# NAG Library Routine Document

## 1Purpose

g07dcf computes an $M$-estimate of location with (optional) simultaneous estimation of scale, where you provide the weight functions.

## 2Specification

Fortran Interface
 Subroutine g07dcf ( chi, psi, n, x, beta, tol, rs, nit, wrk,
 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)
#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)

## 3Description

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,
 $Fxi-θ/σ$
where $\theta$ is a location parameter, and $\sigma$ is a scale parameter. $M$-estimators of $\theta$ and $\sigma$ are given by the solution to the following system of equations;
 $∑i=1nψxi-θ^/σ^ = 0 ∑i=1nχxi-θ^/σ^ = n-1β$
where $\psi$ and $\chi$ are user-supplied weight functions, and $\beta$ is a constant. Optionally the second equation can be omitted and the first equation is solved for $\stackrel{^}{\theta }$ using an assigned value of $\sigma ={\sigma }_{c}$.
The constant $\beta$ should be chosen so that $\stackrel{^}{\sigma }$ is an unbiased estimator when ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$ has a Normal distribution. To achieve this the value of $\beta$ is calculated as:
 $β=Eχ=∫-∞∞χz12πexp-z22dz$
The values of $\psi \left(\frac{{x}_{i}-\stackrel{^}{\theta }}{\stackrel{^}{\sigma }}\right)\stackrel{^}{\sigma }$ are known as the Winsorized residuals.
The equations are solved by a simple iterative procedure, suggested by Huber:
 $σ^k=1βn-1 ∑i=1nχ xi-θ^k-1σ^k-1 σ^k-12$
and
 $θ^k=θ^k- 1+1n ∑i= 1nψ xi-θ^k- 1σ^k σ^k$
or
 $σ^k=σc$
if $\sigma$ is fixed.
The initial values for $\stackrel{^}{\theta }$ and $\stackrel{^}{\sigma }$ may be user-supplied or calculated within g07dbf as the sample median and an estimate of $\sigma$ based on the median absolute deviation respectively.
g07dcf is based upon subroutine LYHALG within the ROBETH library, see Marazzi (1987).

## 4References

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

## 5Arguments

1:     $\mathbf{chi}$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure
chi must return the value of the weight function $\chi$ for a given value of its argument. The value of $\chi$ must be non-negative.
The specification of chi is:
Fortran Interface
 Function chi ( t)
 Real (Kind=nag_wp) :: chi Real (Kind=nag_wp), Intent (In) :: t
#include <nagmk26.h>
 double chi (const double *t)
1:     $\mathbf{t}$ – 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:     $\mathbf{psi}$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure
psi must return the value of the weight function $\psi$ for a given value of its argument.
The specification of psi is:
Fortran Interface
 Function psi ( t)
 Real (Kind=nag_wp) :: psi Real (Kind=nag_wp), Intent (In) :: t
#include <nagmk26.h>
 double psi (const double *t)
1:     $\mathbf{t}$ – 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:     $\mathbf{isigma}$ – IntegerInput
On entry: the value assigned to isigma determines whether $\stackrel{^}{\sigma }$ is to be simultaneously estimated.
${\mathbf{isigma}}=0$
The estimation of $\stackrel{^}{\sigma }$ is bypassed and sigma is set equal to ${\sigma }_{c}$.
${\mathbf{isigma}}=1$
$\stackrel{^}{\sigma }$ is estimated simultaneously.
4:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of observations.
Constraint: ${\mathbf{n}}>1$.
5:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the vector of observations, ${x}_{1},{x}_{2},\dots ,{x}_{n}$.
6:     $\mathbf{beta}$ – Real (Kind=nag_wp)Input
On entry: the value of the constant $\beta$ of the chosen chi function.
Constraint: ${\mathbf{beta}}>0.0$.
7:     $\mathbf{theta}$ – Real (Kind=nag_wp)Input/Output
On entry: if ${\mathbf{sigma}}>0$, theta must be set to the required starting value of the estimate of the location parameter $\stackrel{^}{\theta }$. A reasonable initial value for $\stackrel{^}{\theta }$ will often be the sample mean or median.
On exit: the $M$-estimate of the location parameter $\stackrel{^}{\theta }$.
8:     $\mathbf{sigma}$ – Real (Kind=nag_wp)Input/Output
On entry: the role of sigma depends on the value assigned to isigma as follows.
If ${\mathbf{isigma}}=1$, sigma must be assigned a value which determines the values of the starting points for the calculation of $\stackrel{^}{\theta }$ and $\stackrel{^}{\sigma }$. If ${\mathbf{sigma}}\le 0.0$, g07dcf will determine the starting points of $\stackrel{^}{\theta }$ and $\stackrel{^}{\sigma }$. Otherwise, the value assigned to sigma will be taken as the starting point for $\stackrel{^}{\sigma }$, and theta must be assigned a relevant value before entry, see above.
If ${\mathbf{isigma}}=0$, sigma must be assigned a value which determines the values of ${\sigma }_{c}$, which is held fixed during the iterations, and the starting value for the calculation of $\stackrel{^}{\theta }$. If ${\mathbf{sigma}}\le 0$, g07dcf will determine the value of ${\sigma }_{c}$ as the median absolute deviation adjusted to reduce bias (see g07daf) and the starting point for $\theta$. Otherwise, the value assigned to sigma will be taken as the value of ${\sigma }_{c}$ and theta must be assigned a relevant value before entry, see above.
On exit: the $M$-estimate of the scale parameter $\stackrel{^}{\sigma }$, if isigma was assigned the value $1$ on entry, otherwise sigma will contain the initial fixed value ${\sigma }_{c}$.
9:     $\mathbf{maxit}$ – IntegerInput
On entry: the maximum number of iterations that should be used during the estimation.
Suggested value: ${\mathbf{maxit}}=50$.
Constraint: ${\mathbf{maxit}}>0$.
10:   $\mathbf{tol}$ – 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 ${\mathbf{tol}}×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1.0,{\sigma }_{k-1}\right)$.
Constraint: ${\mathbf{tol}}>0.0$.
11:   $\mathbf{rs}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the Winsorized residuals.
12:   $\mathbf{nit}$ – IntegerOutput
On exit: the number of iterations that were used during the estimation.
13:   $\mathbf{wrk}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: if ${\mathbf{sigma}}\le 0.0$ on entry, wrk will contain the $n$ observations in ascending order.
14:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . 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  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  is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{isigma}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{isigma}}=0$ or $1$.
On entry, ${\mathbf{maxit}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{maxit}}>0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>1$.
On entry, ${\mathbf{tol}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{tol}}>0.0$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{beta}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{beta}}>0.0$.
${\mathbf{ifail}}=3$
All elements of x are equal.
${\mathbf{ifail}}=4$
Current estimate of sigma is zero or negative: ${\mathbf{sigma}}=〈\mathit{\text{value}}〉$. This error exit is very unlikely, although it may be caused by too large an initial value of sigma.
${\mathbf{ifail}}=5$
Number of iterations required exceeds maxit: ${\mathbf{maxit}}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=6$
All winsorized residuals are zero. This may occur when using the ${\mathbf{isigma}}=0$ option with a redescending $\psi$ function, i.e., Hampel's piecewise linear function, Andrew's sine wave, and Tukey's biweight.
If the given value of $\sigma$ is too small, the standardized residuals $\frac{{x}_{i}-{\stackrel{^}{\theta }}_{k}}{{\sigma }_{c}}$, will be large and all the residuals may fall into the region for which $\psi \left(t\right)=0$. This may incorrectly terminate the iterations thus making theta and sigma invalid.
Re-enter the routine with a larger value of ${\sigma }_{c}$ or with ${\mathbf{isigma}}=1$.
${\mathbf{ifail}}=7$
The chi function returned a negative value: ${\mathbf{chi}}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

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

## 8Parallelism 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.

Standard forms of the functions $\psi$ and $\chi$ are given in Hampel et al. (1986), Huber (1981) and Marazzi (1987). g07dbf calculates $M$-estimates using some standard forms for $\psi$ and $\chi$.
When you supply the initial values, care has to be taken over the choice of the initial value of $\sigma$. If too small a value is chosen then initial values of the standardized residuals $\frac{{x}_{i}-{\stackrel{^}{\theta }}_{k}}{\sigma }$ will be large. If the redescending $\psi$ functions are used, i.e., $\psi =0$ if $\left|t\right|>\tau$, for some positive constant $\tau$, 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).

## 10Example

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 $\theta$ and $\sigma$ are calculated and printed along with the number of iterations used:
 (a) g07dcf determined the starting values, $\sigma$ is estimated simultaneously. (b) You must supply the starting values, $\sigma$ is estimated simultaneously. (c) g07dcf determined the starting values, $\sigma$ is fixed. (d) You must supply the starting values, $\sigma$ is fixed.

### 10.1Program Text

Program Text (g07dcfe.f90)

### 10.2Program Data

Program Data (g07dcfe.d)

### 10.3Program Results

Program Results (g07dcfe.r)