naginterfaces.library.univar.robust_​1var_​mestim_​wgt

naginterfaces.library.univar.robust_1var_mestim_wgt(chi, psi, isigma, x, beta, theta, sigma, tol, maxit=50, data=None)

robust_1var_mestim_wgt computes an $M$-estimate of location with (optional) simultaneous estimation of scale, where you provide the weight functions.

For full information please refer to the NAG Library document for g07dc

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g07/g07dcf.html

Parameters

chicallable retval = chi(t, data=None)

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

The value of $\chi$ must be non-negative.

Parameters

tfloat

The argument for which $chi$ must be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

retvalfloat

The value of the weight function $\varphi$ evaluated at $t$.

psicallable retval = psi(t, data=None)

$psi$ must return the value of the weight function $\psi$ 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 weight function $\psi$ evaluated at $t$.

isigmaint

The value assigned to $isigma$ determines whether $\hat{\sigma }$ is to be simultaneously estimated.

$isigma=0$

The estimation of $\hat{\sigma }$ is bypassed and $sigma$ is set equal to ${\sigma }_c$.

$isigma=1$

$\hat{\sigma }$ is estimated simultaneously.

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

The vector of observations, $x_1,x_2,\dots ,x_n$.

betafloat

The value of the constant $\beta$ of the chosen $chi$ function.

thetafloat

If $sigma>0$, $theta$ must be set to the required starting value of the estimate of the location parameter $\hat{\theta }$. A reasonable initial value for $\hat{\theta }$ will often be the sample mean or median.

sigmafloat

The role of $sigma$ depends on the value assigned to $isigma$ as follows.

If $isigma=1$, $sigma$ must be assigned a value which determines the values of the starting points for the calculation of $\hat{\theta }$ and $\hat{\sigma }$.

If $sigma\le 0.0$, robust_1var_mestim_wgt will determine the starting points of $\hat{\theta }$ and $\hat{\sigma }$.

Otherwise, the value assigned to $sigma$ will be taken as the starting point for $\hat{\sigma }$, and $theta$ must be assigned a relevant value before entry, see above.

If $isigma=0$, $sigma$ must be assigned a value which determines the values of ${\sigma }_c$, which is held fixed during the iterations, and the starting value for the calculation of $\hat{\theta }$.

If $sigma\le 0$, robust_1var_mestim_wgt will determine the value of ${\sigma }_c$ as the median absolute deviation adjusted to reduce bias (see robust_1var_median()) and the starting point for $\theta$.

Otherwise, the value assigned to $sigma$ will be taken as the value of ${\sigma }_c$ and $theta$ must be assigned a relevant value before entry, see above.

tolfloat

The relative precision for the final estimates. Convergence is assumed when the increments for $theta$, and $sigma$ are less than $tol\times max(1.0,{\sigma }_{k-1})$.

maxitint, optional

The maximum number of iterations that should be used during the estimation.

dataarbitrary, optional

User-communication data for callback functions.

Returns

thetafloat

The $M$-estimate of the location parameter $\hat{\theta }$.

sigmafloat

The $M$-estimate of the scale parameter $\hat{\sigma }$, if $isigma$ was assigned the value $1$ on entry, otherwise $sigma$ will contain the initial fixed value ${\sigma }_c$.

rsfloat, ndarray, shape $\left(n\right)$

The Winsorized residuals.

nitint

The number of iterations that were used during the estimation.

wrkfloat, ndarray, shape $\left(n\right)$

If $sigma\le 0.0$ on entry, $wrk$ will contain the $n$ observations in ascending order.

Raises

NagValueError

(errno $1$)

On entry, $isigma=⟨value⟩$.

Constraint: $isigma=0$ or $1$.

(errno $1$)

On entry, $maxit=⟨value⟩$.

Constraint: $maxit>0$.

(errno $1$)

On entry, $tol=⟨value⟩$.

Constraint: $tol>0.0$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n>1$.

(errno $2$)

On entry, $beta=⟨value⟩$.

Constraint: $beta>0.0$.

(errno $3$)

All elements of $x$ are equal.

(errno $4$)

Current estimate of $sigma$ is zero or negative: $sigma=⟨value⟩$.

(errno $5$)

Number of iterations required exceeds $maxit$: $maxit=⟨value⟩$.

(errno $6$)

All winsorized residuals are zero.

(errno $7$)

The $chi$ function returned a negative value: $chi=⟨value⟩$.

Notes

The data consists of a sample of size $n$, denoted by $x_1,x_2,\dots ,x_n$, drawn from a random variable $X$.

The $x_i$ are assumed to be independent with an unknown distribution function of the form,

$$F\left((x_i-\theta )/\sigma \right)$$

where $\theta$ is a location parameter, and $\sigma$ is a scale parameter. $M$-estimators of $\theta$ and $\sigma$ are given by the solution to the following system of equations;

$$\begin{array}{c}\begin{array}{ccc}{\sum }_{i=1}^n\psi \left((x_i-\hat{\theta })/\hat{\sigma }\right)& =& 0\\ & & \\ & & \\ {\sum }_{i=1}^n\chi \left((x_i-\hat{\theta })/\hat{\sigma }\right)& =& (n-1)\beta \end{array}\end{array}$$

where $\psi$ and $\chi$ are user-supplied weight functions, and $\beta$ is a constant. Optionally the second equation can be omitted and the first equation is solved for $\hat{\theta }$ using an assigned value of $\sigma ={\sigma }_c$.

The constant $\beta$ should be chosen so that $\hat{\sigma }$ is an unbiased estimator when $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$ has a Normal distribution. To achieve this the value of $\beta$ is calculated as:

$$\beta =E\left(\chi \right)={\int }_{-\infty }^{\infty }\chi \left(z\right)\frac{1}{\sqrt{2\pi }}exp\left\{\frac{-z^2}{2}\right\}dz$$

The values of $\psi \left(\frac{x_i-\hat{\theta }}{\hat{\sigma }}\right)\hat{\sigma }$ are known as the Winsorized residuals.

The equations are solved by a simple iterative procedure, suggested by Huber:

$${\hat{\sigma }}_k=\sqrt{\frac{1}{\beta (n-1)}\left(\sum _{i=1}^n\chi \left(\frac{x_i-{\hat{\theta }}_{k-1}}{{\hat{\sigma }}_{k-1}}\right)\right){\hat{\sigma }}_{k-1}^2}$$

and

$${\hat{\theta }}_k={\hat{\theta }}_{k-1}+\frac{1}{n}\sum _{i=1}^n\psi \left(\frac{x_i-{\hat{\theta }}_{k-1}}{{\hat{\sigma }}_k}\right){\hat{\sigma }}_k$$

or

$${\hat{\sigma }}_k={\sigma }_c$$

if $\sigma$ is fixed.

The initial values for $\hat{\theta }$ and $\hat{\sigma }$ may be user-supplied or calculated within robust_1var_mestim() as the sample median and an estimate of $\sigma$ based on the median absolute deviation respectively.

robust_1var_mestim_wgt is based upon function LYHALG within the ROBETH library, 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 estimation of location and scale in ROBETH, Cah. Rech. Doc. IUMSP, No. 3 ROB 1, Institut Universitaire de Médecine Sociale et Préventive, Lausanne