g02hac (Nag_RegType regtype, Nag_PsiFun psifun, Nag_SigmaEst sigma_est, Nag_CovMatrixEst covmat_est, Integer n, Integer m, double x[], Integer tdx, double y[], double cpsi, const double hpsi[], double cucv, double dchi, double theta[], double *sigma, double c[], Integer tdc, double rs[], double wt[], double tol, Integer max_iter, Integer print_iter, const char *outfile, double info[], NagError *fail)

The function may be called by the names: g02hac, nag_correg_robustm or nag_robust_m_regsn_estim.

3 Description

For the linear regression model

y = X θ + ε

where	$y$ is a vector of length $n$ of the dependent variable,
	$X$ is a $n \times m$ 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 $var (ε_{i}) = σ^{2}$ :

g02hac 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:
	$\hat{σ} = {med}_{i} [\| r_{i} \|] / β_{1}$

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

$w_{i}^{*} \leftarrow \sqrt{w_{i}}$
$y_{i}^{*} \leftarrow y_{i} \sqrt{w_{i}}$
$x_{i j}^{*} \leftarrow x_{i j} \sqrt{w_{i}}, j = 1, 2, \dots, m$

(see 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}}) = . 75

where

Φ

is the standard Normal cumulative distribution function.

β_{2}

is given by:

\begin{array}{l} β_{2} = \int_{- \infty}^{\infty} χ (z) ϕ (z) d z, & in Huber case; \\ β_{2} = \frac{1}{n} \sum_{i = 1}^{n} w_{i} \int_{- \infty}^{\infty} χ (z) ϕ (z) d z, & in 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 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{matrix} \frac{ψ (r_{i} / (σ w_{i}))}{(r_{i} / (σ w_{i}))}, & r_{i} \neq 0 \\ ψ^{'} (0), & r_{i} = 0 \end{matrix}

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

${\tilde{y}}_{i} = y_{i} \sqrt{G_{i i}}$
${\tilde{x}}_{i j} = x_{i j} \sqrt{G_{i i}}, j = 1, 2, \dots, m$

and then obtaining the solution of the resulting least squares problem. If

X

is of full column rank then an orthogonal-triangular (QR) decomposition is used, if not, a singular value decomposition is used.

The following functions are available for

ψ

and

χ

in g02hac.

(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{matrix} \frac{t^{2}}{2}, & | t | \leq d \\ \frac{d^{2}}{2}, & | t | > d \end{matrix}

(c)Hampel's Piecewise Linear Function

ψ_{h_{1}, h_{2}, h_{3}} (t) = - ψ_{h_{1}, h_{2}, h_{3}} (- t) = {\begin{matrix} 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{matrix} .

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

(d)Andrew's Sine Wave Function

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

(e)Tukey's Bi-weight

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

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 a $m \times m$ 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

g02hac finds

A

using the iterative procedure

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

where

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

s_{j l} = {\begin{matrix} - \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{matrix} .

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 g02hac:

(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
$u (t) = \begin{matrix} c / t^{2} & | t | > c \\ 1 & | t | \leq c \end{matrix}$

$f (t) = \sqrt{u (t)} .$
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 g02hac:

1.Average over the

r_{i}

\begin{array}{l} 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} \end{array}

2.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.

g02hac is based on routines in ROBETH, see Marazzi (1987a).

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 (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

5 Arguments

1: $regtype$ – Nag_RegType Input

On entry: specifies the type of regression to be performed.

$regtype = Nag_HuberReg$: Huber type regression.
$regtype = Nag_MallowsReg$: Mallows type regression with Maronna's proposed weights.
$regtype = Nag_SchweppeReg$: Schweppe type regression with Krasker–Welsch weights.

Constraint:

regtype = Nag_HuberReg

Nag_MallowsReg

Nag_SchweppeReg

2: $psifun$ – Nag_PsiFun Input

On entry: specifies which

ψ

function is to be used.

$psifun = Nag_Lsq$: $ψ (t) = t$ , i.e., least squares.
$psifun = Nag_HuberFun$: Huber's function.
$psifun = Nag_HampelFun$: Hampel's piecewise linear function.
$psifun = Nag_AndrewFun$: Andrew's sine wave.
$psifun = Nag_TukeyFun$: Tukey's bi-weight.

Constraint:

psifun = Nag_Lsq

Nag_HuberFun

Nag_HampelFun

Nag_AndrewFun

Nag_TukeyFun

3: $sigma_est$ – Nag_SigmaEst Input

On entry: specifies how

σ

is to be estimated.

$sigma_est = Nag_SigmaRes$: $σ$ is estimated by median absolute deviation of residuals.
$sigma_est = Nag_SigmaConst$: $σ$ is held constant at its initial value.
$sigma_est = Nag_SigmaChi$: $σ$ is estimated using the $χ$ function.

Constraint:

sigma_est = Nag_SigmaRes

Nag_SigmaConst

Nag_SigmaChi

4: $covmat_est$ – Nag_CovMatrixEst Input

On entry: if

regtype \neq Nag_HuberReg

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

\hat{θ}

covmat_est = Nag_CovMatAve

, averaging over residuals.

covmat_est = Nag_CovMatObs

, replacing expected by observed.

regtype = Nag_HuberReg

then covmat_est is not referenced.

Constraint:

covmat_est = Nag_CovMatAve

Nag_CovMatObs

5: $n$ – Integer Input

On entry: the number of observations,

n

Constraint:

n > 1

6: $m$ – Integer Input

On entry: the number

m

, of independent variables.

Constraint:

1 \leq m < n

7: $x [n \times tdx]$ – double Input/Output

Note: the

(i, j)

th element of the matrix

X

is stored in

x [(i - 1) \times tdx + j - 1]

On entry: the values of the

X

matrix, i.e., the independent variables.

x [i - 1] [j - 1]

must contain the

i j

th element of

X

, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, m

On exit: if

regtype = Nag_MallowsReg

, then during calculations the elements of x will be transformed as described in Section 3. Before exit the inverse transformation will be applied. As a result there may be slight differences between the input x and the output x. Otherwise x is unchanged.

8: $tdx$ – Integer Input

On entry: the stride separating matrix column elements in the array x.

Constraint:

tdx \geq m

9: $y [n]$ – double Input/Output

On entry: the data values of the dependent variable.

y [i - 1]

must contain the value of

y

for the

i

th observation, for

i = 1, 2, \dots, n

On exit: if

regtype = Nag_MallowsReg

, then during calculations the elements of y will be transformed as described in Section 3. Before exit the inverse transformation will be applied. As a result there may be slight differences between the input y and the output y. Otherwise y is unchanged.

10: $cpsi$ – double Input

On entry: if

psifun = Nag_HuberFun

, cpsi must specify the argument,

c

, of Huber's

ψ

function. Otherwise cpsi is not referenced.

Constraint: if

psifun = Nag_HuberFun

then

cpsi > 0.0

11: $hpsi [3]$ – const double Input

On entry: if

psifun = Nag_HampelFun

then

hpsi [0]

hpsi [1]

and

hpsi [2]

must specify the arguments

h_{1}

h_{2}

, and

h_{3}

, of Hampel's piecewise linear

ψ

function. Otherwise the elements of hpsi are not referenced.

Constraint: if

psifun = Nag_HampelFun

0.0 \leq hpsi [0] \leq hpsi [1] \leq hpsi [2]

and

hpsi [2] > 0.0

12: $cucv$ – double Input

On entry: if

regtype = Nag_MallowsReg

then cucv must specify the value of the constant,

c

, of the function

u

for Maronna's proposed weights.

regtype = Nag_SchweppeReg

then cucv must specify the value of the function

u

for the Krasker–Welsch weights.

regtype = Nag_HuberReg

then cucv is not referenced.

Constraints:

if $regtype = Nag_MallowsReg$ , $cucv \geq m$ ;
if $regtype = Nag_SchweppeReg$ , $cucv \geq \sqrt{m}$ .

13: $dchi$ – double Input

On entry: the constant,

d

, of the

χ

function.

dchi is referenced only if

psifun \neq Nag_Lsq

and

sigma_est = Nag_SigmaChi

Constraint: if

psifun \neq Nag_Lsq

and

sigma_est = Nag_SigmaChi

dchi > 0.0

14: $theta [m]$ – double Input/Output

On entry: starting values of the argument vector

θ

. These may be obtained from least squares regression.

Alternatively if

sigma_est = Nag_SigmaRes

and

sigma = 1

or if

sigma_est = Nag_SigmaChi

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

y

, then

theta [i - 1] = 0.0

, for

i = 1, 2, \dots, m

may provide reasonable starting values.

On exit:

theta [i - 1]

contains the M-estimate of

θ_{i}

, for

i = 1, 2, \dots, m

15: $sigma$ – double * Input/Output

On entry: 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.

On exit: sigma contains the final estimate of

σ

, unless

sigma_est = Nag_SigmaConst

Constraint:

sigma > 0.0

16: $c [m \times tdc]$ – double Output

On exit: the diagonal elements of c contain the estimated asymptotic standard errors of the estimates of

θ

, i.e.,

c [(i - 1) \times tdc + i - 1]

contains the estimated asymptotic standard error of the estimate contained in

theta [i - 1]

, for

i = 1, 2, \dots, m

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

θ

, i.e.,

c [(i - 1) \times tdc + j - 1]

1 \leq i < j \leq m

contains the asymptotic correlation between the estimates contained in

theta [i - 1]

and

theta [j - 1]

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

θ

, i.e.,

c [(i - 1) \times tdc + j - 1]

1 \leq j < i \leq m

contains the estimated asymptotic covariance between the estimates contained in

theta [i - 1]

and

theta [j - 1]

17: $tdc$ – Integer Input

On entry: the stride separating matrix column elements in the array c.

Constraint:

tdc \geq m

18: $rs [n]$ – double Output

On exit: contains the residuals from the model evaluated at final value of theta, i.e.,

rs [i - 1]

, for

i = 1, 2, \dots, n

, contains the vector

(y - X \hat{θ})

19: $wt [n]$ – double Output

On exit: contains the vector of weights.

wt [i - 1]

contains the weight for the

i

th observation, for

i = 1, 2, \dots, n

20: $tol$ – double Input

On entry: the relative precision for the calculation of

A

(if

regtype \neq Nag_HuberReg

), the estimates of

θ

and the estimate of

σ

(if

sigma_est \neq Nag_SigmaConst

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

regtype = Nag_MallowsReg

and

sigma_est = Nag_SigmaRes

, 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

21: $max_iter$ – Integer Input

On entry: the maximum number of iterations that should be used in the calculation of

A

(if

regtype \neq Nag_HuberReg

), and of the estimates of

θ

and

σ

, and of

β_{1}

(if

regtype = Nag_MallowsReg

and

sigma_est = Nag_SigmaRes

)

Suggested value: A value of

max_iter = 50

should be adequate for most uses.

Constraint:

max_iter > 0

22: $print_iter$ – Integer Input

On entry: the amount of information that is printed on each iteration.

$print_iter = 0$: No information is printed.
$print_iter \neq 0$: 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 $abs (print_iter)$ iterations.

Also, if

regtype \neq Nag_HuberReg

and

print_iter > 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 Section 3).

23: $outfile$ – const char * Input

On entry: a null terminated character string giving the name of the file to which results should be printed. If outfile is NULL or an empty string then the stdout stream is used. Note that the file will be opened in the append mode.

24: $info [4]$ – double Output

On exit: elements of info contain the following values:

$info [0] = β_{1}$ if $sigma_est = Nag_SigmaRes$ ,
or $info [0] = β_{2}$ if $sigma_est = Nag_SigmaChi$ ,
$info [1] =$ number of iterations used to calculate $A$ .
$info [2] =$ number of iterations used to calculate final estimates of $θ$ and $σ$ .
$info [3] = k$ , the rank of the weighted least squares equations.

25: $fail$ – 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_2_INT_ARG_GE

On entry,

m = ⟨ value ⟩

while

n = ⟨ value ⟩

. These arguments must satisfy

m < n

NE_2_INT_ARG_LT

On entry,

tdc = ⟨ value ⟩

while

m = ⟨ value ⟩

. These arguments must satisfy

tdc \geq m

On entry,

tdx = ⟨ value ⟩

while

m = ⟨ value ⟩

. These arguments must satisfy

tdx \geq m

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_BAD_HAMPEL_PSI_FUN

On entry,

psifun = Nag_HampelFun

and

hpsi [0] = ⟨ value ⟩

hpsi [1] = ⟨ value ⟩

and

hpsi [2] = ⟨ value ⟩

. For this value of psifun, the elements of hpsi must satisfy the condition

0.0 \leq hpsi [0] \leq hpsi [1] \leq hpsi [2]

and

hpsi [2] > 0.0

NE_BAD_PARAM

On entry, argument covmat_est had an illegal value.

On entry, argument psifun had an illegal value.

On entry, argument regtype had an illegal value.

On entry, argument sigma_est had an illegal value.

NE_BETA1_ITER_EXCEEDED

The number of iterations required to calculate

β_{1}

exceeds max_iter. This is only applicable if

regtype = Nag_MallowsReg

and

sigma_est = Nag_SigmaRes

NE_COV_MAT_FACTOR_ZERO

In calculating the correlation factor for the asymptotic variance-covariance matrix, the factor for covariance matrix

= 0

.
For this error, 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 Section 9. In this case c is returned as ${(X^{T} X)}^{- 1}$ . (This is only applicable if $regtype = Nag_HuberReg$ ).

NE_ERR_DOF_LEQ_ZERO

n = ⟨ value ⟩

, rank of

x = ⟨ value ⟩

. The degrees of freedom for error,

n -

(rank of x) must be

> 0.0

NE_ESTIM_SIGMA_ZERO

The estimated value of

σ

was

0.0

during an iteration.

NE_INT_ARG_LE

On entry, max_iter must not be less than or equal to 0:

max_iter = ⟨ value ⟩

NE_INT_ARG_LT

On entry,

m = ⟨ value ⟩

.
Constraint:

m \geq 1

On entry,

n = ⟨ value ⟩

.
Constraint:

n \geq 2

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.

NE_INVALID_DCHI_FUN

On entry,

psifun \neq Nag_Lsq

sigma_est = Nag_SigmaChi

and

dchi = ⟨ value ⟩

. For these values of psifun and sigma_est, dchi must be

> 0.0

NE_INVALID_HUBER_FUN

On entry,

psifun = Nag_HuberFun

and

cpsi = ⟨ value ⟩

. For this value of psifun, cpsi must be

> 0.0

NE_INVALID_MALLOWS_REG_C

On entry,

regtype = Nag_MallowsReg

cucv = ⟨ value ⟩

and

m = ⟨ value ⟩

. For this value of regtype, cucv must be

\geq m

NE_INVALID_SCHWEPPE_REG_C

On entry,

regtype = Nag_SchweppeReg

cucv = ⟨ value ⟩

and

m = ⟨ value ⟩

. For this value of regtype, cucv must be

\geq \sqrt{m}

NE_LSQ_FAIL_CONV

The iterations to solve the weighted least squares equations failed to converge.

NE_NOT_APPEND_FILE

Cannot open file

⟨ string ⟩

for appending.

NE_NOT_CLOSE_FILE

Cannot close file

⟨ string ⟩

NE_REAL_ARG_LE

On entry, sigma must not be less than or equal to 0.0:

sigma = ⟨ value ⟩

On entry, tol must not be less than or equal to 0.0:

tol = ⟨ value ⟩

NE_REG_MAT_SINGULAR

Failure to invert matrix while calculating covariance.
If

regtype = Nag_HuberReg

, then

(X^{T} X)

is almost singular.
If

regtype \neq Nag_HuberReg

, then

S_{1}

is singular or almost singular. This may be due to too many diagonal elements of the matrix being zero, see Section 9.

NE_THETA_ITER_EXCEEDED

The number of iterations required to calculate

θ

and

σ

exceeds max_iter. In this case,

info [2] = max_iter

on exit.

NE_VAR_THETA_LEQ_ZERO

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 the 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. See Section 9.

NE_WT_ITER_EXCEEDED

The number of iterations required to calculate the weights exceeds max_iter. This is only applicable if

regtype \neq Nag_HuberReg

NE_WT_LSQ_NOT_FULL_RANK

The weighted least squares equations are not of full rank.

7 Accuracy

The precision of the estimates is determined by tol, see Section 5. 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.

8 Parallelism and Performance

g02hac is not threaded in any implementation.

9 Further Comments

In cases when

sigma_est \neq Nag_SigmaRes

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 with fail set to one of the following values:

NE_WT_LSQ_NOT_FULL_RANK,
NE_REG_MAT_SINGULAR (if $regtype \neq Nag_HuberReg$ ),
NE_COV_MAT_FACTOR_ZERO (if $regtype = Nag_HuberReg$ ),
NE_VAR_THETA_LEQ_ZERO.

10 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 g02hac,

\hat{θ}

, its standard error and

\hat{σ}

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

g02ha: FL CL CPP AD

NAG CL Interfaceg02hac (robustm)

▸▿ 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

NAG CL Interface
g02hac (robustm)