Constants for Huber and Bisquare Psi

DESCRIPTION:

Computes tuning constants or efficiencies for Huber and bisquare psi functions. The constants can be used to obtain a location M-estimate with specified asymptotic efficiency at the Gaussian model.

USAGE:

chb(eff=NULL, ch=<<see below>>, cb=<<see below>>) 

OPTIONAL ARGUMENTS:

eff
parameter (vector of length 1) giving the desired asymptotic efficiency of a location M-estimate at the Gaussian model. Only values between 0.7 and 1.0 are allowed. If eff is supplied, then the input values of ch and cb are ignored.
ch
parameter giving tuning constant for Huber psi. If ch is supplied, then cb must also be supplied.
cb
parameter giving tuning constant for bisquare psi. If cb is supplied, then ch must also be supplied.

VALUE:

if no arguments are supplied, then a 34 by 5 matrix is returned. The columns are: efficiency (1), Huber tuning constant (2), Huber beta value (3), bisquare tuning constant (4), and bisquare beta (5).

If arguments are supplied, then a list with the following components is returned:
ch
tuning constant for the Huber psi function.
bh
beta for the Huber psi function used to compute a Huber proposal 2 scale estimate.
eh
estimate of the asymptotic efficiency of the location M-estimate using the Huber psi with tuning constant ch.
cb
tuning constant for Tukey's bisquare psi function.
bb
beta for the bisquare psi function used to compute a Huber proposal 2 scale estimate.
eb
estimate of the asymptotic efficiency for the bisquare location estimate.

BACKGROUND:

A beta parameter is the expectation of the square of a psi function under the Gaussian distribution, and is used to achieve statistical consistency at the Gaussian model for the scale estimate in robloc.

REFERENCES:

Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J. and Stahel, W. A. (1986). Robust Statistics: The Approach Based on Influence Functions. Wiley, New York.

Huber, P. J. (1981). Robust Statistics. Wiley, New York.

SEE ALSO:

, .

EXAMPLES:

chb(.95)$ch 
 # tuning constant for the 95% efficient Huber M-estimate of location. 
chb(ch=1.345, cb=4.685)