NAG CL Interface
g07dcc (robust_1var_mestim_wgt)
1
Purpose
g07dcc computes an $M$estimate of location with (optional) simultaneous estimation of scale, where you provide the weight functions.
2
Specification
void 
g07dcc (
double 
(*chi)(double t,
Nag_Comm *comm),


double 
(*psi)(double t,
Nag_Comm *comm),


Integer isigma,
Integer n,
const double x[],
double beta,
double *theta,
double *sigma,
Integer maxit,
double tol,
double rs[],
Integer *nit,
Nag_Comm *comm,
NagError *fail) 

The function may be called by the names: g07dcc, nag_univar_robust_1var_mestim_wgt or nag_robust_m_estim_1var_usr.
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
g07dbc as the sample median and an estimate of
$\sigma $ based on the median absolute deviation respectively.
g07dcc is based upon function 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:
$\mathbf{chi}$ – function, supplied by the user
External Function

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:
double 
chi (double t,
Nag_Comm *comm)



1:
$\mathbf{t}$ – double
Input

On entry: the argument for which
chi must be evaluated.

2:
$\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
chi.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling
g07dcc you may allocate memory and initialize these pointers with various quantities for use by
chi when called from
g07dcc (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: chi should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
g07dcc. If your code inadvertently
does return any NaNs or infinities,
g07dcc is likely to produce unexpected results.

2:
$\mathbf{psi}$ – function, supplied by the user
External Function

psi must return the value of the weight function
$\psi $ for a given value of its argument.
The specification of
psi is:
double 
psi (double t,
Nag_Comm *comm)



1:
$\mathbf{t}$ – double
Input

On entry: the argument for which
psi must be evaluated.

2:
$\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
psi.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling
g07dcc you may allocate memory and initialize these pointers with various quantities for use by
psi when called from
g07dcc (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: psi should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
g07dcc. If your code inadvertently
does return any NaNs or infinities,
g07dcc 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:
$\mathbf{x}\left[{\mathbf{n}}\right]$ – const double
Input

On entry: the vector of observations, ${x}_{1},{x}_{2},\dots ,{x}_{n}$.

6:
$\mathbf{beta}$ – double
Input

On entry: the value of the constant
$\beta $ of the chosen
chi function.
Constraint:
${\mathbf{beta}}>0.0$.

7:
$\mathbf{theta}$ – double *
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:
$\mathbf{sigma}$ – double *
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
${\mathbf{sigma}}\le 0.0$,
g07dcc 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
${\mathbf{sigma}}\le 0$,
g07dcc will determine the value of
${\sigma}_{c}$ as the median absolute deviation adjusted to reduce bias (see
g07dac) 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:
$\mathbf{tol}$ – double
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:
$\mathbf{rs}\left[{\mathbf{n}}\right]$ – double
Output

On exit: the Winsorized residuals.

12:
$\mathbf{nit}$ – Integer *
Output

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

13:
$\mathbf{comm}$ – Nag_Comm *

The NAG communication argument (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).

14:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_FUN_RET_VAL

The
chi function returned a negative value:
${\mathbf{chi}}=\u2329\mathit{\text{value}}\u232a$.
 NE_INT

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: ${\mathbf{n}}>1$.
 NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
 NE_REAL

On entry, ${\mathbf{beta}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{beta}}>0.0$.
On entry, ${\mathbf{tol}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{tol}}>0.0$.
 NE_REAL_ARRAY_ELEM_CONS

All elements of
x are equal.
 NE_SIGMA_NEGATIVE

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.
 NE_TOO_MANY_ITER

Number of iterations required exceeds
maxit:
${\mathbf{maxit}}=\u2329\mathit{\text{value}}\u232a$.
 NE_ZERO_RESID

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 function with a larger value of ${\sigma}_{c}$ or with ${\mathbf{isigma}}=1$.
7
Accuracy
On successful exit the accuracy of the results is related to the value of
tol, see
Section 5.
8
Parallelism and Performance
g07dcc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g07dcc 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 function. 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).
g07dbc 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)g07dcc determined the starting values, $\sigma $ is estimated simultaneously.

(b)You must supply the starting values, $\sigma $ is estimated simultaneously.

(c)g07dcc 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