NAG Library Routine Document

Integer, Intent (In)	::	isigma, n, maxit
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	nit
Real (Kind=nag_wp), External	::	chi, psi
Real (Kind=nag_wp), Intent (In)	::	x(n), beta, tol
Real (Kind=nag_wp), Intent (Inout)	::	theta, sigma
Real (Kind=nag_wp), Intent (Out)	::	rs(n), wrk(n)

C Header Interface

#include <nagmk26.h>

void	g07dcf_ ( double (NAG_CALL chi)(const double t), double (NAG_CALL psi)(const double t), const Integer isigma, const Integer n, const double x[], const double beta, double theta, double sigma, const Integer maxit, const double tol, double rs[], Integer nit, double wrk[], 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

The

x_{i}

are assumed to be independent with an unknown distribution function of the form,

F ((x_{i} - θ) / σ)

where

θ

is a location parameter, and

σ

is a scale parameter.

M

-estimators of

θ

and

σ

are given by the solution to the following system of equations;

\begin{array}{lcl} \sum_{i = 1}^{n} ψ ((x_{i} - \hat{θ}) / \hat{σ}) & = & 0 \\ \sum_{i = 1}^{n} χ ((x_{i} - \hat{θ}) / \hat{σ}) & = & (n - 1) β \end{array}

where

ψ

and

χ

are user-supplied weight functions, and

β

is a constant. Optionally the second equation can be omitted and the first equation is solved for

\hat{θ}

using an assigned value of

σ = σ_{c}

The constant

β

should be chosen so that

\hat{σ}

is an unbiased estimator when

x_{i}

, for

i = 1, 2, \dots, n

has a Normal distribution. To achieve this the value of

β

is calculated as:

β = E (χ) = \int_{- \infty}^{\infty} χ (z) \frac{1}{\sqrt{2 π}} \exp \{\frac{- z^{2}}{2}\} d z

The values of

ψ (\frac{x_{i} - \hat{θ}}{\hat{σ}}) \hat{σ}

are known as the Winsorized residuals.

The equations are solved by a simple iterative procedure, suggested by Huber:

{\hat{σ}}_{k} = \sqrt{\frac{1}{β (n - 1)} (\sum_{i = 1}^{n} χ (\frac{x_{i} - {\hat{θ}}_{k - 1}}{{\hat{σ}}_{k - 1}})) {\hat{σ}}_{k - 1}^{2}}

and

{\hat{θ}}_{k} = {\hat{θ}}_{k - 1} + \frac{1}{n} \sum_{i = 1}^{n} ψ (\frac{x_{i} - {\hat{θ}}_{k - 1}}{{\hat{σ}}_{k}}) {\hat{σ}}_{k}

{\hat{σ}}_{k} = σ_{c}

σ

is fixed.

The initial values for

\hat{θ}

and

\hat{σ}

may be user-supplied or calculated within g07dbf as the sample median and an estimate of

σ

based on the median absolute deviation respectively.

g07dcf is based upon subroutine LYHALG within the ROBETH library, see Marazzi (1987).

4

References

Hampel F R, Ronchetti E M, Rousseeuw P J and Stahel W A (1986) Robust Statistics. The Approach Based on Influence Functions Wiley

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: $chi$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure

chi must return the value of the weight function

χ

for a given value of its argument. The value of

χ

must be non-negative.

The specification of chi is:

Fortran Interface

Function chi (

Real (Kind=nag_wp)	::	chi
Real (Kind=nag_wp), Intent (In)	::	t

C Header Interface

#include <nagmk26.h>

double

chi (const double *t)

1: $t$ – Real (Kind=nag_wp)Input: On entry: the argument for which chi must be evaluated.

chi must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which g07dcf is called. Arguments denoted as Input must not be changed by this procedure.

Note: chi should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by g07dcf. If your code inadvertently does return any NaNs or infinities, g07dcf is likely to produce unexpected results.

2: $psi$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure

psi must return the value of the weight function

ψ

for a given value of its argument.

The specification of psi is:

Fortran Interface

Function psi (

Real (Kind=nag_wp)	::	psi
Real (Kind=nag_wp), Intent (In)	::	t

C Header Interface

#include <nagmk26.h>

double

psi (const double *t)

1: $t$ – Real (Kind=nag_wp)Input: On entry: the argument for which psi must be evaluated.

psi must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which g07dcf is called. Arguments denoted as Input must not be changed by this procedure.

Note: psi should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by g07dcf. If your code inadvertently does return any NaNs or infinities, g07dcf is likely to produce unexpected results.

3: $isigma$ – IntegerInput

On entry: the value assigned to isigma determines whether

\hat{σ}

is to be simultaneously estimated.

$isigma = 0$: The estimation of $\hat{σ}$ is bypassed and sigma is set equal to $σ_{c}$ .
$isigma = 1$: $\hat{σ}$ is estimated simultaneously.

4: $n$ – IntegerInput

On entry:

n

, the number of observations.

Constraint:

n > 1

5: $x (n)$ – Real (Kind=nag_wp) arrayInput

On entry: the vector of observations,

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

6: $beta$ – Real (Kind=nag_wp)Input

On entry: the value of the constant

β

of the chosen chi function.

Constraint:

beta > 0.0

7: $theta$ – Real (Kind=nag_wp)Input/Output

On entry: if

sigma > 0

, theta must be set to the required starting value of the estimate of the location parameter

\hat{θ}

. A reasonable initial value for

\hat{θ}

will often be the sample mean or median.

On exit: the

M

-estimate of the location parameter

\hat{θ}

8: $sigma$ – Real (Kind=nag_wp)Input/Output

On entry: the role of sigma depends on the value assigned to isigma as follows.

isigma = 1

, sigma must be assigned a value which determines the values of the starting points for the calculation of

\hat{θ}

and

\hat{σ}

. If

sigma \leq 0.0

, g07dcf will determine the starting points of

\hat{θ}

and

\hat{σ}

. Otherwise, the value assigned to sigma will be taken as the starting point for

\hat{σ}

, and theta must be assigned a relevant value before entry, see above.

isigma = 0

, sigma must be assigned a value which determines the values of

σ_{c}

, which is held fixed during the iterations, and the starting value for the calculation of

\hat{θ}

. If

sigma \leq 0

, g07dcf will determine the value of

σ_{c}

as the median absolute deviation adjusted to reduce bias (see g07daf) and the starting point for

θ

. Otherwise, the value assigned to sigma will be taken as the value of

σ_{c}

and theta must be assigned a relevant value before entry, see above.

On exit: the

M

-estimate of the scale parameter

\hat{σ}

, if isigma was assigned the value

1

on entry, otherwise sigma will contain the initial fixed value

σ_{c}

9: $maxit$ – IntegerInput

On entry: the maximum number of iterations that should be used during the estimation.

Suggested value:

maxit = 50

Constraint:

maxit > 0

10: $tol$ – Real (Kind=nag_wp)Input

On entry: the relative precision for the final estimates. Convergence is assumed when the increments for theta, and sigma are less than

tol \times \max (1.0, σ_{k - 1})

Constraint:

tol > 0.0

11: $rs (n)$ – Real (Kind=nag_wp) arrayOutput

On exit: the Winsorized residuals.

12: $nit$ – IntegerOutput

On exit: the number of iterations that were used during the estimation.

13: $wrk (n)$ – Real (Kind=nag_wp) arrayOutput

On exit: if

sigma \leq 0.0

on entry, wrk will contain the

n

observations in ascending order.

14: $ifail$ – 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, $isigma = 〈value〉$ .
Constraint: $isigma = 0$ or $1$ .

On entry, $maxit = 〈value〉$ .
Constraint: $maxit > 0$ .

On entry, $n = 〈value〉$ .
Constraint: $n > 1$ .

On entry, $tol = 〈value〉$ .
Constraint: $tol > 0.0$ .

$ifail = 2$: On entry, $beta = 〈value〉$ .
Constraint: $beta > 0.0$ .

$ifail = 3$: All elements of x are equal.

$ifail = 4$: Current estimate of sigma is zero or negative: $sigma = 〈value〉$ . This error exit is very unlikely, although it may be caused by too large an initial value of sigma.

$ifail = 5$: Number of iterations required exceeds maxit: $maxit = 〈value〉$ .

$ifail = 6$: All winsorized residuals are zero. This may occur when using the $isigma = 0$ option with a redescending $ψ$ function, i.e., Hampel's piecewise linear function, Andrew's sine wave, and Tukey's biweight.
If the given value of $σ$ is too small, the standardized residuals $\frac{x_{i} - {\hat{θ}}_{k}}{σ_{c}}$ , will be large and all the residuals may fall into the region for which $ψ (t) = 0$ . This may incorrectly terminate the iterations thus making theta and sigma invalid.
Re-enter the routine with a larger value of $σ_{c}$ or with $isigma = 1$ .

$ifail = 7$: The chi function returned a negative value: $chi = 〈value〉$ .

$ifail = - 99$: 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

On successful exit the accuracy of the results is related to the value of tol, see Section 5.

8

Parallelism and Performance

g07dcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g07dcf 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

Standard forms of the functions

ψ

and

χ

are given in Hampel et al. (1986), Huber (1981) and Marazzi (1987). g07dbf calculates

M

-estimates using some standard forms for

ψ

and

χ

When you supply the initial values, care has to be taken over the choice of the initial value of

σ

. If too small a value is chosen then initial values of the standardized residuals

\frac{x_{i} - {\hat{θ}}_{k}}{σ}

will be large. If the redescending

ψ

functions are used, i.e.,

ψ = 0

|t| > τ

, for some positive constant

τ

, then these large values are Winsorized as zero. If a sufficient number of the residuals fall into this category then a false solution may be returned, see page 152 of Hampel et al. (1986).

10

Example

The following program reads in a set of data consisting of eleven observations of a variable

X

The psi and chi functions used are Hampel's Piecewise Linear Function and Hubers chi function respectively.

Using the following starting values various estimates of

θ

and

σ

are calculated and printed along with the number of iterations used:

(a)	g07dcf determined the starting values, $σ$ is estimated simultaneously.
(b)	You must supply the starting values, $σ$ is estimated simultaneously.
(c)	g07dcf determined the starting values, $σ$ is fixed.
(d)	You must supply the starting values, $σ$ is fixed.

NAG Library Routine Document

g07dcf (robust_1var_mestim_wgt)

▸▿ 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

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