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NAG Toolbox

NAG Toolbox: nag_univar_robust_1var_median (g07da)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_univar_robust_1var_median (g07da) finds the median, median absolute deviation, and a robust estimate of the standard deviation for a set of ungrouped data.

Syntax

[y, xme, xmd, xsd, ifail] = g07da(x, 'n', n)
[y, xme, xmd, xsd, ifail] = nag_univar_robust_1var_median(x, 'n', n)

Description

The data consists of a sample of size n, denoted by x1,x2,,xn, drawn from a random variable X.
nag_univar_robust_1var_median (g07da) first computes the median,
θmed=medixi,  
and from this the median absolute deviation can be computed,
σmed=medixi-θmed.  
Finally, a robust estimate of the standard deviation is computed,
σmed=σmed/Φ-10.75  
where Φ-10.75 is the value of the inverse standard Normal function at the point 0.75.
nag_univar_robust_1var_median (g07da) is based upon function LTMDDV within the ROBETH library, see Marazzi (1987).

References

Huber P J (1981) Robust Statistics Wiley
Marazzi A (1987) Subroutines for robust estimation of location and scale in ROBETH Cah. Rech. Doc. IUMSP, No. 3 ROB 1 Institut Universitaire de Médecine Sociale et Préventive, Lausanne

Parameters

Compulsory Input Parameters

1:     xn – double array
The vector of observations, x1,x2,,xn.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the dimension of the array x.
n, the number of observations.
Constraint: n>1.

Output Parameters

1:     yn – double array
The observations sorted into ascending order.
2:     xme – double scalar
The median, θmed.
3:     xmd – double scalar
The median absolute deviation, σmed.
4:     xsd – double scalar
The robust estimate of the standard deviation, σmed.
5:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
   ifail=1
On entry,n1.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

Accuracy

The computations are believed to be stable.

Further Comments

None.

Example

The following program reads in a set of data consisting of eleven observations of a variable X. The median, median absolute deviation and a robust estimate of the standard deviation are calculated and printed along with the sorted data in output array y.
function g07da_example


fprintf('g07da example results\n\n');

x = [13;  11;  16;  5;  3;  18;  9;  8;  6;  27;  7];
fprintf('Original Data\n ');
fprintf('%7.3f',x)
fprintf('\n\n');

% Sort Data abd compute estimates
[y, xme, xmd, xsd, ifail] = g07da(x);

fprintf('Sorted Data\n ');
fprintf('%7.3f',y)
fprintf('\n\n');
fprintf('Median                             = %6.3f\n', xme);
fprintf('Median absolute deviation          = %6.3f\n', xmd);
fprintf('Robust estimate standard deviation = %6.3f\n', xsd);


g07da example results

Original Data
  13.000 11.000 16.000  5.000  3.000 18.000  9.000  8.000  6.000 27.000  7.000

Sorted Data
   3.000  5.000  6.000  7.000  8.000  9.000 11.000 13.000 16.000 18.000 27.000

Median                             =  9.000
Median absolute deviation          =  4.000
Robust estimate standard deviation =  5.930

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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