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In [[statistics]], a '''robust measure of scale''' is a [[robust statistics|robust statistic]] that quantifies the [[statistical dispersion]] in a set of [[interval scale|numerical]] [[data]]. The most common such statistics are the [[interquartile range]] (IQR) and the [[median absolute deviation]] (MAD). These are contrasted with conventional measures of scale, such as sample [[variance]] or sample [[standard deviation]], which are non-robust, meaning greatly influenced by [[outlier]]s.
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These robust statistics are particularly used as [[estimator]]s of a [[scale parameter]], and have the advantages of both robustness and superior efficiency on contaminated data, at the cost of inferior efficiency on clean data from distributions such as the normal distribution. To illustrate robustness, the standard deviation can be made arbitrarily large by increasing exactly one observation (it has a [[breakdown point]] of 0, as it can be contaminated by a single point), a defect that is not shared by robust statistics.
 
== IQR and MAD ==
The most familiar robust measures of scale are the [[interquartile range]] (IQR) and the [[median absolute deviation]] (MAD). The IQR is the difference between the 75th [[percentile]] and the 25th [[percentile]] of a sample; this is the 25% [[trimmed estimator|trimmed]] [[Range (statistics)|range]], an example of an [[L-estimator]]. Other trimmed ranges, such as the [[interdecile range]] (10% trimmed range) can also be used. The MAD is the [[median]] of the absolute values of the differences between the data values and the overall median of the data set; as explained in [[median absolute deviation#relation to standard deviation]], MAD is related to σ as:
 
:<math>\sigma \approx 1.4826\ \operatorname{MAD}. \, </math>
 
== Estimation ==
Robust measures of scale can be used as [[estimator]]s of properties of the population, either for [[parameter estimation]] or as estimators of their own [[expected value]].
 
For example, robust estimators of scale are used to estimate the [[population variance]] or population [[standard deviation]], generally by multiplying by a [[scale factor]] to make it an [[unbiased estimator|unbiased]] [[consistent estimator]]; see [[scale parameter#Estimation|scale parameter: estimation]]. For example, dividing the IQR by by 2√2 erf<sup>-1</sup>(1/2) (approximately 1.349), makes it an unbiased, consistent estimator for the population variance if the data follow a [[normal distribution]].
 
In other situations, it makes more sense to think of a robust measure of scale as an estimator of its own [[expected value]], interpreted as an alternative to the population variance or standard deviation as a measure of scale. For example, the MAD of a sample from a standard [[Cauchy distribution]] is an estimator of the population MAD, which in this case is 1, whereas the population variance does not exist.
 
==Efficiency==
These robust estimators typically have inferior [[efficiency (statistics)|statistical efficiency]] compared to conventional estimators for data drawn from a distribution without outliers (such as a normal distribution), but have superior efficiency for data drawn from a [[mixture distribution]] or from a [[heavy-tailed distribution]], for which non-robust measures such as the standard deviation should not be used.
 
For example, for data drawn from the normal distribution, the MAD is 37% as efficient as the sample standard deviation, while the Rousseeuw–Croux estimator ''Q''<sub>''n''</sub> is 88% as efficient as the sample standard deviation.
 
== Absolute pairwise differences ==
Rousseeuw and Croux<ref>{{citation | title=Alternatives to the Median Absolute Deviation | last1=Rousseeuw | first1=Peter J. | authorlink1=Peter Rousseeuw|last2=Croux | first2=Christophe | journal=Journal of the American Statistical Association | volume=88 | issue=424 | month=December | year=1993 | pages=1273–1283 | doi=10.2307/2291267 | publisher=American Statistical Association | jstor=2291267}}</ref> propose alternatives to the MAD, motivated by two weaknesses of it:
# It is [[efficiency (statistics)|inefficient]] (37% efficiency) at [[Gaussian distribution]]s.
# it computes a symmetric statistic about a location estimate, thus not dealing with [[skewness]].
 
They propose two alternative statistics based on pairwise differences: ''S<sub>n</sub>'' and ''Q<sub>n</sub>'', defined as:
:<math>
\begin{align}
    S_n &:= 1.1926 \, \operatorname{med}_i \left( \operatorname{med}_j (\,\left| x_i - x_j \right|\,) \right) ,\\
    Q_n & := c_n \text{first quartile of} \left( \left| x_i - x_j \right| : i < j
                                      \right),
\end{align} </math>
where <math>c_n</math> is a constant depending on <math>n</math>.
 
These can be computed in ''O''(''n'' log ''n'') time and ''O''(''n'') space.
 
Neither of these requires [[Location parameter|location]] estimation, as they are based only on differences between values. They are both more efficient than the MAD under a Gaussian distribution: ''S<sub>n</sub>'' is 58% efficient, while ''Q<sub>n</sub>'' is 82% efficient.
 
For a sample from a normal distribution, ''S''<sub>n</sub> is approximately unbiased  for the population standard deviation even down to very modest sample sizes (<1% bias for ''n''&nbsp;=&nbsp;10). For a large sample from a normal distribution, 2.219144465985075864722''Q''<sub>n</sub> is approximately unbiased for the population standard deviation.  For small or moderate samples, the expected value of ''Q''<sub>n</sub> under a normal distribution depends markedly on the sample size, so finite-sample correction factors (obtained from a table or from simulations) are used to calibrate the scale of ''Q''<sub>n</sub>.
 
== The biweight midvariance ==
 
Like ''S''<sub>n</sub> and ''Q''<sub>n</sub>, the biweight midvariance aims to be robust without sacrificing too much efficiency. It is defined as
 
:<math>
\frac{n*\sum_{i=1}^n (x_i-Q)^2(1-u_i^2)^4 I(|u_i|<1)}{(\sum_i(1-u_i^2)(1-5u_i^2)I(|u_i|<1))^2} ,
</math>
 
where ''I'' is the [[indicator function]], ''Q'' is the sample median of the ''X''<sub>i</sub>, and
 
:<math>
u_i = \frac{x_i-Q}{9\cdot{\rm MAD}}.
</math>
 
Its square root is a robust estimator of scale, since data points are downweighted as their distance from the median increases, with points more than 9 MAD units from the median having no influence at all.
 
==Simultaneous estimation of location and scale==
{{stub section|date=October 2013}}
{{harvtxt|Mizera|Müller|2004}} propose a robust depth-based estimator for location and scale simultaneously.<ref>{{citation|last1=Mizera|first1=I.|last2=Müller|first2=C. H.|title=Location-scale depth|journal=Journal of the American Statistical Association|year=2004|volume=99|issue=468|pages=949–966|doi=10.1198/016214504000001312}}.</ref>
 
== References ==
<references/>
 
[[Category:Robust statistics]]
[[Category:Statistical terminology]]
[[Category:Statistical deviation and dispersion]]
[[Category:Scale statistics]]

Latest revision as of 02:21, 23 November 2014

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