naginterfaces.library.correg.robustm_​user_​varmat

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

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://support.nag.com/numeric/nl/nagdoc_30/flhtml/g02/g02hff.html

Parameters

psicallable retval = psi(t, data=None)

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

Parameters

tfloat

The argument for which $psi$ must be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

retvalfloat

The value of the function $\psi$ evaluated at $t$.

pspcallable retval = psp(t, data=None)

$psp$ must return the value of ${\psi }^{\prime }\left(t\right)=\frac{d}{dt}\psi \left(t\right)$ for a given value of its argument.

Parameters

tfloat

The argument for which $psp$ must be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

retvalfloat

The value of ${\psi }^{\prime }\left(t\right)$ evaluated at $t$.

indwint

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

$indw=-1$

Mallows type regression.

$indw=0$

Huber type regression.

$indw=1$

Schweppe type regression.

indcint

If $indw\ne 0$, $indc$ must specify the approximation to be used.

If $indc=1$, averaging over residuals.

If $indc=0$, replacing expected by observed.

If $indw=0$, $indc$ is not referenced.

sigmafloat

The value of $\hat{\sigma }$, as given by robustm_user().

xfloat, array-like, shape $(n,m)$

The values of the $X$ matrix, i.e., the independent variables. $x[\text{i}-1,\text{j}-1]$ must contain the $\text{i}\text{j}$th element of $X$, for $\text{j}=1,2,\dots ,m$, for $\text{i}=1,2,\dots ,n$.

rsfloat, array-like, shape $\left(n\right)$

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

wgtfloat, array-like, shape $\left(n\right)$

If $indw\ne 0$, $wgt$ must contain the vector of weights used by the bounded influence regression. These should be used with robustm_user().

If $indw=0$, $wgt$ is not referenced.

dataarbitrary, optional

User-communication data for callback functions.

Returns

cfloat, ndarray, shape $(m,m)$

The estimate of the variance-covariance matrix.

wkfloat, ndarray, shape $(m\times (n+m+1)+2\times n)$

If $indw\ne 0$, $wk[\text{i}-1]$, for $\text{i}=1,2,\dots ,n$, will contain the diagonal elements of the matrix $D$ and $wk[\text{i}-1]$, for $\text{i}=n+1,\dots ,2n$, will contain the diagonal elements of matrix $P$.

The rest of the array is used as workspace.

Raises

NagValueError

(errno $1$)

On entry, $m=⟨value⟩$ and $\text{ldc}=⟨value⟩$.

Constraint: $\text{ldc}\ge m$.

(errno $1$)

On entry, $n=⟨value⟩$ and $\text{ldx}=⟨value⟩$.

Constraint: $\text{ldx}\ge n$.

(errno $1$)

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

Constraint: $n>m$.

(errno $1$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 1$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 2$.

(errno $2$)

On entry, $sigma=⟨value⟩$.

Constraint: $sigma\ge 0.0$.

(errno $3$)

${\text{S}}_1$ matrix is singular or almost singular.

(errno $3$)

$X^{T}X$ matrix not positive definite.

(errno $4$)

Correction factor $=0$ (Huber type regression).

Notes

For a description of bounded influence regression see robustm_user(). Let $\theta$ be the regression parameters and let $C$ be the asymptotic variance-covariance matrix of $\hat{\theta }$. Then for Huber type regression

$$C=f_H{\left(X^{T}X\right)}^{-1}{\hat{\sigma }}^2\text{,}$$

where

$$f_H=\frac{1}{n-m}\frac{{\sum }_{i=1}^n{\psi }^2\left(r_i/\hat{\sigma }\right)}{{(\frac{1}{n}\sum {\psi }^{\prime }\left(\frac{r_i}{\hat{\sigma }}\right))}^2}{\kappa }^2$$

$${\kappa }^2=1+\frac{m}{n}\frac{\frac{1}{n}{\sum }_{i=1}^n{({\psi }^{\prime }\left(r_i/\hat{\sigma }\right)-\frac{1}{n}{\sum }_{i=1}^n{\psi }^{\prime }\left(r_i/\hat{\sigma }\right))}^2}{{\left(\frac{1}{n}{\sum }_{i=1}^n{\psi }^{\prime }\left(\frac{r_i}{\hat{\sigma }}\right)\right)}^2}\text{,}$$

see Huber (1981) and Marazzi (1987).

For Mallows and Schweppe type regressions, $C$ is of the form

$${\frac{\hat{\sigma }}{n}}^2{S}_1^{-1}{S}_2{S}_1^{-1}\text{,}$$

where $S_1=\frac{1}{n}{X}^{T}DX$ and $S_2=\frac{1}{n}{X}^{T}PX$.

$D$ is a diagonal matrix such that the $i$th element approximates $E\left({\psi }^{\prime }\left(r_i/\left(\sigma w_i\right)\right)\right)$ in the Schweppe case and $E\left({\psi }^{\prime }\left(r_i/\sigma \right)w_i\right)$ in the Mallows case.

$P$ is a diagonal matrix such that the $i$th element approximates $E\left({\psi }^2\left(r_i/\left(\sigma w_i\right)\right)w_i^2\right)$ in the Schweppe case and $E\left({\psi }^2\left(r_i/\sigma \right)w_i^2\right)$ in the Mallows case.

Two approximations are available in robustm_user_varmat:

1. Average over the $r_i$

   $$\begin{array}{c}\begin{array}{cc}\text{Schweppe}& \text{Mallows}\\ & \\ D_i=\left(\frac{1}{n}{\sum }_{j=1}^n{\psi }^{\prime }\left(\frac{r_j}{\hat{\sigma }{w}_i}\right)\right)w_i& D_i=\left(\frac{1}{n}{\sum }_{j=1}^n{\psi }^{\prime }\left(\frac{r_j}{\hat{\sigma }}\right)\right)w_i\\ & \\ P_i=\left(\frac{1}{n}{\sum }_{j=1}^n{\psi }^2\left(\frac{r_j}{\hat{\sigma }{w}_i}\right)\right)w_i^2& P_i=\left(\frac{1}{n}{\sum }_{j=1}^n{\psi }^2\left(\frac{r_j}{\hat{\sigma }}\right)\right)w_i^2\end{array}\end{array}$$

2. Replace expected value by observed

   $$\begin{array}{c}\begin{array}{cc}\text{Schweppe}& \text{Mallows}\\ & \\ D_i={\psi }^{\prime }\left(\frac{r_i}{\hat{\sigma }{w}_i}\right)w_i& D_i={\psi }^{\prime }\left(\frac{r_i}{\hat{\sigma }}\right)w_i\\ & \\ P_i={\psi }^2\left(\frac{r_i}{\hat{\sigma }{w}_i}\right)w_i^2& P_i={\psi }^2\left(\frac{r_i}{\hat{\sigma }}\right)w_i^2\end{array}\end{array}$$

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

In all cases $\hat{\sigma }$ is a robust estimate of $\sigma$.

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