NAG Library Function Document

nag_robust_m_regsn_param_var (g02hfc)

+− 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

1 Purpose

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

2 Specification

#include <nag.h>

#include <nagg02.h>

void

nag_robust_m_regsn_param_var (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)

3 Description

For a description of bounded influence regression see nag_robust_m_regsn_user_fn (g02hdc). Let

θ

be the regression arguments 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 nag_robust_m_regsn_param_var (g02hfc):

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 (1987).

In all cases

\hat{σ}

is a robust estimate of

σ

nag_robust_m_regsn_param_var (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_OrderTypeInput

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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: psi – function, supplied by the userExternal 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 – doubleInput

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 nag_robust_m_regsn_param_var (g02hfc) you may allocate memory and initialize these pointers with various quantities for use by psi when called from nag_robust_m_regsn_param_var (g02hfc) (see Section 3.2.1.1 in the Essential Introduction).

3: psp – function, supplied by the userExternal 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 – doubleInput

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 nag_robust_m_regsn_param_var (g02hfc) you may allocate memory and initialize these pointers with various quantities for use by psp when called from nag_robust_m_regsn_param_var (g02hfc) (see Section 3.2.1.1 in the Essential Introduction).

4: regtype – Nag_RegTypeInput

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_CovMatrixEstInput

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 – doubleInput

On entry: the value of

\hat{σ}

, as given by nag_robust_m_regsn_user_fn (g02hdc).

Constraint:

sigma > 0.0

7: n – IntegerInput

On entry:

n

, the number of observations.

Constraint:

n > 1

8: m – IntegerInput

On entry:

m

, the number of independent variables.

Constraint:

1 \leq m < n

9: x[ $\dim$ ] – const doubleInput

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 – IntegerInput

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 doubleInput

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

12: wgt[n] – const doubleInput

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 nag_robust_m_regsn_user_fn (g02hdc).

regtype = Nag_HuberReg

, wgt is not referenced.

13: cov[ $\dim$ ] – doubleOutput

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 – IntegerInput

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[ $\dim$ ] – doubleOutput

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 *Communication Structure

The NAG communication argument (see Section 3.2.1.1 in the Essential Introduction).

17: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_CORRECTION_FACTOR: Correction factor = 0 (Huber type regression).
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.
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 nag_robust_m_regsn_user_fn (g02hdc).

8 Parallelism and Performance

nag_robust_m_regsn_param_var (g02hfc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_robust_m_regsn_param_var (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 Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

nag_robust_m_regsn_param_var (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.

NAG Library Function Documentnag_robust_m_regsn_param_var (g02hfc)

+− 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 Library Function Document

nag_robust_m_regsn_param_var (g02hfc)