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NAG Library Routine Document

g07daf (robust_1var_median)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 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

Fortran Interface

Subroutine g07daf (

n, x, y, xme, xmd, xsd, ifail)

Integer, Intent (In)	::	n
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x(n)
Real (Kind=nag_wp), Intent (Out)	::	y(n), xme, xmd, xsd

C Header Interface

#include nagmk26.h

void	g07daf_ ( const Integer n, const double x[], double y[], double xme, double xmd, double xsd, Integer *ifail)

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

Arguments

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 argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

The computations are believed to be stable.

8

Parallelism and Performance

g07daf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g07daf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

Further Comments

Unless otherwise stated in the Users' Note, the routine may be called with the same actual array supplied for arguments 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.

10

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 Document

g07daf (robust_1var_median)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results