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.

2 Specification

#include <nag.h>

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.

3 Description

For a description of bounded influence regression see g02hdc. Let

θ

be the regression parameters and let

C

be the asymptotic variance-covariance matrix of

\hat{θ}

. Then 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 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 (1987).

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 g02hfc:

1.Average over the $r_{i}$

$\begin{matrix} Schweppe & Mallows \\ D_{i} = (\frac{1}{n} \sum_{j = 1}^{n} ψ^{'} (\frac{r_{j}}{\hat{σ} w_{i}})) w_{i} & D_{i} = (\frac{1}{n} \sum_{j = 1}^{n} ψ^{'} (\frac{r_{j}}{\hat{σ}})) w_{i} \\ P_{i} = (\frac{1}{n} \sum_{j = 1}^{n} ψ^{2} (\frac{r_{j}}{\hat{σ} w_{i}})) w_{i}^{2} & P_{i} = (\frac{1}{n} \sum_{j = 1}^{n} ψ^{2} (\frac{r_{j}}{\hat{σ}})) w_{i}^{2} \end{matrix}$
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 (1987).

In all cases

\hat{σ}

is a robust estimate of

σ

g02hfc is based on routines in ROBETH; 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 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: $order$ – 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

order = 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:

order = Nag_RowMajor

Nag_ColMajor

2: $psi$ – function, supplied by the user External Function

psi must return the value of the

ψ

function for a given value of its argument.

The specification of psi is:

double

psi (double t, Nag_Comm *comm)

1: $t$ – double Input

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

2: $comm$ – 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 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: $psp$ – function, supplied by the user External Function

psp must return the value of

ψ^{'} (t) = \frac{d}{d t} ψ (t)

for a given value of its argument.

The specification of psp is:

double

psp (double t, Nag_Comm *comm)

1: $t$ – double Input

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

2: $comm$ – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to psp.

user – double *
iuser – Integer *
p – Pointer: 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: $regtype$ – Nag_RegType Input

On entry: the type of regression for which the asymptotic variance-covariance matrix is to be calculated.

$regtype = Nag_MallowsReg$: Mallows type regression.
$regtype = Nag_HuberReg$: Huber type regression.
$regtype = Nag_SchweppeReg$: Schweppe type regression.

Constraint:

regtype = Nag_MallowsReg

Nag_HuberReg

Nag_SchweppeReg

5: $covmat_est$ – Nag_CovMatrixEst Input

On entry: if

regtype \neq Nag_HuberReg

, covmat_est must specify the approximation to be used.

covmat_est = Nag_CovMatAve

, averaging over residuals.

covmat_est = Nag_CovMatObs

, replacing expected by observed.

regtype = Nag_HuberReg

, covmat_est is not referenced.

Constraint:

covmat_est = Nag_CovMatAve

Nag_CovMatObs

6: $sigma$ – double Input

On entry: the value of

\hat{σ}

, as given by g02hdc.

Constraint:

sigma > 0.0

7: $n$ – Integer Input

On entry:

n

, the number of observations.

Constraint:

n > 1

8: $m$ – Integer Input

On entry:

m

, the number of independent variables.

Constraint:

1 \leq m < n

9: $x [\dim]$ – const double Input

Note: the dimension, dim, of the array x must be at least

$\max (1, pdx \times m)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdx)$ when $order = Nag_RowMajor$ .

where

X (i, j)

appears in this document, it refers to the array element

$x [(j - 1) \times pdx + i - 1]$ when $order = Nag_ColMajor$ ;
$x [(i - 1) \times pdx + j - 1]$ when $order = Nag_RowMajor$ .

On entry: 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

10: $pdx$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array x.

Constraints:

if $order = Nag_ColMajor$ , $pdx \geq n$ ;
if $order = Nag_RowMajor$ , $pdx \geq m$ .

11: $rs [n]$ – const double Input

On entry: the residuals from the bounded influence regression. These are given by g02hdc.

12: $wgt [n]$ – const double Input

On entry: if

regtype \neq Nag_HuberReg

, wgt must contain the vector of weights used by the bounded influence regression. These should be used with g02hdc.

regtype = Nag_HuberReg

, wgt is not referenced.

13: $cov [\dim]$ – double Output

Note: the dimension, dim, of the array cov must be at least

pdc \times m

the

(i, j)

th element of the matrix is stored in

$cov [(j - 1) \times pdc + i - 1]$ when $order = Nag_ColMajor$ ;
$cov [(i - 1) \times pdc + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the estimate of the variance-covariance matrix.

14: $pdc$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array cov.

Constraint:

pdc \geq m

15: $comm_arr [\dim]$ – double Output

Note: the dimension, dim, of the array comm_arr must be at least

m \times (n + m + 1) + 2 \times n

On exit: if

regtype \neq Nag_HuberReg

comm_arr [i - 1]

, for

i = 1, 2, \dots, n

, will contain the diagonal elements of the matrix

D

and

comm_arr [i - 1]

, for

i = n + 1, \dots, 2 n

, will contain the diagonal elements of matrix

P

16: $comm$ – Nag_Comm *

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

17: $fail$ – 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 $⟨ value ⟩$ had an illegal value.
NE_CORRECTION_FACTOR: Either the value of $\frac{1}{n} \sum_{i = 1}^{n} ψ^{'} (\frac{r_{i}}{\hat{σ}}) = 0$ ,
or $κ = 0$ ,
or $\sum_{i = 1}^{n} ψ^{2} (\frac{r_{i}}{\hat{σ}}) = 0$ .
In this situation g02hfc returns $C$ as ${(X^{T} X)}^{−1}$ .
NE_INT: On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 1$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n > 1$ .

On entry, $pdc = ⟨ value ⟩$ .
Constraint: $pdc > 0$ .

On entry, $pdx = ⟨ value ⟩$ .
Constraint: $pdx > 0$ .
NE_INT_2: On entry, $m = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $1 \leq m < n$ .

On entry, $m = ⟨ value ⟩$ and $pdc = ⟨ value ⟩$ .
Constraint: $pdc \geq m$ .

On entry, $n = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $n > m$ .

On entry, $pdc = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $pdc \geq m$ .

On entry, $pdx = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $pdx \geq m$ .
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: $X^{T} X$ matrix not positive definite.
NE_REAL: On entry, $sigma = ⟨ value ⟩$ .
Constraint: $sigma \geq 0.0$ .
NE_SINGULAR: $S_{1}$ matrix is singular or almost singular.

7 Accuracy

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

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

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.

9 Further Comments

g02hfc is only for situations in which

X

has full column rank.

Care has to be taken in the choice of the

ψ

function since if

ψ^{'} (t) = 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

10 Example

The asymptotic variance-covariance matrix is calculated for a Schweppe type regression. The values of

X

\hat{σ}

and the residuals and weights are read in. The averaging over residuals approximation is used.

g02hf: FL CL CPP AD PY MB

NAG CL Interfaceg02hfc (robustm_​user_​varmat)

▸▿ 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
g02hfc (robustm_user_varmat)