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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_correg_robustm (g02ha)

## Purpose

nag_correg_robustm (g02ha) performs bounded influence regression (M$M$-estimates). Several standard methods are available.

## Syntax

[x, y, theta, sigma, c, rs, wgt, work, ifail] = g02ha(indw, ipsi, isigma, indc, x, y, cpsi, h1, h2, h3, cucv, dchi, theta, sigma, 'n', n, 'm', m, 'tol', tol, 'maxit', maxit, 'nitmon', nitmon)
[x, y, theta, sigma, c, rs, wgt, work, ifail] = nag_correg_robustm(indw, ipsi, isigma, indc, x, y, cpsi, h1, h2, h3, cucv, dchi, theta, sigma, 'n', n, 'm', m, 'tol', tol, 'maxit', maxit, 'nitmon', nitmon)
Note: the interface to this routine has changed since earlier releases of the toolbox:
Mark 23: nitmon, tol, maxit now optional
.

## Description

For the linear regression model
 y = Xθ + ε , $y = Xθ+ε ,$
 where y$y$ is a vector of length n$n$ of the dependent variable, X$X$ is a n$n$ by m$m$ matrix of independent variables of column rank k$k$, θ$\theta$ is a vector of length m$m$ of unknown parameters, and ε$\epsilon$ is a vector of length n$n$ of unknown errors with var(εi) = σ2$\mathrm{var}\left({\epsilon }_{i}\right)={\sigma }^{2}$,
nag_correg_robustm (g02ha) calculates the M-estimates given by the solution, θ̂$\stackrel{^}{\theta }$, to the equation
 n ∑ ψ(ri / (σwi))wixij = 0,  j = 1,2, … ,m, i = 1
$∑ i=1 n ψ ( ri / (σwi) ) wi xij = 0 , j=1,2,…,m ,$
(1)
where ri${r}_{i}$ is the i$i$th residual, i.e., the i$i$th element of r = yXθ̂$r=y-X\stackrel{^}{\theta }$,
ψ$\psi$ is a suitable weight function,
wi${w}_{i}$ are suitable weights,
and σ$\sigma$ may be estimated at each iteration by the median absolute deviation of the residuals
 σ̂ = medi  [|ri|] / β1 $σ^ = medi⁡ [|ri|] / β1$
or as the solution to
 n ∑ χ(ri / (σ̂wi))wi2 = (n − k)β2 i = 1
$∑ i= 1 n χ ( ri / ( σ ^ wi ) ) wi2 = (n-k) β2$
for suitable weight function χ$\chi$, where β1${\beta }_{1}$ and β2${\beta }_{2}$ are constants, chosen so that the estimator of σ$\sigma$ is asymptotically unbiased if the errors, εi${\epsilon }_{i}$, have a Normal distribution. Alternatively σ$\sigma$ may be held at a constant value.
The above describes the Schweppe type regression. If the wi${w}_{i}$ are assumed to equal 1$1$ for all i$i$ then Huber type regression is obtained. A third type, due to Mallows, replaces (1) by
 n ∑ ψ(ri / σ)wixij = 0,  j = 1,2, … ,m. i = 1
$∑ i=1 n ψ ( ri / σ ) wi xij = 0 , j=1,2,…,m .$
This may be obtained by use of the transformations
 wi * ← sqrt(wi) yi * ← yi sqrt(wi) xij * ← xij sqrt(wi), j = 1,2, … ,m
$wi*←wi yi*←yi wi xij*←xij wi, j= 1,2,…,m$
(see Section 3 of Marazzi (1987a)).
For Huber and Schweppe type regressions, β1${\beta }_{1}$ is the 75th percentile of the standard Normal distribution. For Mallows type regression β1${\beta }_{1}$ is the solution to
 n 1/n ∑ Φ(β1 / sqrt(wi)) = 0.75, i = 1
$1n ∑ i=1 n Φ ( β1 / wi ) = 0.75 ,$
where Φ$\Phi$ is the standard Normal cumulative distribution function (see nag_specfun_cdf_normal (s15ab)).
β2${\beta }_{2}$ is given by
 ∞ β2 = ∫ χ(z)φ(z)dz − ∞
in the Huber case;
 n ∞ β2 = 1/n ∑ wi ∫ χ(z)φ(z)dz i = 1 − ∞
in the Mallows case;
 n ∞ β2 = 1/n ∑ wi2 ∫ χ(z / wi)φ(z)dz i = 1 − ∞
in the Schweppe case;
$β2 = ∫ -∞ ∞ χ(z) ϕ(z) dz in the Huber case; β2 = 1 n ∑ i=1 n wi ∫ -∞ ∞ χ(z) ϕ(z) dz in the Mallows case; β2 = 1n ∑ i=1 n wi2 ∫ -∞ ∞ χ (z/wi) ϕ(z) dz in the Schweppe case;$
where φ$\varphi$ is the standard Normal density, i.e., 1/(sqrt(2π))exp((1/2)x2) .$\frac{1}{\sqrt{2\pi }}\mathrm{exp}\left(-\frac{1}{2}{x}^{2}\right)\text{.}$
The calculation of the estimates of θ$\theta$ can be formulated as an iterative weighted least squares problem with a diagonal weight matrix G$G$ given by
Gii =
 { ( ψ (ri / (σwi)) )/( (ri / (σwi)) ) , ri ≠ 0 ψ′ (0) , ri = 0 ,
$Gii = { ψ ( ri / (σwi) ) ( ri / (σwi) ) , ri≠0 ψ′ (0) , ri=0 ,$
where ψ(t)${\psi }^{\prime }\left(t\right)$ is the derivative of ψ$\psi$ at the point t$t$.
The value of θ$\theta$ at each iteration is given by the weighted least squares regression of y$y$ on X$X$. This is carried out by first transforming the y$y$ and X$X$ by
 ỹi = yisqrt(Gii) x̃ij = xijsqrt(Gii), j = 1,2, … ,m
$y~i=yiGii x~ij=xijGii, j=1,2,…,m$
and then using nag_linsys_real_gen_solve (f04jg). If X$X$ is of full column rank then an orthogonal-triangular (QR$QR$) decomposition is used; if not, a singular value decomposition is used.
The following functions are available for ψ$\psi$ and χ$\chi$ in nag_correg_robustm (g02ha).
(a) Unit Weights
 ψ(t) = t ,   χ(t) = (t2)/2 . $ψ(t) = t , χ(t) = t22 .$
This gives least squares regression.
(b) Huber's Function
ψ(t) = max ( − c,min (c,t)) ,   χ(t) =
 { (t2)/2 , |t| ≤ d (d2)/2 , |t| > d
$ψ(t) = max(-c,min(c,t)) , χ(t) = { t2 2 , |t|≤d d2 2 , |t|>d$
(c) Hampel's Piecewise Linear Function
ψ h_1 , h_2 , h_3 (t) = − ψ h_1 , h_2 , h_3 ( − t) =
 { t, 0 ≤ t ≤ h1 h1, h1 ≤ t ≤ h2 h1 (h3 − t) / (h3 − h2) , h2 ≤ t ≤ h3 0, h3 < t
$ψ h1 , h2 , h3 (t) = - ψ h1 , h2 , h3 (-t) = { t, 0≤t≤ h1 h1, h1 ≤t≤ h2 h1 ( h3 - t ) / ( h3 - h2 ) , h2 ≤t≤ h3 0, h3
χ(t) =
 { (t2)/2 , |t| ≤ d (d2)/2 , |t| > d
$χ(t) = { t2 2 , |t|≤d d2 2 , |t|>d$
(d) Andrew's Sine Wave Function
ψ(t) =
 { sint, − π ≤ t ≤ π 0, |t| > π
χ(t) =
 { (t2)/2 , |t| ≤ d (d2)/2 , |t| > d
$ψ(t) = { sin⁡t, -π≤t≤π 0, |t|>π χ(t) = { t2 2 , |t|≤d d2 2 , |t|>d$
(e) Tukey's Bi-weight
ψ(t) =
 { t (1 − t2)2 , |t| ≤ 1 0, |t| > 1
χ(t) =
 { (t2)/2 , |t| ≤ d (d2)/2 , |t| > d
$ψ(t) = { t ( 1 - t2 ) 2 , |t|≤ 1 0, |t|> 1 χ(t) = { t2 2 , |t|≤d d2 2 , |t|>d$
where c$c$, h1${h}_{1}$, h2${h}_{2}$, h3${h}_{3}$, and d$d$ 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$A$ has to be found such that:
 n 1/n ∑ u(‖zi‖2)ziziT = I i = 1
$1n ∑ i=1 n u (‖zi‖2) zi ziT = I$
and
 zi = Axi , $zi = Axi ,$
 where xi${x}_{i}$ is a vector of length m$m$ containing the i$i$th row of X$X$, A$A$ is an m$m$ by m$m$ lower triangular matrix, and u$u$ is a suitable function.
The weights are then calculated as
 wi = f (‖zi‖2) $wi = f (‖zi‖2)$
for a suitable function f$f$.
nag_correg_robustm (g02ha) finds A$A$ using the iterative procedure
 Ak = (Sk + I) Ak − 1 , $Ak = (Sk+I) Ak-1 ,$
where Sk = (sjl) ${S}_{k}=\left({s}_{jl}\right)$,
sjl =
 { − min [max ( hjl / n , − BL),BL] , j > l − min [max ( (1/2) (hjj / n − 1) , − BD),BD] , j = l
$sjl = { - min[max( h jl / n ,-BL),BL] , j>l - min[max( 12 (hjj/n-1) ,-BD),BD] , j=l$
and
 n hjl = ∑ u(‖zi‖2)zijzil i = 1
$hjl = ∑ i= 1 n u (‖zi‖2) zij zil$
and BL$BL$ and BD$BD$ are bounds set at 0.9$0.9$.
Two weights are available in nag_correg_robustm (g02ha):
 u (t) = g1 (c/t) , $u (t) = g1 (ct) ,$
 where g1(t) = t2 + (1 − t2)(2Φ(t) − 1) − 2tφ(t)${g}_{1}\left(t\right)={t}^{2}+\left(1-{t}^{2}\right)\left(2\Phi \left(t\right)-1\right)-2t\varphi \left(t\right)$, Φ(t)$\Phi \left(t\right)$ is the standard Normal cumulative distribution function, φ(t)$\varphi \left(t\right)$ is the standard Normal probability density function, and f(t) = 1/t $f\left(t\right)=\frac{1}{t}$.
These are for use with Schweppe type regression.
(ii) Maronna's Proposed Weights
f(t) = sqrt(u(t)) .
$u(t) = { c t2 |t|>c 1 |t|≤c f(t)=u(t) .$
 u(t) = { c/(t2) |t| > c 1 |t| ≤ c
These are for use with Mallows type regression.
Finally the asymptotic variance-covariance matrix, C$C$, of the estimates θ$\theta$ is calculated.
For Huber type regression
 C = fH (XTX) − 1 σ̂2 , $C = fH (XTX) -1 σ^2 ,$
where
fH = 1/(n − m) ( ∑ i = 1n ψ2 (ri / σ̂) )/(
 ( n )1/n ∑ ψ′((ri)/(σ̂)) i = 1 2
) κ2
$fH = 1 n-m ∑ i= 1 n ψ2 ( ri / σ ^ ) ( 1n ∑ i= 1 n ψ′ ( ri σ^ ) ) 2 κ2$
κ2 = 1 + m/n (1/n ∑ i = 1n
 ( n )ψ′(ri / σ̂) − 1/n ∑ ψ′(ri / σ̂) i = 1 2
)/(
 ( n )1/n ∑ ψ′((ri)/(σ̂)) i = 1 2
) .
$κ2 = 1 + mn 1n ∑ i=1 n ( ψ′ ( ri / σ^ ) - 1n ∑ i=1 n ψ′ ( ri / σ^ ) ) 2 ( 1n ∑ i=1 n ψ′ ( ri σ^ ) ) 2 .$
See Huber (1981) and Marazzi (1987b).
For Mallows and Schweppe type regressions C$C$ is of the form
 (σ̂)/n2 S1 − 1 S2 S1 − 1 , $σ^n 2 S1-1 S2 S1-1 ,$
where S1 = 1/nXTDX${S}_{1}=\frac{1}{n}{X}^{\mathrm{T}}DX$ and S2 = 1/nXTPX${S}_{2}=\frac{1}{n}{X}^{\mathrm{T}}PX$.
D$D$ is a diagonal matrix such that the i$i$th element approximates E(ψ(ri / (σwi)))$E\left({\psi }^{\prime }\left({r}_{i}/\left(\sigma {w}_{i}\right)\right)\right)$ in the Schweppe case and E(ψ(ri / σ)wi)$E\left({\psi }^{\prime }\left({r}_{i}/\sigma \right){w}_{i}\right)$ in the Mallows case.
P$P$ is a diagonal matrix such that the i$i$th element approximates E(ψ2(ri / (σwi))wi2)$E\left({\psi }^{2}\left({r}_{i}/\left(\sigma {w}_{i}\right)\right){w}_{i}^{2}\right)$ in the Schweppe case and E(ψ2(ri / σ)wi2)$E\left({\psi }^{2}\left({r}_{i}/\sigma \right){w}_{i}^{2}\right)$ in the Mallows case.
Two approximations are available in nag_correg_robustm (g02ha):
1. Average over the ri${r}_{i}$
$Schweppe Mallows Di = ( 1n ∑ j=1 n ψ′ ( rj σ^ wi ) ) wi Di = ( 1n ∑ j=1 n ψ′ ( rj σ^ ) ) wi Pi = ( 1n ∑ j=1 n ψ2 ( rj σ^ wi ) ) wi2 Pi = ( 1n ∑ j=1 n ψ2 ( rj σ^ ) ) wi2$
Schweppe Mallows
 Di = ( n )1/n ∑ ψ′((rj)/( σ̂ wi )) j = 1 wi
 Di = ( n )1/n ∑ ψ′((rj)/(σ̂)) j = 1 wi
 Pi = ( n )1/n ∑ ψ2((rj)/( σ̂ wi )) j = 1 wi2
 Pi = ( n )1/n ∑ ψ2((rj)/(σ̂)) j = 1 wi2
2. Replace expected value by observed
 Schweppe Mallows Di = ψ′ ((ri)/( σ̂ wi )) wi Di = ψ′ ((ri)/(σ̂)) wi Pi = ψ2 ((ri)/( σ̂ wi )) wi2 Pi = ψ2 ((ri)/(σ̂)) wi2
.
$Schweppe Mallows Di = ψ′ ( ri σ^ wi ) wi Di = ψ′ ( ri σ ^ ) wi Pi = ψ2 ( ri σ ^ wi ) wi2 Pi = ψ2 ( ri σ ^ ) wi2 .$
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$1.0$ for all observations will produce a value of θ̂$\stackrel{^}{\theta }$ 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${\mathbf{indw}}<0$
Mallows type regression with Maronna's proposed weights.
indw = 0${\mathbf{indw}}=0$
Huber type regression.
indw > 0${\mathbf{indw}}>0$
Schweppe type regression with Krasker–Welsch weights.
2:     ipsi – int64int32nag_int scalar
Specifies which ψ$\psi$ function is to be used.
ipsi = 0${\mathbf{ipsi}}=0$
ψ(t) = t$\psi \left(t\right)=t$, i.e., least squares.
ipsi = 1${\mathbf{ipsi}}=1$
Huber's function.
ipsi = 2${\mathbf{ipsi}}=2$
Hampel's piecewise linear function.
ipsi = 3${\mathbf{ipsi}}=3$
Andrew's sine wave.
ipsi = 4${\mathbf{ipsi}}=4$
Tukey's bi-weight.
Constraint: 0ipsi4$0\le {\mathbf{ipsi}}\le 4$.
3:     isigma – int64int32nag_int scalar
Specifies how σ$\sigma$ is to be estimated.
isigma < 0${\mathbf{isigma}}<0$
σ$\sigma$ is estimated by median absolute deviation of residuals.
isigma = 0${\mathbf{isigma}}=0$
σ$\sigma$ is held constant at its initial value.
isigma > 0${\mathbf{isigma}}>0$
σ$\sigma$ is estimated using the χ$\chi$ function.
4:     indc – int64int32nag_int scalar
If indw0${\mathbf{indw}}\ne 0$, indc specifies the approximations used in estimating the covariance matrix of θ̂$\stackrel{^}{\theta }$.
indc = 1${\mathbf{indc}}=1$
Averaging over residuals.
indc1${\mathbf{indc}}\ne 1$
Replacing expected by observed.
indw = 0${\mathbf{indw}}=0$
indc is not referenced.
5:     x(ldx,m) – double array
ldx, the first dimension of the array, must satisfy the constraint ldxn$\mathit{ldx}\ge {\mathbf{n}}$.
The values of the X$X$ matrix, i.e., the independent variables. x(i,j)${\mathbf{x}}\left(\mathit{i},\mathit{j}\right)$ must contain the ij$\mathit{i}\mathit{j}$th element of X$X$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$ and j = 1,2,,m$\mathit{j}=1,2,\dots ,m$.
If indw < 0${\mathbf{indw}}<0$, then during calculations the elements of x will be transformed as described in Section [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
n, the dimension of the array, must satisfy the constraint n > 1${\mathbf{n}}>1$.
The data values of the dependent variable.
y(i)${\mathbf{y}}\left(\mathit{i}\right)$ must contain the value of y$y$ for the i$\mathit{i}$th observation, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$.
If indw < 0${\mathbf{indw}}<0$, then during calculations the elements of y will be transformed as described in Section [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
If ipsi = 1${\mathbf{ipsi}}=1$, cpsi must specify the parameter, c$c$, of Huber's ψ$\psi$ function.
If ipsi1${\mathbf{ipsi}}\ne 1$ on entry, cpsi is not referenced.
Constraint: if cpsi > 0.0${\mathbf{cpsi}}>0.0$, ipsi = 1${\mathbf{ipsi}}=1$.
8:     h1 – double scalar
9:     h2 – double scalar
10:   h3 – double scalar
If ipsi = 2${\mathbf{ipsi}}=2$, h1, h2, and h3 must specify the parameters h1${h}_{1}$, h2${h}_{2}$, and h3${h}_{3}$, of Hampel's piecewise linear ψ$\psi$ function. h1, h2, and h3 are not referenced if ipsi2${\mathbf{ipsi}}\ne 2$.
Constraint: if ipsi = 2${\mathbf{ipsi}}=2$, 0.0h1h2h3$0.0\le {\mathbf{h1}}\le {\mathbf{h2}}\le {\mathbf{h3}}$ and h3 > 0.0${\mathbf{h3}}>0.0$.
11:   cucv – double scalar
If indw < 0${\mathbf{indw}}<0$, must specify the value of the constant, c$c$, of the function u$u$ for Maronna's proposed weights.
If indw > 0${\mathbf{indw}}>0$, must specify the value of the function u$u$ for the Krasker–Welsch weights.
If indw = 0${\mathbf{indw}}=0$, is not referenced.
Constraints:
• if indw < 0${\mathbf{indw}}<0$, ${\mathbf{cucv}}\ge {\mathbf{m}}$;
• if indw > 0${\mathbf{indw}}>0$, cucvsqrt(m)${\mathbf{cucv}}\ge \sqrt{{\mathbf{m}}}$.
12:   dchi – double scalar
d$d$, the constant of the χ$\chi$ function. dchi is not referenced if ipsi = 0${\mathbf{ipsi}}=0$, or if isigma0${\mathbf{isigma}}\le 0$.
Constraint: if ipsi0${\mathbf{ipsi}}\ne 0$ and isigma > 0${\mathbf{isigma}}>0$, dchi > 0.0${\mathbf{dchi}}>0.0$.
13:   theta(m) – double array
m, the dimension of the array, must satisfy the constraint 1m < n$1\le {\mathbf{m}}<{\mathbf{n}}$.
Starting values of the parameter vector θ$\theta$. These may be obtained from least squares regression. Alternatively if isigma < 0${\mathbf{isigma}}<0$ and sigma = 1${\mathbf{sigma}}=1$ or if isigma > 0${\mathbf{isigma}}>0$ and sigma approximately equals the standard deviation of the dependent variable, y$y$, then theta(i) = 0.0${\mathbf{theta}}\left(\mathit{i}\right)=0.0$, for i = 1,2,,m$\mathit{i}=1,2,\dots ,m$ may provide reasonable starting values.
14:   sigma – double scalar
A starting value for the estimation of σ$\sigma$. sigma should be approximately the standard deviation of the residuals from the model evaluated at the value of θ$\theta$ given by theta on entry.
Constraint: sigma > 0.0${\mathbf{sigma}}>0.0$.

### 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$n$, the number of observations.
Constraint: n > 1${\mathbf{n}}>1$.
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$m$, the number of independent variables.
Constraint: 1m < n$1\le {\mathbf{m}}<{\mathbf{n}}$.
3:     tol – double scalar
The relative precision for the calculation of A$A$ (if indw0${\mathbf{indw}}\ne 0$), the estimates of θ$\theta$ and the estimate of σ$\sigma$ (if isigma0${\mathbf{isigma}}\ne 0$). Convergence is assumed when the relative change in all elements being considered is less than tol.
If indw < 0${\mathbf{indw}}<0$ and isigma < 0${\mathbf{isigma}}<0$, tol is also used to determine the precision of β1${\beta }_{1}$.
It is advisable for tol to be greater than 100 × machine precision.
Default: 5e-5$5e-5$
Constraint: tol > 0.0${\mathbf{tol}}>0.0$.
4:     maxit – int64int32nag_int scalar
The maximum number of iterations that should be used in the calculation of A$A$ (if indw0${\mathbf{indw}}\ne 0$), and of the estimates of θ$\theta$ and σ$\sigma$, and of β1${\beta }_{1}$ (if indw < 0${\mathbf{indw}}<0$ and isigma < 0${\mathbf{isigma}}<0$).
A value of maxit = 50${\mathbf{maxit}}=50$ should be adequate for most uses.
Default: 50$50$
Constraint: maxit > 0${\mathbf{maxit}}>0$.
5:     nitmon – int64int32nag_int scalar
The amount of information that is printed on each iteration.
nitmon = 0${\mathbf{nitmon}}=0$
No information is printed.
nitmon0${\mathbf{nitmon}}\ne 0$
The current estimate of θ$\theta$, the change in θ$\theta$ during the current iteration and the current value of σ$\sigma$ are printed on the first and every abs(nitmon)$\mathrm{abs}\left({\mathbf{nitmon}}\right)$ iterations.
Also, if indw0${\mathbf{indw}}\ne 0$ and nitmon > 0${\mathbf{nitmon}}>0$ then information on the iterations to calculate A$A$ is printed. This is the current estimate of A$A$ and the maximum value of Sij${S}_{ij}$ (see Section [Description]).
When printing occurs the output is directed to the current advisory message unit (see nag_file_set_unit_advisory (x04ab)).
Default: 0$0$

ldx ldc

### Output Parameters

1:     x(ldx,m) – double array
ldxn$\mathit{ldx}\ge {\mathbf{n}}$.
Unchanged, except as described above.
2:     y(n) – double array
Unchanged, except as described above.
3:     theta(m) – double array
theta(i)${\mathbf{theta}}\left(\mathit{i}\right)$ contains the M-estimate of θi${\theta }_{\mathit{i}}$, for i = 1,2,,m$\mathit{i}=1,2,\dots ,m$.
4:     sigma – double scalar
Contains the final estimate of σ$\sigma$ if isigma0${\mathbf{isigma}}\ne 0$ or the value assigned on entry if isigma = 0${\mathbf{isigma}}=0$.
5:     c(ldc,m) – double array
ldcm$\mathit{ldc}\ge {\mathbf{m}}$.
The diagonal elements of c contain the estimated asymptotic standard errors of the estimates of θ$\theta$, i.e., c(i,i)${\mathbf{c}}\left(i,i\right)$ contains the estimated asymptotic standard error of the estimate contained in theta(i)${\mathbf{theta}}\left(i\right)$.
The elements above the diagonal contain the estimated asymptotic correlation between the estimates of θ$\theta$, i.e., c(i,j)${\mathbf{c}}\left(i,j\right)$, 1i < jm$1\le i contains the asymptotic correlation between the estimates contained in theta(i)${\mathbf{theta}}\left(i\right)$ and theta(j)${\mathbf{theta}}\left(j\right)$.
The elements below the diagonal contain the estimated asymptotic covariance between the estimates of θ$\theta$, i.e., c(i,j)${\mathbf{c}}\left(i,j\right)$, 1j < im$1\le j contains the estimated asymptotic covariance between the estimates contained in theta(i)${\mathbf{theta}}\left(i\right)$ and theta(j)${\mathbf{theta}}\left(j\right)$.
6:     rs(n) – double array
The residuals from the model evaluated at final value of theta, i.e., rs contains the vector (yXθ̂)$\left(y-X\stackrel{^}{\theta }\right)$.
7:     wgt(n) – double array
The vector of weights. wgt(i)${\mathbf{wgt}}\left(\mathit{i}\right)$ contains the weight for the i$\mathit{i}$th observation, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$.
8:     work(4 × n + m × (n + m)$4×{\mathbf{n}}+{\mathbf{m}}×\left({\mathbf{n}}+{\mathbf{m}}\right)$) – double array
The following values are assigned to work:
• work(1) = β1${\mathbf{work}}\left(1\right)={\beta }_{1}$ if isigma < 0${\mathbf{isigma}}<0$, or work(1) = β2${\mathbf{work}}\left(1\right)={\beta }_{2}$ if isigma > 0${\mathbf{isigma}}>0$.
• work(2) = ${\mathbf{work}}\left(2\right)=\text{}$ number of iterations used to calculate A$A$.
• work(3) = ${\mathbf{work}}\left(3\right)=\text{}$ number of iterations used to calculate final estimates of θ$\theta$ and σ$\sigma$.
• work(4) = k${\mathbf{work}}\left(4\right)=k$, the rank of the weighted least squares equations.
The rest of the array is used as workspace.
9:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ 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${\mathbf{ifail}}=1$
 On entry, n ≤ 1${\mathbf{n}}\le 1$, or m < 1${\mathbf{m}}<1$, or n ≤ m${\mathbf{n}}\le {\mathbf{m}}$, or ldx < n$\mathit{ldx}<{\mathbf{n}}$, or ldc < m$\mathit{ldc}<{\mathbf{m}}$.
ifail = 2${\mathbf{ifail}}=2$
 On entry, ipsi < 0${\mathbf{ipsi}}<0$, or ipsi > 4${\mathbf{ipsi}}>4$.
ifail = 3${\mathbf{ifail}}=3$
 On entry, sigma ≤ 0.0${\mathbf{sigma}}\le 0.0$, or ipsi = 1${\mathbf{ipsi}}=1$ and cpsi ≤ 0.0${\mathbf{cpsi}}\le 0.0$, or ipsi = 2${\mathbf{ipsi}}=2$ and h1 < 0.0${\mathbf{h1}}<0.0$, or ipsi = 2${\mathbf{ipsi}}=2$ and h1 > h2${\mathbf{h1}}>{\mathbf{h2}}$, or ipsi = 2${\mathbf{ipsi}}=2$ and h2 > h3${\mathbf{h2}}>{\mathbf{h3}}$, or ipsi = 2${\mathbf{ipsi}}=2$ and h1 = h2 = h3 = 0.0${\mathbf{h1}}={\mathbf{h2}}={\mathbf{h3}}=0.0$, or ipsi ≠ 0${\mathbf{ipsi}}\ne 0$ and isigma > 0${\mathbf{isigma}}>0$ and dchi ≤ 0.0${\mathbf{dchi}}\le 0.0$, or indw > 0${\mathbf{indw}}>0$ and cucv < sqrt(m)${\mathbf{cucv}}<\sqrt{{\mathbf{m}}}$, or indw < 0${\mathbf{indw}}<0$ and ${\mathbf{cucv}}<{\mathbf{m}}$.
ifail = 4${\mathbf{ifail}}=4$
 On entry, tol ≤ 0.0${\mathbf{tol}}\le 0.0$, or maxit ≤ 0${\mathbf{maxit}}\le 0$.
ifail = 5${\mathbf{ifail}}=5$
The number of iterations required to calculate the weights exceeds maxit. (Only if indw0${\mathbf{indw}}\ne 0$.)
ifail = 6${\mathbf{ifail}}=6$
The number of iterations required to calculate β1${\beta }_{1}$ exceeds maxit. (Only if indw < 0${\mathbf{indw}}<0$ and isigma < 0${\mathbf{isigma}}<0$.)
ifail = 7${\mathbf{ifail}}=7$
Either the number of iterations required to calculate θ$\theta$ and σ$\sigma$ exceeds maxit (note that, in this case work(3) = maxit${\mathbf{work}}\left(3\right)={\mathbf{maxit}}$ 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${\mathbf{ifail}}=8$
The weighted least squares equations are not of full rank.
W ifail = 9${\mathbf{ifail}}=9$
If indw = 0${\mathbf{indw}}=0$ then (XTX)$\left({X}^{\mathrm{T}}X\right)$ is almost singular.
If indw0${\mathbf{indw}}\ne 0$ then S1${S}_{1}$ is singular or almost singular. This may be due to too many diagonal elements of the matrix being zero, see Section [Further Comments].
W ifail = 10${\mathbf{ifail}}=10$
In calculating the correlation factor for the asymptotic variance-covariance matrix either the value of
 n n 1/n ∑ ψ′(ri / σ̂) = 0,   or  κ = 0,   or ∑ ψ2(ri / σ̂) = 0. i = 1 i = 1
$1n ∑ i=1 n ψ′ ( ri / σ^ ) = 0 , or κ = 0 , or ∑ i=1 n ψ2 ( ri / σ^ ) = 0 .$
See Section [Further Comments]. In this case c is returned as XTX${X}^{\mathrm{T}}X$.
(Only if indw = 0${\mathbf{indw}}=0$.)
W ifail = 11${\mathbf{ifail}}=11$
The estimated variance for an element of θ0$\theta \le 0$.
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$P$ being zero, where P$P$ is defined in Section [Description]. See Section [Further Comments].
ifail = 12${\mathbf{ifail}}=12$
The degrees of freedom for error, nk0$n-k\le 0$ (this is an unlikely error exit), or the estimated value of σ$\sigma$ was 0$0$ during an iteration.

## Accuracy

The precision of the estimates is determined by tol. As a more stable method is used to calculate the estimates of θ$\theta$ than is used to calculate the covariance matrix, it is possible for the least squares equations to be of full rank but the (XTX)$\left({X}^{\mathrm{T}}X\right)$ matrix to be too nearly singular to be inverted.

In cases when isigma0${\mathbf{isigma}}\ge 0$ 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(ri / σ)$\psi \left({r}_{i}/\sigma \right)$, to be zero or a value of ψ(ri / σ)${\psi }^{\prime }\left({r}_{i}/\sigma \right)$, used to estimate the asymptotic covariance matrix, to be zero. This can lead to errors ${\mathbf{ifail}}={\mathbf{8}}$ or 9${\mathbf{9}}$ (if indw0${\mathbf{indw}}\ne 0$), ${\mathbf{ifail}}={\mathbf{10}}$ (if indw = 0${\mathbf{indw}}=0$) and ${\mathbf{ifail}}={\mathbf{11}}$.
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$\psi$, χ$\chi$, ψ${\psi }^{\prime }\text{}$ and u$u$.

## Example

```function nag_correg_robustm_example
indw = int64(1);
ipsi = int64(2);
isigma = int64(1);
indc = int64(0);
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];
cpsi = 0;
h1 = 1.5;
h2 = 3;
h3 = 4.5;
cucv = 3;
dchi = 1.5;
theta = [0;
0;
0];
sigma = 1;
[xOut, yOut, thetaOut, sigmaOut, c, rs, wgt, work, ifail] = ...
nag_correg_robustm(indw, ipsi, isigma, indc, x, y, cpsi, h1, h2, h3, cucv, dchi, theta, sigma)
```
```

xOut =

1    -1    -1
1    -1     1
1     1    -1
1     1     1
1    -2     0
1     0    -2
1     2     0
1     0     2

yOut =

2.1000
3.6000
4.5000
6.1000
1.3000
1.9000
6.7000
5.5000

thetaOut =

4.0423
1.3083
0.7519

sigmaOut =

0.2026

c =

0.0384   -0.5299   -0.5929
-0.0006    0.0272    0.0546
-0.0007    0.0000    0.0311

rs =

0.1179
0.1141
-0.0987
-0.0026
-0.1256
-0.6385
0.0410
-0.0462

wgt =

0.5783
0.5783
0.5783
0.5783
0.4603
0.4603
0.4603
0.4603

work =

0.1848
10.0000
14.0000
3.0000
1.0000
0
1.0000
1.0000
0.3388
0.3171
0.2374
0.0002
0.3845
0
0.0410
0.0519
1.2308
0
-0.3077
0
0.6667
0
-0.3077
0
1.0769
-1.0000
1.0000
1.0000
0.3388
0.2169
-0.0921
-0.0736
-0.1618
0.2163
0.0135
-0.0662
-0.0180
0.1542
0.2374
0.0002
-0.7689
0
0.0820
0
-0.3388
0.3171
-0.2374
0.0002
0
0
0
0.1038
0.1367
0
-0.2371
-0.7840
0
0
0
0
0
0
0
0
0

ifail =

0

```
```function g02ha_example
indw = int64(1);
ipsi = int64(2);
isigma = int64(1);
indc = int64(0);
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];
cpsi = 0;
h1 = 1.5;
h2 = 3;
h3 = 4.5;
cucv = 3;
dchi = 1.5;
theta = [0;
0;
0];
sigma = 1;
[xOut, yOut, thetaOut, sigmaOut, c, rs, wgt, work, ifail] = ...
g02ha(indw, ipsi, isigma, indc, x, y, cpsi, h1, h2, h3, cucv, dchi, theta, sigma)
```
```

xOut =

1    -1    -1
1    -1     1
1     1    -1
1     1     1
1    -2     0
1     0    -2
1     2     0
1     0     2

yOut =

2.1000
3.6000
4.5000
6.1000
1.3000
1.9000
6.7000
5.5000

thetaOut =

4.0423
1.3083
0.7519

sigmaOut =

0.2026

c =

0.0384   -0.5299   -0.5929
-0.0006    0.0272    0.0546
-0.0007    0.0000    0.0311

rs =

0.1179
0.1141
-0.0987
-0.0026
-0.1256
-0.6385
0.0410
-0.0462

wgt =

0.5783
0.5783
0.5783
0.5783
0.4603
0.4603
0.4603
0.4603

work =

0.1848
10.0000
14.0000
3.0000
1.0000
0
1.0000
1.0000
0.3388
0.3171
0.2374
0.0002
0.3845
0
0.0410
0.0519
1.2308
0
-0.3077
0
0.6667
0
-0.3077
0
1.0769
-1.0000
1.0000
1.0000
0.3388
0.2169
-0.0921
-0.0736
-0.1618
0.2163
0.0135
-0.0662
-0.0180
0.1542
0.2374
0.0002
-0.7689
0
0.0820
0
-0.3388
0.3171
-0.2374
0.0002
0
0
0
0.1038
0.1367
0
-0.2371
-0.7840
0
0
0
0
0
0
0
0
0

ifail =

0

```