scaleTau2: Robust Tau-Estimate of Scale

Description

Computes the robust $τ$ -estimate of univariate scale, as proposed by Maronna and Zamar (2002); improved by a consistency factor,

Usage


scaleTau2(x, c1 = 4.5, c2 = 3.0, na.rm = FALSE, consistency = TRUE,
          mu0 = median(x),
          sigma0 = median(x.), mu.too = FALSE, iter = 1, tol.iter = 1e-7)

Value

numeric vector of length one (if mu.too is FALSE as by default) or two (when mu.too = TRUE) with robust scale or (location,scale) estimators

$\hat{σ} (x)$ or

$(\hat{μ} (x), \hat{σ} (x))$ .

Arguments

x: numeric vector
c1,c2: non-negative numbers, specifying cutoff values for the biweighting of the mean and the rho function respectively.
na.rm: a logical value indicating whether NA values should be stripped before the computation proceeds.
consistency: logical indicating if the consistency correction factor (for the scale) should be applied.
mu0: the initial location estimate $μ_{0}$ , defaulting to the median.
sigma0: the initial scale estimate $s_{0}$ , defaulting to the MAD; may be set to a positive value when the MAD is zero.
mu.too: logical indicating if both location and scale should be returned or just the scale (when mu.too=FALSE as by default).
iter: positive integer or logical indicating if and how many iterations should be done. The default, iter = 1 computes the “traditional” tau-estimate of scale.
tol.iter: if iter is true, or iter > 1, stop the iterations when $| s_{n} - s_{o} | \leq ϵ s_{n}$ , where $ϵ :=$ tol.iter, and $s_{o}$ and $s_{n}$ are the previous and current estimates of $σ$ .

Author

Original by Kjell Konis with substantial modifications by Martin Maechler.

Details

First, $s_{0}$ := MAD, i.e. the equivalent of mad(x, constant=1) is computed. Robustness weights $w_{i} := w_{c 1} ((x_{i} - m e d (X)) / s_{0})$ are computed, where $w_{c} (u) = m a x (0, (1 - (u / c)^{2})^{2})$ . The robust location measure is defined as $μ (X) := (\sum_{i} w_{i} x_{i}) / (\sum_{i} w_{i})$ , and the robust $τ (t a u)$ -estimate is $s (X)^{2} := s_{0}^{2} * (1 / n) \sum_{i} ρ_{c 2} ((x_{i} - μ (X)) / s_{0})$ , where $ρ_{c} (u) = m i n (c^{2}, u^{2})$ .
When iter=TRUE or iter > 1, the above estimate is iterated in a fixpoint iteration, setting $s_{0}$ to the current estimate $s (X)$ and iterating until the number of iterations is larger than iter or the fixpoint is found in the sense that \
scaleTau2(*, consistency=FALSE) returns $s (X)$ , whereas this value is divided by its asymptotic limit when consistency = TRUE as by default.

Note that for n = length(x) == 2, all equivariant scale estimates are proportional, and specifically, scaleTau2(x, consistency=FALSE) == mad(x, constant=1). See also the reference.

References

Maronna, R.A. and Zamar, R.H. (2002) Robust estimates of location and dispersion of high-dimensional datasets; Technometrics 44(4), 307--317.

Yohai, V.J., and Zamar, R.H. (1988). High breakdown-point estimates of regression by means of the minimization of an efficient scale. Journal of the American Statistical Association 83, 406--413.

Examples

Run this code

x <- c(1:7, 1000)
sd(x) # non-robust std.deviation
scaleTau2(x)
scaleTau2(x, mu.too = TRUE)
(sI  <- scaleTau2(c(x,Inf), mu.too = TRUE))
(sIN <- scaleTau2(c(x,Inf,NA), mu.too = TRUE, na.rm=TRUE))
stopifnot({
  identical(sI, sIN)
  all.equal(scaleTau2(c(x, 999), mu.too = TRUE), sIN,
            tol = 1e-15)
})

if(doExtras <- robustbase:::doExtras()) {
 set.seed(11)
 ## show how much faster this is, compared to Qn
 x <- sample(c(rnorm(1e6), rt(5e5, df=3)))
 (system.time(Qx <- Qn(x)))         ## 2.04 [2017-09, lynne]
 (system.time(S2x <- scaleTau2(x))) ## 0.25    (ditto)
 cbind(Qn = Qx, sTau2 = S2x)
}##       Qn    sTau2
##  1.072556 1.071258

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