naginterfaces.library.correg.robustm_​user_​varmat

naginterfaces.library.correg.robustm_user_varmat(psi, psp, indw, indc, sigma, x, rs, wgt, data=None)[source]

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

For full information please refer to the NAG Library document for g02hf

https://www.nag.com/numeric/nl/nagdoc_27.1/flhtml/g02/g02hff.html

Parameters
psicallable retval = psi(t, data=None)

must return the value of the function for a given value of its argument.

Parameters
tfloat

The argument for which must be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns
retvalfloat

The value of the function evaluated at .

pspcallable retval = psp(t, data=None)

must return the value of for a given value of its argument.

Parameters
tfloat

The argument for which must be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns
retvalfloat

The value of evaluated at .

indwint

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

Mallows type regression.

Huber type regression.

Schweppe type regression.

indcint

If , must specify the approximation to be used.

If , averaging over residuals.

If , replacing expected by observed.

If , is not referenced.

sigmafloat

The value of , as given by robustm_user().

xfloat, array-like, shape

The values of the matrix, i.e., the independent variables. must contain the th element of , for , for .

rsfloat, array-like, shape

The residuals from the bounded influence regression. These are given by robustm_user().

wgtfloat, array-like, shape

If , must contain the vector of weights used by the bounded influence regression. These should be used with robustm_user().

If , is not referenced.

dataarbitrary, optional

User-communication data for callback functions.

Returns
cfloat, ndarray, shape

The estimate of the variance-covariance matrix.

wkfloat, ndarray, shape

If , , for , will contain the diagonal elements of the matrix and , for , will contain the diagonal elements of matrix .

The rest of the array is used as workspace.

Raises
NagValueError
(errno )

On entry, and .

Constraint: .

(errno )

On entry, and .

Constraint: .

(errno )

On entry, and .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

matrix is singular or almost singular.

(errno )

matrix not positive definite.

(errno )

Correction factor (Huber type regression).

Notes

For a description of bounded influence regression see robustm_user(). Let be the regression parameters and let be the asymptotic variance-covariance matrix of . Then for Huber type regression

where

see Huber (1981) and Marazzi (1987).

For Mallows and Schweppe type regressions, is of the form

where and .

is a diagonal matrix such that the th element approximates in the Schweppe case and in the Mallows case.

is a diagonal matrix such that the th element approximates in the Schweppe case and in the Mallows case.

Two approximations are available in robustm_user_varmat:

  1. Average over the

  2. Replace expected value by observed

See Hampel et al. (1986) and Marazzi (1987).

In all cases is a robust estimate of .

robustm_user_varmat is based on routines in ROBETH; see Marazzi (1987).

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