@stats.ucl.ac.uk 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

[Package smoothmest version 0.1-2 Index]