R: Auxiliary functions for smoothm

smpsi {smoothmest} | R Documentation

## Auxiliary functions for smoothm

### Description

Psi-functions, derivatives and further auxiliary functions used for computing the estimators in `smoothm`.

### Usage

      psicauchy(x)
      psidcauchy(x)
      likcauchy(x,mu)
      flikcauchy(y,x,mu,sn)
      smtfcauchy(x,mu,sn)
      smcipsi(y, x, sn=sqrt(2/length(x)))
      smcipsid(y, x, sn=sqrt(2/length(x)))
      smcpsi(x, sn=sqrt(2/length(x)))
      smcpsid(x, sn=sqrt(2/length(x)))
      smbpsi(y, x, k=4.685, sn=sqrt(2/length(x)))
      smbpsid(y, x, k=4.685, sn=sqrt(2/length(x)))
      smbpsii(x, k=4.685, sn=sqrt(2/length(x)))
      smbpsidi(x, k=4.685, sn=sqrt(2/length(x)))
      smpsi(x,k=0.862,sn=sqrt(2/length(x)))
      smpmed(x,sn=sqrt(1/5))

### Arguments

`x` | numeric vector.
`mu` | numeric.
`y` | numeric vector.
`sn` | numeric. Smoothing constant. See `smoothm`.
`k` | numeric. Tuning constant. See `smoothm`.

### Details

psicauchy

  psi-function for Cauchy ML-estimator at `x`.

psidcauchy

  derivative of `psicauchy` at `x`.

likcauchy

  Cauchy likelihood of data `x` for mode parameter `mu`.

flikcauchy

  vector of Gaussian density at `y` with mean 0 and st. dev. `sn` times Cauchy log-likelihood of `x` with mode parameter `mu+y`.

smtfcauchy

  integral of `flikcauchy` with `y` running from `-Inf` to `Inf`.

smcipsi

  `psicauchy(x-y)*dnorm(y,sd=sn)`.

smcipsid

  derivative of `smcipsi` w.r.t. `x`.

smcpsi

  psi-function for smoothed Cauchy ML-estimator. Integral of `smpcipsi` with `y` running from `-Inf` to `Inf`.

smcpsid

  integral of `smpcipsid` with `y` running from `-Inf` to `Inf`.

smbpsi

  `(x-y)*psi.bisquare(x-y,c=k)*dnorm(y,sd=sn)`.

smbpsid

  `psi.bisquare(x-y,c=k,deriv=1)*dnorm(y,sd=sn)`.

smbpsii

  psi-function for smoothed bisquare M-estimator. Integral of `smbpsi` with `y` running from `-Inf` to `Inf`.

smbpsidi

  integral of `smbpsid` with `y` running from `-Inf` to `Inf`.

smpsi

  psi-function for smoothed Huber-estimator at `x`.

smpmed

  psi-function for smoothed median at `x`.

### Value

A numeric vector.

### 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

`smoothm`, `psi.huber`, `psi.bisquare`

### Examples

    psicauchy(1:5)
    psidcauchy(1:5)
    likcauchy(1:5,0)
    flikcauchy(3,1:5,0,1)
    smtfcauchy(1:5,0,1)
    smcipsi(1,1:3)
    smcipsid(1,1:3)
    smcpsi(1:5)
    smcpsid(1:5)
    smbpsi(1,1:5)
    smbpsid(0:4,1:5)
    smbpsii(1:5)
    smbpsidi(1:5)
    smpsi(1:5)
    smpmed(1:5)

[Package smoothmest version 0.1-2 Index]