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

x_{1}, x_{2}, \dots, x_{n}

, drawn from a random variable

X

nag_univar_robust_1var_median (g07da) first computes the median,

θ_{med} = {med}_{i} \{x_{i}\},

and from this the median absolute deviation can be computed,

σ_{med} = {med}_{i} \{|x_{i} - θ_{med}|\} .

Finally, a robust estimate of the standard deviation is computed,

σ_{med}^{'} = σ_{med} / Φ^{- 1} (0.75)

where

Φ^{- 1} (0.75)

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: $x (n)$ – double array: The vector of observations, $x_{1}, x_{2}, \dots, x_{n}$ .

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: $y (n)$ – double array: The observations sorted into ascending order.
2: $xme$ – double scalar: The median, $θ_{med}$ .
3: $xmd$ – double scalar: The median absolute deviation, $σ_{med}$ .
4: $xsd$ – double scalar: The robust estimate of the standard deviation, $σ_{med}^{'}$ .
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,

n \leq 1

$ifail = - 99$: 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.

Open in the MATLAB editor: g07da_example

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