MAD Function

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MAD Function

finds the univariate (scaled) median-absolute-deviation

MAD( (x<,spt>))

where

x

is an n ×p input data matrix.

spt

is an optional string argument with the following values:

"MAD": for computing the MAD (which is the default)
"NMAD": for computing the normalized version of MAD
"SN": for computing S_n
"QN": for computing Q_n

The MAD function can be used for computing one of the following three robust scale estimates:

Median Absolute Deviation (MAD) or normalized form of MAD:
$MAD_n = b * med_i^n \; | x_i - med_j^n \; x_j|$
where b=1 is the unscaled default and b=1.4826 is used for the scaled version (consistency with the Gaussian distribution).
S_n which is a more efficient alternative to MAD:
$S_n = c_n * med_i \; med_{j \neq i} \; | x_i - x_j|$
where the outer median is a low median (order statistic of rank [[(n+1)/2]]) and the inner median is a high median (order statistic of rank [[n/2]+1]), and where c_n is a scalar depending on sample size n.
Q_n is another efficient alternative to MAD. It is based on the kth order statistic of the $( n \ 2 )$ inter-point distances:
$Q_n = d_n * \{ | x_i - x_j|; i \lt j \}_{(k)} {with} k \approx ( n \ 2 ) / 4$
where d_n is a scalar similar to, but different from c_n. See Rousseeuw and Croux (1993) for more details.

The scalars c_n and d_n are defined as follows:

$c_n = 1.1926 * \{ .743 & {for n=2} \ 1.851 & {for n=3} \ .954 & {for n=4} \ 1.35... ...n=8} \ .872 & {for n=9} \ n/(n + 1.4) & {uneven n} \ n/(n + 3.8) & {even n} .$

Example

This example uses the univariate data set of Barnett & Lewis (1978) that is used above to illustrate the univariate LMS and LTS estimates:

  b = { 3, 4, 7, 8, 10, 949, 951 };

  rmad1 = mad(b);
  rmad2 = mad(b,"mad");
  rmad3 = mad(b,"nmad");
  rmad4 = mad(b,"sn");
  rmad5 = mad(b,"qn");
  print "Default MAD=" rmad1,
        "Common MAD =" rmad2,
        "MAD*1.4826 =" rmad3,
        "Robust S_n =" rmad4,
        "Robust Q_n =" rmad5;

This program produces the following:

                       Default MAD=         4
                       Common MAD =         4
                       MAD*1.4826 = 5.9304089
                       Robust S_n =  7.143674
                       Robust Q_n = 5.7125049

References

Barnett, V. and Lewis, T. (1978), Outliers in Statistical Data, New York: John Wiley & Sons, Inc.
Rousseeuw, P.J. and Croux, C. (1993), "Alternatives to the Median Absolute Deviation," Journal of the American Statistical Association, 88, 1273 -1283.

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