NAG Library Routine Document

G07DAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G07DAF finds the median, median absolute deviation, and a robust estimate of the standard deviation for a set of ungrouped data.

2 Specification

SUBROUTINE G07DAF (

N, X, Y, XME, XMD, XSD, IFAIL)

INTEGER	N, IFAIL
REAL (KIND=nag_wp)	X(N), Y(N), XME, XMD, XSD

3 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

G07DAF 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

G07DAF is based upon subroutine LTMDDV within the ROBETH library, see Marazzi (1987).

4 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

5 Parameters

1: N – INTEGERInput: On entry: $n$ , the number of observations.

Constraint: $N > 1$ .
2: X(N) – REAL (KIND=nag_wp) arrayInput: On entry: the vector of observations, $x_{1}, x_{2}, \dots, x_{n}$ .
3: Y(N) – REAL (KIND=nag_wp) arrayOutput: On exit: the observations sorted into ascending order.
4: XME – REAL (KIND=nag_wp)Output: On exit: the median, $θ_{med}$ .
5: XMD – REAL (KIND=nag_wp)Output: On exit: the median absolute deviation, $σ_{med}$ .
6: XSD – REAL (KIND=nag_wp)Output: On exit: the robust estimate of the standard deviation, $σ_{med}^{'}$ .
7: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit: $IFAIL = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,

N \leq 1

7 Accuracy

The computations are believed to be stable.

8 Further Comments

Unless otherwise stated in the Users' Note, the routine may be called with the same actual array supplied for parameters X and Y, in which case the sorted data values will overwrite the original contents of X. However this is not standard Fortran, and may not work on all systems.

9 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.

NAG Library Routine DocumentG07DAF