robustbase (version 0.92-6)

Qn: Robust Location-Free Scale Estimate More Efficient than MAD

Description

Compute the robust scale estimator $Qn$, an efficient alternative to the MAD.

See the references for more.

Usage

Qn(x, constant = 2.21914, finite.corr = missing(constant))
s_Qn(x, mu.too = FALSE, ...)

Arguments

x

numeric vector of observations.

constant

number by which the result is multiplied; the default achieves consistency for normally distributed data. Note that until Nov. 2010, “thanks” to a typo in the very first papers, a slightly wrong default constant, 2.2219, was used instead of the correct one which is equal to 1 / (sqrt(2) * qnorm(5/8)) (as mentioned already on p.1277, after (3.7) in Rousseeuw and Croux (1993)).

If you need the old slightly off version for historical reproducibility, you can use Qn.old().

Note that the relative difference is only about 1 in 1000, and that the correction should not affect the finite sample corrections for $n

finite.corr

logical indicating if the finite sample bias correction factor should be applied. Defaults to TRUE unless constant is specified.

mu.too

logical indicating if the median(x) should also be returned for s_Qn().

...

potentially further arguments for s_Qn() passed to Qn().

Value

Qn() returns a number, the $Qn$ robust scale estimator, scaled to be consistent for $\sigma^2$ and i.i.d. Gaussian observatsions, optionally bias corrected for finite samples.s_Qn(x, mu.too=TRUE) returns a length-2 vector with location ($\mu$) and scale; this is typically only useful for covOGK(*, sigmamu = s_Qn).

Details

As the (default, consistency) constant needed to be corrected, the finite sample correction has been based on a much more extensive simulation, and on a 3rd or 4th degree polynomial model in $1/n$ for odd or even n, respectively.

References

Rousseeuw, P.J. and Croux, C. (1993) Alternatives to the Median Absolute Deviation, Journal of the American Statistical Association 88, 1273--1283.

Christophe Croux and Peter J. Rousseeuw (1992) Time-Efficient Algorithms for Two Highly Robust Estimators of Scale, Computational Statistics, Vol. 1, ed. Dodge and Whittaker, Physica-Verlag Heidelberg, 411--428; available via Springer Link.

About the typo in the constant: Christophe Croux (2010) Private e-mail, Fri Jul 16, w/ Subject Re: Slight inaccuracy of Qn implementation .......

See Also

mad for the ‘most robust’ but much less efficient scale estimator; Sn for a similar faster but less efficient alternative. Finally, scaleTau2 which some consider “uniformly” better than Qn or competitors.

Examples

set.seed(153)
x <- sort(c(rnorm(80), rt(20, df = 1)))
s_Qn(x, mu.too = TRUE)
Qn(x, finite.corr = FALSE)