For rank-based analyses of linear models, the estimator \(\hat{\tau}\) of the scale parameter \(\tau\) plays a standardizing role in the standard errors (SE) of the rank-based estimators of the regression coefficients and in the denominator of Wald-type and the drop-in-dispersion test statistics of linear hypotheses.
rfit currently implements the KSM (Koul, Sievers, and McKean 1987) estimator of tau.
The functions gettau and gettauF0 are both available to compute the KSM estimate and may be call from rfit and used for inference. The default is to use the faster FORTRAN version gettauF0 via the to option TAU='F0'.
The R version, gettau, may be much slower especially when sample sizes are large; this version may be called from rfit using the option TAU='R'.
The KSM estimator tauhat is a density type estimator that has the bandwidth given by \(t_\delta/sqrt{n}\),
where \(t_\delta\) is the \(\delta-th\) quantile of the cdf \(H(y)\) given in expression (3.7.2) of Hettmansperger and McKean (2011), with the corresponding estimator \(\hat{H}\), given in expression (3.7.7) of Hettmansperger and McKean (2011).
Based on simulation studies, most situations where (n/p >= 6), the default delta = 0.80 provides a valid rank-based
analysis (McKean and Sheather, 1991).
For situations with n/p < 6, caution is needed as the KSM estimate is sensitive to choice of bandwidth. McKean and Sheather (1991) recommend using a value of 0.95 for delta in such situations.
To correct for heavy-tailed random errors, Huber (1973) proposed a degree of freedom correction for the M-estimate scale parameter. The correction is given by \(K = 1 + [p*(1-h_c)/n*h_c]\) where \(h_c\) is the proportion of standardized residuals in absolute value less than the parameter hparm. This correction \(K\) is used as a multiplicative factor to tauhat. The default value of hparm is set at 2.
The usual degrees of freedom correction, \(\sqrt{n/(n-p)}\), is also used as a multiplicative factor to tauhat.