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 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 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

or 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 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 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, $n = 〈value〉$ and $pdx = 〈value〉$ .
Constraint: $pdx \geq n$ .; 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 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

9 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 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_robust_m_regsn_param_var (g02hfc)