NAG Library Routine Document

G02HFF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

G02HFF calculates an estimate of the asymptotic variance-covariance matrix for the bounded influence regression estimates (M-estimates). It is intended for use with G02HDF.

2 Specification

SUBROUTINE G02HFF (

PSI, PSP, INDW, INDC, SIGMA, N, M, X, LDX, RS, WGT, C, LDC, WK, IFAIL)

INTEGER	INDW, INDC, N, M, LDX, LDC, IFAIL
REAL (KIND=nag_wp)	PSI, PSP, SIGMA, X(LDX,M), RS(N), WGT(N), C(LDC,M), WK(M(N+M+1)+2N)
EXTERNAL	PSI, PSP

3 Description

For a description of bounded influence regression see G02HDF. 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 G02HFF:

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

G02HFF 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 Parameters

1: PSI – REAL (KIND=nag_wp) FUNCTION, supplied by the user.External Procedure

PSI must return the value of the

ψ

function for a given value of its argument.

The specification of PSI is:

FUNCTION PSI (

REAL (KIND=nag_wp) PSI

REAL (KIND=nag_wp)

1: T – REAL (KIND=nag_wp)Input: On entry: the argument for which PSI must be evaluated.

PSI must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which G02HFF is called. Parameters denoted as Input must not be changed by this procedure.

2: PSP – REAL (KIND=nag_wp) FUNCTION, supplied by the user.External Procedure

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:

FUNCTION PSP (

REAL (KIND=nag_wp) PSP

REAL (KIND=nag_wp)

1: T – REAL (KIND=nag_wp)Input: On entry: the argument for which PSP must be evaluated.

PSP must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which G02HFF is called. Parameters denoted as Input must not be changed by this procedure.

3: INDW – INTEGERInput

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

$INDW < 0$: Mallows type regression.
$INDW = 0$: Huber type regression.
$INDW > 0$: Schweppe type regression.

4: INDC – INTEGERInput

On entry: if

INDW \neq 0

, INDC must specify the approximation to be used.

INDC = 1

, averaging over residuals.

INDC \neq 1

, replacing expected by observed.

INDW = 0

, INDC is not referenced.

5: SIGMA – REAL (KIND=nag_wp)Input

On entry: the value of

\hat{σ}

, as given by G02HDF.

Constraint:

SIGMA > 0.0

6: N – INTEGERInput

On entry:

n

, the number of observations.

Constraint:

N > 1

7: M – INTEGERInput

On entry:

m

, the number of independent variables.

Constraint:

1 \leq M < N

8: X(LDX,M) – REAL (KIND=nag_wp) arrayInput

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

9: LDX – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which G02HFF is called.

Constraint:

LDX \geq N

10: RS(N) – REAL (KIND=nag_wp) arrayInput

On entry: the residuals from the bounded influence regression. These are given by G02HDF.

11: WGT(N) – REAL (KIND=nag_wp) arrayInput

On entry: if

INDW \neq 0

, WGT must contain the vector of weights used by the bounded influence regression. These should be used with G02HDF.

INDW = 0

, WGT is not referenced.

12: C(LDC,M) – REAL (KIND=nag_wp) arrayOutput

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

13: LDC – INTEGERInput

On entry: the first dimension of the array C as declared in the (sub)program from which G02HFF is called.

Constraint:

LDC \geq M

14: WK( $M \times (N + M + 1) + 2 \times N$ ) – REAL (KIND=nag_wp) arrayOutput

On exit: if

INDW \neq 0

WK (i)

, for

i = 1, 2, \dots, n

, will contain the diagonal elements of the matrix

D

and

WK (i)

, for

i = n + 1, \dots, 2 n

, will contain the diagonal elements of matrix

P

The rest of the array is used as workspace.

15: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$N \leq 1$ ,
or	$M < 1$ ,
or	$N \leq M$ ,
or	$LDC < M$ ,
or	$LDX < N$ .

$IFAIL = 2$

On entry,

SIGMA \leq 0.0

$IFAIL = 3$: If $INDW = 0$ then the matrix $X^{T} X$ is either not positive definite, possibly due to rounding errors, or is ill-conditioned.

If $INDW \neq 0$ then the matrix $S_{1}$ is singular or almost singular. This may be due to many elements of $D$ being zero.

$IFAIL = 4$: 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 G02HFF returns $C$ as ${(X^{T} X)}^{- 1}$ .

7 Accuracy

In general, the accuracy of the variance-covariance matrix will depend primarily on the accuracy of the results from G02HDF.

8 Further Comments

G02HFF 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 Routine DocumentG02HFF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Routine Document

G02HFF