# NAG CL Interfaceg02hfc (robustm_​user_​varmat)

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## 1Purpose

g02hfc calculates an estimate of the asymptotic variance-covariance matrix for the bounded influence regression estimates (M-estimates). It is intended for use with g02hdc.

## 2Specification

 #include
void  g02hfc (Nag_OrderType order,
 double (*psi)(double t, Nag_Comm *comm),
 double (*psp)(double t, Nag_Comm *comm),
Nag_RegType regtype, Nag_CovMatrixEst covmat_est, double sigma, Integer n, Integer m, const double x[], Integer pdx, const double rs[], const double wgt[], double cov[], Integer pdc, double comm_arr[], Nag_Comm *comm, NagError *fail)
The function may be called by the names: g02hfc, nag_correg_robustm_user_varmat or nag_robust_m_regsn_param_var.

## 3Description

For a description of bounded influence regression see g02hdc. Let $\theta$ be the regression parameters and let $C$ be the asymptotic variance-covariance matrix of $\stackrel{^}{\theta }$. Then for Huber type regression
 $C=fH(XTX)−1σ^2,$
where
 $fH=1n-m ∑i= 1nψ2 (ri/σ^) (1n∑ψ′(riσ^)) 2 κ2$
 $κ2=1+mn 1n ∑i=1n (ψ′(ri/σ^)-1n∑i=1nψ′(ri/σ^)) 2 (1n∑i=1nψ′(riσ^)) 2 ,$
see Huber (1981) and Marazzi (1987).
For Mallows and Schweppe type regressions, $C$ is of the form
 $σ^n2S1−1S2S1−1,$
where ${S}_{1}=\frac{1}{n}{X}^{\mathrm{T}}DX$ and ${S}_{2}=\frac{1}{n}{X}^{\mathrm{T}}PX$.
$D$ is a diagonal matrix such that the $i$th element approximates $E\left({\psi }^{\prime }\left({r}_{i}/\left(\sigma {w}_{i}\right)\right)\right)$ in the Schweppe case and $E\left({\psi }^{\prime }\left({r}_{i}/\sigma \right){w}_{i}\right)$ in the Mallows case.
$P$ is a diagonal matrix such that the $i$th element approximates $E\left({\psi }^{2}\left({r}_{i}/\left(\sigma {w}_{i}\right)\right){w}_{i}^{2}\right)$ in the Schweppe case and $E\left({\psi }^{2}\left({r}_{i}/\sigma \right){w}_{i}^{2}\right)$ in the Mallows case.
Two approximations are available in g02hfc:
1. 1.Average over the ${r}_{i}$
 $Schweppe Mallows Di=(1n∑j=1nψ′(rjσ^wi )) wi Di=(1n∑j=1nψ′(rjσ^)) wi Pi=(1n∑j=1nψ2(rjσ^wi )) wi2 Pi=(1n∑j=1nψ2(rjσ^)) wi2$
2. 2.Replace expected value by observed
 $Schweppe Mallows Di=ψ′ ( riσ ^wi ) wi Di=ψ′ ( riσ ^) wi Pi=ψ2 ( riσ ^wi ) wi2 Pi=ψ2 ( riσ ^) wi2$
In all cases $\stackrel{^}{\sigma }$ is a robust estimate of $\sigma$.
g02hfc is based on routines in ROBETH; see Marazzi (1987).

## 4References

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 and bounded influence regression in ROBETH Cah. Rech. Doc. IUMSP, No. 3 ROB 2 Institut Universitaire de Médecine Sociale et Préventive, Lausanne

## 5Arguments

1: $\mathbf{order}$Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{psi}$function, supplied by the user External Function
psi must return the value of the $\psi$ function 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.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling g02hfc you may allocate memory and initialize these pointers with various quantities for use by psi when called from g02hfc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: psi should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by g02hfc. If your code inadvertently does return any NaNs or infinities, g02hfc is likely to produce unexpected results.
3: $\mathbf{psp}$function, supplied by the user External Function
psp must return the value of ${\psi }^{\prime }\left(t\right)=\frac{d}{dt}\psi \left(t\right)$ for a given value of its argument.
The specification of psp is:
 double psp (double t, Nag_Comm *comm)
1: $\mathbf{t}$double Input
On entry: the argument for which psp must be evaluated.
2: $\mathbf{comm}$Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to psp.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling g02hfc you may allocate memory and initialize these pointers with various quantities for use by psp when called from g02hfc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: psp should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by g02hfc. If your code inadvertently does return any NaNs or infinities, g02hfc is likely to produce unexpected results.
4: $\mathbf{regtype}$Nag_RegType Input
On entry: the type of regression for which the asymptotic variance-covariance matrix is to be calculated.
${\mathbf{regtype}}=\mathrm{Nag_MallowsReg}$
Mallows type regression.
${\mathbf{regtype}}=\mathrm{Nag_HuberReg}$
Huber type regression.
${\mathbf{regtype}}=\mathrm{Nag_SchweppeReg}$
Schweppe type regression.
Constraint: ${\mathbf{regtype}}=\mathrm{Nag_MallowsReg}$, $\mathrm{Nag_HuberReg}$ or $\mathrm{Nag_SchweppeReg}$.
5: $\mathbf{covmat_est}$Nag_CovMatrixEst Input
On entry: if ${\mathbf{regtype}}\ne \mathrm{Nag_HuberReg}$, covmat_est must specify the approximation to be used.
If ${\mathbf{covmat_est}}=\mathrm{Nag_CovMatAve}$, averaging over residuals.
If ${\mathbf{covmat_est}}=\mathrm{Nag_CovMatObs}$, replacing expected by observed.
If ${\mathbf{regtype}}=\mathrm{Nag_HuberReg}$, covmat_est is not referenced.
Constraint: ${\mathbf{covmat_est}}=\mathrm{Nag_CovMatAve}$ or $\mathrm{Nag_CovMatObs}$.
6: $\mathbf{sigma}$double Input
On entry: the value of $\stackrel{^}{\sigma }$, as given by g02hdc.
Constraint: ${\mathbf{sigma}}>0.0$.
7: $\mathbf{n}$Integer Input
On entry: $n$, the number of observations.
Constraint: ${\mathbf{n}}>1$.
8: $\mathbf{m}$Integer Input
On entry: $m$, the number of independent variables.
Constraint: $1\le {\mathbf{m}}<{\mathbf{n}}$.
9: $\mathbf{x}\left[\mathit{dim}\right]$const double Input
Note: the dimension, dim, of the array x must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdx}}×{\mathbf{m}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pdx}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
where ${\mathbf{X}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{x}}\left[\left(j-1\right)×{\mathbf{pdx}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{x}}\left[\left(i-1\right)×{\mathbf{pdx}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the values of the $X$ matrix, i.e., the independent variables. ${\mathbf{X}}\left(\mathit{i},\mathit{j}\right)$ must contain the $\mathit{i}\mathit{j}$th element of $X$, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,m$.
10: $\mathbf{pdx}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdx}}\ge {\mathbf{n}}$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdx}}\ge {\mathbf{m}}$.
11: $\mathbf{rs}\left[{\mathbf{n}}\right]$const double Input
On entry: the residuals from the bounded influence regression. These are given by g02hdc.
12: $\mathbf{wgt}\left[{\mathbf{n}}\right]$const double Input
On entry: if ${\mathbf{regtype}}\ne \mathrm{Nag_HuberReg}$, wgt must contain the vector of weights used by the bounded influence regression. These should be used with g02hdc.
If ${\mathbf{regtype}}=\mathrm{Nag_HuberReg}$, wgt is not referenced.
13: $\mathbf{cov}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array cov must be at least ${\mathbf{pdc}}×{\mathbf{m}}$.
the $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{cov}}\left[\left(j-1\right)×{\mathbf{pdc}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{cov}}\left[\left(i-1\right)×{\mathbf{pdc}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: the estimate of the variance-covariance matrix.
14: $\mathbf{pdc}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array cov.
Constraint: ${\mathbf{pdc}}\ge {\mathbf{m}}$.
15: $\mathbf{comm_arr}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array comm_arr must be at least ${\mathbf{m}}×\left({\mathbf{n}}+{\mathbf{m}}+1\right)+2×{\mathbf{n}}$.
On exit: if ${\mathbf{regtype}}\ne \mathrm{Nag_HuberReg}$, ${\mathbf{comm_arr}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,n$, will contain the diagonal elements of the matrix $D$ and ${\mathbf{comm_arr}}\left[\mathit{i}-1\right]$, for $\mathit{i}=n+1,\dots ,2n$, will contain the diagonal elements of matrix $P$.
16: $\mathbf{comm}$Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
17: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error 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.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_CORRECTION_FACTOR
Either the value of $\frac{1}{n}\sum _{i=1}^{n}{\psi }^{\prime }\left(\frac{{r}_{i}}{\stackrel{^}{\sigma }}\right)=0$,
or $\kappa =0$,
or $\sum _{i=1}^{n}{\psi }^{2}\left(\frac{{r}_{i}}{\stackrel{^}{\sigma }}\right)=0$.
In this situation g02hfc returns $C$ as ${\left({X}^{\mathrm{T}}X\right)}^{-1}$.
NE_INT
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 1$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}>1$.
On entry, ${\mathbf{pdc}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdc}}>0$.
On entry, ${\mathbf{pdx}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdx}}>0$.
NE_INT_2
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: $1\le {\mathbf{m}}<{\mathbf{n}}$.
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{pdc}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdc}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}>{\mathbf{m}}$.
On entry, ${\mathbf{pdc}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdc}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{pdx}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{m}}$.
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_POS_DEF
${X}^{\mathrm{T}}X$ matrix not positive definite.
NE_REAL
On entry, ${\mathbf{sigma}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{sigma}}\ge 0.0$.
NE_SINGULAR
${S}_{1}$ matrix is singular or almost singular.

## 7Accuracy

In general, the accuracy of the variance-covariance matrix will depend primarily on the accuracy of the results from g02hdc.

## 8Parallelism and Performance

g02hfc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02hfc 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 implementation-specific information.

g02hfc is only for situations in which $X$ has full column rank.
Care has to be taken in the choice of the $\psi$ function since if ${\psi }^{\prime }\left(t\right)=0$ for too wide a range then either the value of ${f}_{H}$ will not exist or too many values of ${D}_{i}$ will be zero and it will not be possible to calculate $C$.

## 10Example

The asymptotic variance-covariance matrix is calculated for a Schweppe type regression. The values of $X$, $\stackrel{^}{\sigma }$ and the residuals and weights are read in. The averaging over residuals approximation is used.

### 10.1Program Text

Program Text (g02hfce.c)

### 10.2Program Data

Program Data (g02hfce.d)

### 10.3Program Results

Program Results (g02hfce.r)