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

Qn {robustbase} | R Documentation

## 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 <= 9.
`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()`.

### 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.

### Value

`Qn()` returns a number, the Qn robust scale estimator, scaled to be consistent for σ^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 (μ) and scale; this is typically only useful for `covOGK(*, sigmamu = s_Qn)`.

### Author(s)

Original Fortran code: Christophe Croux and Peter Rousseeuw rousse@wins.uia.ac.be.
Port to C and R: Martin Maechler, maechler@R-project.org

### 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)

[Package robustbase version 0.92-7 Index]