NAG FL Interface
g07dcf (robust_1var_mestim_wgt)
1
Purpose
g07dcf computes an $M$estimate of location with (optional) simultaneous estimation of scale, where you provide the weight functions.
2
Specification
Fortran Interface
Subroutine g07dcf ( 
chi, psi, isigma, n, x, beta, theta, sigma, maxit, tol, rs, nit, wrk, ifail) 
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 <nag.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) 

C++ Header Interface
#include <nag.h> extern "C" {
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) 
}

The routine may be called by the names g07dcf or nagf_univar_robust_1var_mestim_wgt.
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,
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;
where
$\psi $ and
$\chi $ are usersupplied weight functions, and
$\beta $ is a constant. Optionally the second equation can be omitted and the first equation is solved for
$\hat{\theta}$ using an assigned value of
$\sigma ={\sigma}_{c}$.
The constant
$\beta $ should be chosen so that
$\hat{\sigma}$ 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:
The values of
$\psi \left(\frac{{x}_{i}\hat{\theta}}{\hat{\sigma}}\right)\hat{\sigma}$ are known as the Winsorized residuals.
The equations are solved by a simple iterative procedure, suggested by Huber:
and
or
if
$\sigma $ is fixed.
The initial values for
$\hat{\theta}$ and
$\hat{\sigma}$ may be usersupplied 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).
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:
$\mathbf{chi}$ – 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 nonnegative.
The specification of
chi is:
Fortran Interface
Real (Kind=nag_wp) 
:: 
chi 
Real (Kind=nag_wp), Intent (In) 
:: 
t 

C Header Interface
double 
chi_ (const double *t) 

C++ Header Interface
#include <nag.h> extern "C" {
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 floatingpoint 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
Real (Kind=nag_wp) 
:: 
psi 
Real (Kind=nag_wp), Intent (In) 
:: 
t 

C Header Interface
double 
psi_ (const double *t) 

C++ Header Interface
#include <nag.h> extern "C" {
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 floatingpoint 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:
$\mathbf{isigma}$ – Integer
Input

On entry: the value assigned to
isigma determines whether
$\hat{\sigma}$ is to be simultaneously estimated.
 ${\mathbf{isigma}}=0$
 The estimation of $\hat{\sigma}$ is bypassed and sigma is set equal to ${\sigma}_{c}$.
 ${\mathbf{isigma}}=1$
 $\hat{\sigma}$ is estimated simultaneously.

4:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the number of observations.
Constraint:
${\mathbf{n}}>1$.

5:
$\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) array
Input

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
$\hat{\theta}$. A reasonable initial value for
$\hat{\theta}$ will often be the sample mean or median.
On exit: the $M$estimate of the location parameter $\hat{\theta}$.

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
$\hat{\theta}$ and
$\hat{\sigma}$. If
${\mathbf{sigma}}\le 0.0$,
g07dcf will determine the starting points of
$\hat{\theta}$ and
$\hat{\sigma}$. Otherwise, the value assigned to
sigma will be taken as the starting point for
$\hat{\sigma}$, 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
$\hat{\theta}$. 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
$\hat{\sigma}$, if
isigma was assigned the value
$1$ on entry, otherwise
sigma will contain the initial fixed value
${\sigma}_{c}$.

9:
$\mathbf{maxit}$ – Integer
Input

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
${\mathbf{tol}}\times \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1.0,{\sigma}_{k1}\right)$.
Constraint:
${\mathbf{tol}}>0.0$.

11:
$\mathbf{rs}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) array
Output

On exit: the Winsorized residuals.

12:
$\mathbf{nit}$ – Integer
Output

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

13:
$\mathbf{wrk}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) array
Output

On exit: if
${\mathbf{sigma}}\le 0.0$ on entry,
wrk will contain the
$n$ observations in ascending order.

14:
$\mathbf{ifail}$ – Integer
Input/Output

On entry:
ifail must be set to
$0$,
$1$ or
$1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value
$1$ or
$1$ is recommended. If message printing is undesirable, then the value
$1$ is recommended. Otherwise, the value
$0$ is recommended.
When the value $\mathbf{1}$ or $\mathbf{1}$ 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).
6
Error 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, ${\mathbf{isigma}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{isigma}}=0$ or $1$.
On entry, ${\mathbf{maxit}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{maxit}}>0$.
On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}>1$.
On entry, ${\mathbf{tol}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{tol}}>0.0$.
 ${\mathbf{ifail}}=2$

On entry, ${\mathbf{beta}}=\u2329\mathit{\text{value}}\u232a$.
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:
${\mathbf{sigma}}=\u2329\mathit{\text{value}}\u232a$. 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:
${\mathbf{maxit}}=\u2329\mathit{\text{value}}\u232a$.
 ${\mathbf{ifail}}=6$

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
$\frac{{x}_{i}{\hat{\theta}}_{k}}{{\sigma}_{c}}$, 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.
Reenter the routine with a larger value of ${\sigma}_{c}$ or with ${\mathbf{isigma}}=1$.
 ${\mathbf{ifail}}=7$

The
chi function returned a negative value:
${\mathbf{chi}}=\u2329\mathit{\text{value}}\u232a$.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 7 in the Introduction to the NAG Library FL Interface for further information.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library FL Interface for further information.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
See
Section 9 in the Introduction to the NAG Library FL Interface 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 implementationspecific 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
$\frac{{x}_{i}{\hat{\theta}}_{k}}{\sigma}$ will be large. If the redescending
$\psi $ functions are used, i.e.,
$\psi =0$ if
$\leftt\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).
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
$\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.
10.1
Program Text
10.2
Program Data
10.3
Program Results