where	$y$ is a vector of length $n$ of the dependent variable,
	$X$ is a $n$ by $m$ matrix of independent variables of column rank $k$ ,
	$θ$ is a vector of length $m$ of unknown arguments,
and	$ε$ is a vector of length $n$ of unknown errors with $var (ε_{i}) = σ^{2}$ ,

nag_correg_robustm (g02ha) calculates the M-estimates given by the solution,

\hat{θ}

, to the equation

\sum_{i = 1}^{n} ψ (r_{i} / (σ w_{i})) w_{i} x_{i j} = 0, j = 1, 2, \dots, m,

(1)

where

r_{i}

is the

i

th residual, i.e., the

i

th element of

r = y - X \hat{θ}

ψ

is a suitable weight function,

w_{i}

are suitable weights,

and

σ

may be estimated at each iteration by the median absolute deviation of the residuals

\hat{σ} = \underset{i}{med} [|r_{i}|] / β_{1}

or as the solution to

\sum_{i = 1}^{n} χ (r_{i} / (\hat{σ} w_{i})) w_{i}^{2} = (n - k) β_{2}

for suitable weight function

χ

, where

β_{1}

and

β_{2}

are constants, chosen so that the estimator of

σ

is asymptotically unbiased if the errors,

ε_{i}

, have a Normal distribution. Alternatively

σ

may be held at a constant value.

The above describes the Schweppe type regression. If the

w_{i}

are assumed to equal

1

for all

i

then Huber type regression is obtained. A third type, due to Mallows, replaces (1) by

\sum_{i = 1}^{n} ψ (r_{i} / σ) w_{i} x_{i j} = 0, j = 1, 2, \dots, m .

This may be obtained by use of the transformations

\begin{array}{l} w_{i}^{*} \leftarrow \sqrt{w_{i}} \\ y_{i}^{*} \leftarrow y_{i} \sqrt{w_{i}} \\ x_{i j}^{*} \leftarrow x_{i j} \sqrt{w_{i}}, & j = 1, 2, \dots, m \end{array}

(see Section 3 of Marazzi (1987a)).

For Huber and Schweppe type regressions,

β_{1}

is the 75th percentile of the standard Normal distribution. For Mallows type regression

β_{1}

is the solution to

\frac{1}{n} \sum_{i = 1}^{n} Φ (β_{1} / \sqrt{w_{i}}) = 0.75,

where

Φ

is the standard Normal cumulative distribution function (see nag_specfun_cdf_normal (s15ab)).

β_{2}

is given by

\begin{array}{l} β_{2} = \int_{- \infty}^{\infty} χ (z) ϕ (z) d z & in the Huber case; \\ β_{2} = \frac{1}{n} \sum_{i = 1}^{n} w_{i} \int_{- \infty}^{\infty} χ (z) ϕ (z) d z & in the Mallows case; \\ β_{2} = \frac{1}{n} \sum_{i = 1}^{n} w_{i}^{2} \int_{- \infty}^{\infty} χ (z / w_{i}) ϕ (z) d z & in the Schweppe case; \end{array}

where

ϕ

is the standard Normal density, i.e.,

\frac{1}{\sqrt{2 π}} \exp (- \frac{1}{2} x^{2}) .

The calculation of the estimates of

θ

can be formulated as an iterative weighted least squares problem with a diagonal weight matrix

G

given by

G_{i i} = \{\begin{cases} \frac{ψ (r_{i} / (σ w_{i}))}{(r_{i} / (σ w_{i}))}, & r_{i} \neq 0 \\ ψ^{'} (0), & r_{i} = 0 \end{cases},

where

ψ^{'} (t)

is the derivative of

ψ

at the point

t

The value of

θ

at each iteration is given by the weighted least squares regression of

y

X

. This is carried out by first transforming the

y

and

X

\begin{array}{l} {\tilde{y}}_{i} = y_{i} \sqrt{G_{i i}} \\ {\tilde{x}}_{i j} = x_{i j} \sqrt{G_{i i}}, & j = 1, 2, \dots, m \end{array}

and then using nag_linsys_real_gen_solve (f04jg). If

X

is of full column rank then an orthogonal-triangular (

Q R

) decomposition is used; if not, a singular value decomposition is used.

The following functions are available for

ψ

and

χ

in nag_correg_robustm (g02ha).

(a)

Unit Weights

ψ (t) = t, χ (t) = \frac{t^{2}}{2} .

This gives least squares regression.

(b)

Huber's Function

ψ (t) = \max (- c, \min (c, t)), χ (t) = \{\begin{cases} \frac{t^{2}}{2}, & |t| \leq d \\ \frac{d^{2}}{2}, & |t| > d \end{cases}

(c)

Hampel's Piecewise Linear Function

ψ_{h_{1}, h_{2}, h_{3}} (t) = - ψ_{h_{1}, h_{2}, h_{3}} (- t) = \{\begin{cases} t, & 0 \leq t \leq h_{1} \\ h_{1}, & h_{1} \leq t \leq h_{2} \\ h_{1} (h_{3} - t) / (h_{3} - h_{2}), & h_{2} \leq t \leq h_{3} \\ 0, & h_{3} < t \end{cases}

χ (t) = \{\begin{cases} \frac{t^{2}}{2}, & |t| \leq d \\ \frac{d^{2}}{2}, & |t| > d \end{cases}

(d)

Andrew's Sine Wave Function

ψ (t) = \{\begin{cases} \sin t, & - π \leq t \leq π \\ 0, & |t| > π \end{cases} χ (t) = \{\begin{cases} \frac{t^{2}}{2}, & |t| \leq d \\ \frac{d^{2}}{2}, & |t| > d \end{cases}

(e)

Tukey's Bi-weight

ψ (t) = \{\begin{cases} t {(1 - t^{2})}^{2}, & |t| \leq 1 \\ 0, & |t| > 1 \end{cases} χ (t) = \{\begin{cases} \frac{t^{2}}{2}, & |t| \leq d \\ \frac{d^{2}}{2}, & |t| > d \end{cases}

where

c

h_{1}

h_{2}

h_{3}

, and

d

are given constants.

Several schemes for calculating weights have been proposed, see Hampel et al. (1986) and Marazzi (1987a). As the different independent variables may be measured on different scales, one group of proposed weights aims to bound a standardized measure of influence. To obtain such weights the matrix

A

has to be found such that:

\frac{1}{n} \sum_{i = 1}^{n} u ({‖z_{i}‖}_{2}) z_{i} z_{i}^{T} = I

and

z_{i} = A x_{i},

where	$x_{i}$ is a vector of length $m$ containing the $i$ th row of $X$ ,
	$A$ is an $m$ by $m$ lower triangular matrix,
and	$u$ is a suitable function.

The weights are then calculated as

w_{i} = f ({‖z_{i}‖}_{2})

for a suitable function

f

nag_correg_robustm (g02ha) finds

A

using the iterative procedure

A_{k} = (S_{k} + I) A_{k - 1},

where

S_{k} = (s_{j l})

s_{j l} = \{\begin{cases} - \min [\max (h_{j l} / n, - B L), B L], & j > l \\ - \min [\max (\frac{1}{2} (h_{j j} / n - 1), - B D), B D], & j = l \end{cases}

and

h_{j l} = \sum_{i = 1}^{n} u ({‖z_{i}‖}_{2}) z_{i j} z_{i l}

and

B L

and

B D

are bounds set at

0.9

Two weights are available in nag_correg_robustm (g02ha):

(i)

Krasker–Welsch Weights

u (t) = g_{1} (\frac{c}{t}),

where	$g_{1} (t) = t^{2} + (1 - t^{2}) (2 Φ (t) - 1) - 2 t ϕ (t)$ ,
	$Φ (t)$ is the standard Normal cumulative distribution function,
	$ϕ (t)$ is the standard Normal probability density function,
and	$f (t) = \frac{1}{t}$ .

These are for use with Schweppe type regression.

(ii)

Maronna's Proposed Weights

\begin{array}{l} u (t) = \{\begin{cases} \frac{c}{t^{2}} & |t| > c \\ 1 & |t| \leq c \end{cases} \\ f (t) = \sqrt{u (t)} . \end{array}

These are for use with Mallows type regression.

Finally the asymptotic variance-covariance matrix,

C

, of the estimates

θ

is calculated.

For Huber type regression

C = f_{H} {(X^{T} X)}^{- 1} {\hat{σ}}^{2},

where

f_{H} = \frac{1}{n - m} \frac{\sum_{i = 1}^{n} ψ^{2} (r_{i} / \hat{σ})}{{(\frac{1}{n} \sum_{i = 1}^{n} ψ^{'} (\frac{r_{i}}{\hat{σ}}))}^{2}} κ^{2}

κ^{2} = 1 + \frac{m}{n} \frac{\frac{1}{n} \sum_{i = 1}^{n} {(ψ^{'} (r_{i} / \hat{σ}) - \frac{1}{n} \sum_{i = 1}^{n} ψ^{'} (r_{i} / \hat{σ}))}^{2}}{{(\frac{1}{n} \sum_{i = 1}^{n} ψ^{'} (\frac{r_{i}}{\hat{σ}}))}^{2}} .

See Huber (1981) and Marazzi (1987b).

For Mallows and Schweppe type regressions

C

is of the form

{\frac{\hat{σ}}{n}}^{2} S_{1}^{- 1} S_{2} S_{1}^{- 1},

where

S_{1} = \frac{1}{n} X^{T} D X

and

S_{2} = \frac{1}{n} X^{T} P X

D

is a diagonal matrix such that the

i

th element approximates

E (ψ^{'} (r_{i} / (σ w_{i})))

in the Schweppe case and

E (ψ^{'} (r_{i} / σ) w_{i})

in the Mallows case.

P

is a diagonal matrix such that the

i

th element approximates

E (ψ^{2} (r_{i} / (σ w_{i})) w_{i}^{2})

in the Schweppe case and

E (ψ^{2} (r_{i} / σ) w_{i}^{2})

in the Mallows case.

Two approximations are available in nag_correg_robustm (g02ha):

Average over the

r_{i}

\begin{matrix} Schweppe & Mallows \\ D_{i} = (\frac{1}{n} \sum_{j = 1}^{n} ψ^{'} (\frac{r_{j}}{\hat{σ} w_{i}})) w_{i} & D_{i} = (\frac{1}{n} \sum_{j = 1}^{n} ψ^{'} (\frac{r_{j}}{\hat{σ}})) w_{i} \\ P_{i} = (\frac{1}{n} \sum_{j = 1}^{n} ψ^{2} (\frac{r_{j}}{\hat{σ} w_{i}})) w_{i}^{2} & P_{i} = (\frac{1}{n} \sum_{j = 1}^{n} ψ^{2} (\frac{r_{j}}{\hat{σ}})) w_{i}^{2} \end{matrix}

Replace expected value by observed

\begin{matrix} Schweppe & Mallows \\ D_{i} = ψ^{'} (\frac{r_{i}}{\hat{σ} w_{i}}) w_{i} & D_{i} = ψ^{'} (\frac{r_{i}}{\hat{σ}}) w_{i} \\ P_{i} = ψ^{2} (\frac{r_{i}}{\hat{σ} w_{i}}) w_{i}^{2} & P_{i} = ψ^{2} (\frac{r_{i}}{\hat{σ}}) w_{i}^{2} \end{matrix} .

See Hampel et al. (1986) and Marazzi (1987b).

Note: there is no explicit provision in the function for a constant term in the regression model. However, the addition of a dummy variable whose value is

1.0

for all observations will produce a value of

\hat{θ}

corresponding to the usual constant term.

nag_correg_robustm (g02ha) is based on routines in ROBETH; see Marazzi (1987a).

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 (1987a) Weights for bounded influence regression in ROBETH Cah. Rech. Doc. IUMSP, No. 3 ROB 3 Institut Universitaire de Médecine Sociale et Préventive, Lausanne

Marazzi A (1987b) Subroutines for robust and bounded influence regression in ROBETH Cah. Rech. Doc. IUMSP, No. 3 ROB 2 Institut Universitaire de Médecine Sociale et Préventive, Lausanne

Parameters

Compulsory Input Parameters

1: $indw$ – int64int32nag_int scalar

Specifies the type of regression to be performed.

$indw < 0$: Mallows type regression with Maronna's proposed weights.
$indw = 0$: Huber type regression.
$indw > 0$: Schweppe type regression with Krasker–Welsch weights.

2: $ipsi$ – int64int32nag_int scalar

Specifies which

ψ

function is to be used.

$ipsi = 0$: $ψ (t) = t$ , i.e., least squares.
$ipsi = 1$: Huber's function.
$ipsi = 2$: Hampel's piecewise linear function.
$ipsi = 3$: Andrew's sine wave.
$ipsi = 4$: Tukey's bi-weight.

Constraint:

0 \leq ipsi \leq 4

3: $isigma$ – int64int32nag_int scalar

Specifies how

σ

is to be estimated.

$isigma < 0$: $σ$ is estimated by median absolute deviation of residuals.
$isigma = 0$: $σ$ is held constant at its initial value.
$isigma > 0$: $σ$ is estimated using the $χ$ function.

4: $indc$ – int64int32nag_int scalar

indw \neq 0

, indc specifies the approximations used in estimating the covariance matrix of

\hat{θ}

$indc = 1$: Averaging over residuals.
$indc \neq 1$: Replacing expected by observed.
$indw = 0$: indc is not referenced.

5: $x (ldx, m)$ – double array

ldx, the first dimension of the array, must satisfy the constraint

ldx \geq n

The values of the

X

matrix, i.e., the independent variables.

x (i, j)

must contain the

i j

th element of

X

, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, m

indw < 0

, then during calculations the elements of x will be transformed as described in Description. Before exit the inverse transformation will be applied. As a result there may be slight differences between the input x and the output x.

6: $y (n)$ – double array

The data values of the dependent variable.

y (i)

must contain the value of

y

for the

i

th observation, for

i = 1, 2, \dots, n

indw < 0

, then during calculations the elements of y will be transformed as described in Description. Before exit the inverse transformation will be applied. As a result there may be slight differences between the input y and the output y.

7: $cpsi$ – double scalar

ipsi = 1

, cpsi must specify the argument,

c

, of Huber's

ψ

function.

ipsi \neq 1

on entry, cpsi is not referenced.

Constraint: if

cpsi > 0.0

ipsi = 1

8: $h1$ – double scalar

9: $h2$ – double scalar

10: $h3$ – double scalar

ipsi = 2

, h1, h2, and h3 must specify the arguments

h_{1}

h_{2}

, and

h_{3}

, of Hampel's piecewise linear

ψ

function. h1, h2, and h3 are not referenced if

ipsi \neq 2

Constraint: if

ipsi = 2

0.0 \leq h1 \leq h2 \leq h3

and

h3 > 0.0

11: $cucv$ – double scalar

indw < 0

, must specify the value of the constant,

c

, of the function

u

for Maronna's proposed weights.

indw > 0

, must specify the value of the function

u

for the Krasker–Welsch weights.

indw = 0

, is not referenced.

Constraints:

if $indw < 0$ , $cucv \geq m$ ;
if $indw > 0$ , $cucv \geq \sqrt{m}$ .

12: $dchi$ – double scalar

d

, the constant of the

χ

function. dchi is not referenced if

ipsi = 0

, or if

isigma \leq 0

Constraint: if

ipsi \neq 0

and

isigma > 0

dchi > 0.0

13: $theta (m)$ – double array

Starting values of the argument vector

θ

. These may be obtained from least squares regression. Alternatively if

isigma < 0

and

sigma = 1

or if

isigma > 0

and sigma approximately equals the standard deviation of the dependent variable,

y

, then

theta (i) = 0.0

, for

i = 1, 2, \dots, m

may provide reasonable starting values.

14: $sigma$ – double scalar

A starting value for the estimation of

σ

. sigma should be approximately the standard deviation of the residuals from the model evaluated at the value of

θ

given by theta on entry.

Constraint:

sigma > 0.0

Optional Input Parameters

1: $n$ – int64int32nag_int scalar

Default: the dimension of the array y and the first dimension of the array x. (An error is raised if these dimensions are not equal.)

n

, the number of observations.

Constraint:

n > 1

2: $m$ – int64int32nag_int scalar

Default: the dimension of the array theta and the second dimension of the array x. (An error is raised if these dimensions are not equal.)

m

, the number of independent variables.

Constraint:

1 \leq m < n

3: $tol$ – double scalar

Default:

5e-5

The relative precision for the calculation of

A

(if

indw \neq 0

), the estimates of

θ

and the estimate of

σ

(if

isigma \neq 0

). Convergence is assumed when the relative change in all elements being considered is less than tol.

indw < 0

and

isigma < 0

, tol is also used to determine the precision of

β_{1}

It is advisable for tol to be greater than

100 \times machine precision

Constraint:

tol > 0.0

4: $maxit$ – int64int32nag_int scalar

Default:

50

The maximum number of iterations that should be used in the calculation of

A

(if

indw \neq 0

), and of the estimates of

θ

and

σ

, and of

β_{1}

(if

indw < 0

and

isigma < 0

A value of

maxit = 50

should be adequate for most uses.

Constraint:

maxit > 0

5: $nitmon$ – int64int32nag_int scalar

Default:

0

The amount of information that is printed on each iteration.

$nitmon = 0$: No information is printed.
$nitmon \neq 0$: The current estimate of $θ$ , the change in $θ$ during the current iteration and the current value of $σ$ are printed on the first and every $abs (nitmon)$ iterations.

Also, if

indw \neq 0

and

nitmon > 0

then information on the iterations to calculate

A

is printed. This is the current estimate of

A

and the maximum value of

S_{i j}

(see Description).

When printing occurs the output is directed to the current advisory message unit (see nag_file_set_unit_advisory (x04ab)).

Output Parameters

1: $x (ldx, m)$ – double array

Unchanged, except as described above.

2: $y (n)$ – double array

Unchanged, except as described above.

3: $theta (m)$ – double array

theta (i)

contains the M-estimate of

θ_{i}

, for

i = 1, 2, \dots, m

4: $sigma$ – double scalar

Contains the final estimate of

σ

isigma \neq 0

or the value assigned on entry if

isigma = 0

5: $c (ldc, m)$ – double array

The diagonal elements of c contain the estimated asymptotic standard errors of the estimates of

θ

, i.e.,

c (i, i)

contains the estimated asymptotic standard error of the estimate contained in

theta (i)

The elements above the diagonal contain the estimated asymptotic correlation between the estimates of

θ

, i.e.,

c (i, j)

1 \leq i < j \leq m

contains the asymptotic correlation between the estimates contained in

theta (i)

and

theta (j)

The elements below the diagonal contain the estimated asymptotic covariance between the estimates of

θ

, i.e.,

c (i, j)

1 \leq j < i \leq m

contains the estimated asymptotic covariance between the estimates contained in

theta (i)

and

theta (j)

6: $rs (n)$ – double array

The residuals from the model evaluated at final value of theta, i.e., rs contains the vector

(y - X \hat{θ})

7: $wgt (n)$ – double array

The vector of weights.

wgt (i)

contains the weight for the

i

th observation, for

i = 1, 2, \dots, n

8: $work (4 \times n + m \times (n + m))$ – double array

The following values are assigned to work:

$work (1) = β_{1}$ if $isigma < 0$ , or $work (1) = β_{2}$ if $isigma > 0$ .
$work (2) =$ number of iterations used to calculate $A$ .
$work (3) =$ number of iterations used to calculate final estimates of $θ$ and $σ$ .
$work (4) = k$ , the rank of the weighted least squares equations.

The rest of the array is used as workspace.

9: $ifail$ – int64int32nag_int scalar

ifail = 0

unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Note: nag_correg_robustm (g02ha) may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$ifail = 1$

On entry,	$n \leq 1$ ,
or	$m < 1$ ,
or	$n \leq m$ ,
or	$ldx < n$ ,
or	$ldc < m$ .

$ifail = 2$

On entry,	$ipsi < 0$ ,
or	$ipsi > 4$ .

$ifail = 3$

On entry,	$sigma \leq 0.0$ ,
or	$ipsi = 1$ and $cpsi \leq 0.0$ ,
or	$ipsi = 2$ and $h1 < 0.0$ ,
or	$ipsi = 2$ and $h1 > h2$ ,
or	$ipsi = 2$ and $h2 > h3$ ,
or	$ipsi = 2$ and $h1 = h2 = h3 = 0.0$ ,
or	$ipsi \neq 0$ and $isigma > 0$ and $dchi \leq 0.0$ ,
or	$indw > 0$ and $cucv < \sqrt{m}$ ,
or	$indw < 0$ and $cucv < m$ .

$ifail = 4$

On entry,	$tol \leq 0.0$ ,
or	$maxit \leq 0$ .

$ifail = 5$: The number of iterations required to calculate the weights exceeds maxit. (Only if $indw \neq 0$ .)

$ifail = 6$: The number of iterations required to calculate $β_{1}$ exceeds maxit. (Only if $indw < 0$ and $isigma < 0$ .)

$ifail = 7$: Either the number of iterations required to calculate $θ$ and $σ$ exceeds maxit (note that, in this case $work (3) = maxit$ on exit), or the iterations to solve the weighted least squares equations failed to converge. The latter is an unlikely error exit.

W $ifail = 8$: The weighted least squares equations are not of full rank.

W $ifail = 9$: If $indw = 0$ then $(X^{T} X)$ is almost singular.

If $indw \neq 0$ then $S_{1}$ is singular or almost singular. This may be due to too many diagonal elements of the matrix being zero, see Further Comments.

W $ifail = 10$

In calculating the correlation factor for the asymptotic variance-covariance matrix either the value of

\frac{1}{n} \sum_{i = 1}^{n} ψ^{'} (r_{i} / \hat{σ}) = 0,   or κ = 0,   or \sum_{i = 1}^{n} ψ^{2} (r_{i} / \hat{σ}) = 0 .

See Further Comments. In this case c is returned as

X^{T} X

(Only if

indw = 0

W $ifail = 11$: The estimated variance for an element of $θ \leq 0$ .

In this case the diagonal element of c will contain the negative variance and the above diagonal elements in the row and column corresponding to the element will be returned as zero.

This error may be caused by rounding errors or too many of the diagonal elements of $P$ being zero, where $P$ is defined in Description. See Further Comments.

$ifail = 12$: The degrees of freedom for error, $n - k \leq 0$ (this is an unlikely error exit), or the estimated value of $σ$ was $0$ during an iteration.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The precision of the estimates is determined by tol. As a more stable method is used to calculate the estimates of

θ

than is used to calculate the covariance matrix, it is possible for the least squares equations to be of full rank but the

(X^{T} X)

matrix to be too nearly singular to be inverted.

Further Comments

In cases when

isigma \geq 0

it is important for the value of sigma to be of a reasonable magnitude. Too small a value may cause too many of the winsorized residuals, i.e.,

ψ (r_{i} / σ)

, to be zero or a value of

ψ^{'} (r_{i} / σ)

, used to estimate the asymptotic covariance matrix, to be zero. This can lead to errors

ifail = 8

9

(if

indw \neq 0

ifail = 10

(if

indw = 0

) and

ifail = 11

nag_correg_robustm_wts (g02hb), nag_correg_robustm_user (g02hd) and nag_correg_robustm_user_varmat (g02hf) together carry out the same calculations as nag_correg_robustm (g02ha) but for user-supplied functions for

ψ

χ

ψ^{'}

and

u

Example

The number of observations and the number of

x

variables are read in followed by the data. The option arguments are then read in (in this case giving Schweppe type regression with Hampel's

ψ

function and Huber's

χ

function and then using the ‘replace expected by observed’ option in calculating the covariances). Finally a set of values for the constants are read in.

After a call to nag_correg_robustm (g02ha),

\hat{θ}

, its standard error and

\hat{σ}

are printed. In addition the weight and residual for each observation is printed.

Open in the MATLAB editor: g02ha_example

function g02ha_example


fprintf('g02ha example results\n\n');

x = [1, -1, -1;
     1, -1,  1;
     1,  1, -1;
     1,  1,  1;
     1, -2,  0;
     1,  0, -2;
     1,  2,  0;
     1,  0,  2];
y = [2.1;  3.6;  4.5;  6.1;  1.3;  1.9;   6.7;  5.5];

[n,m] = size(x);

% Control parameters
indw   = int64(1);
ipsi   = int64(2);
isigma = int64(1);
indc   = int64(0);

% Weight function parameters
cpsi = 0;
h1   = 1.5;  h2 = 3;  h3 = 4.5;
cucv = 3;
dchi = 1.5;

% Initial values
sigma = 1;
theta = zeros(m,1);

% Perform M-estimate regression
[x, y, theta, sigma, c, rs, wgt, work, ifail] = ...
  g02ha( ...
         indw, ipsi, isigma, indc, x, y, cpsi, h1, h2, h3, ...
         cucv, dchi, theta, sigma);

% Display results
fprintf('Sigma = %10.4f\n', sigma);
fprintf('\n       Theta      Standard\n');
fprintf('                   errors\n');
for j = 1:m
  fprintf('%12.4f%13.4f\n',theta(j),c(j,j));
end
fprintf('\n     Weights     Residuals\n');
fprintf('%12.4f%13.4f\n',[wgt rs]');

g02ha example results

Sigma =     0.2026

       Theta      Standard
                   errors
      4.0423       0.0384
      1.3083       0.0272
      0.7519       0.0311

     Weights     Residuals
      0.5783       0.1179
      0.5783       0.1141
      0.5783      -0.0987
      0.5783      -0.0026
      0.4603      -0.1256
      0.4603      -0.6385
      0.4603       0.0410
      0.4603      -0.0462

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox