smpsi {smoothmest}

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 [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

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)