# naginterfaces.library.univar.robust_​1var_​mestim¶

naginterfaces.library.univar.robust_1var_mestim(isigma, x, ipsi, c, h1, h2, h3, dchi, theta, sigma, tol, maxit=50)[source]

robust_1var_mestim computes an -estimate of location with (optional) simultaneous estimation of the scale using Huber’s algorithm.

For full information please refer to the NAG Library document for g07db

https://www.nag.com/numeric/nl/nagdoc_28.6/flhtml/g07/g07dbf.html

Parameters
isigmaint

The value assigned to determines whether is to be simultaneously estimated.

The estimation of is bypassed and is set equal to .

is estimated simultaneously.

xfloat, array-like, shape

The vector of observations, .

ipsiint

Which function is to be used.

.

Huber’s function.

Hampel’s piecewise linear function.

Andrew’s sine wave,

Tukey’s bi-weight.

cfloat

If , must specify the parameter, , of Huber’s function. is not referenced if .

h1float

If , , and must specify the parameters, , , and , of Hampel’s piecewise linear function. , and are not referenced if .

h2float

If , , and must specify the parameters, , , and , of Hampel’s piecewise linear function. , and are not referenced if .

h3float

If , , and must specify the parameters, , , and , of Hampel’s piecewise linear function. , and are not referenced if .

dchifloat

, the parameter of the function. is not referenced if .

thetafloat

If then must be set to the required starting value of the estimation of the location parameter . A reasonable initial value for will often be the sample mean or median.

sigmafloat

The role of depends on the value assigned to , as follows:

if , must be assigned a value which determines the values of the starting points for the calculations of and . If then robust_1var_mestim will determine the starting points of and . Otherwise the value assigned to will be taken as the starting point for , and must be assigned a value before entry, see above;

if , must be assigned a value which determines the value of , which is held fixed during the iterations, and the starting value for the calculation of . If , robust_1var_mestim will determine the value of as the median absolute deviation adjusted to reduce bias (see robust_1var_median()) and the starting point for . Otherwise, the value assigned to will be taken as the value of and 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 , and are less than .

maxitint, optional

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

Returns
thetafloat

The -estimate of the location parameter, .

sigmafloat

Contains the -estimate of the scale parameter, , if was assigned the value on entry, otherwise will contain the initial fixed value .

rsfloat, ndarray, shape

The Winsorized residuals.

nitint

The number of iterations that were used during the estimation.

wrkfloat, ndarray, shape

If on entry, will contain the observations in ascending order.

Raises
NagValueError
(errno )

On entry, .

Constraint: , , , or .

(errno )

On entry, .

Constraint: or .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, , and .

Constraint: and .

(errno )

On entry, .

Constraint: .

(errno )

All elements of are equal.

(errno )

Current estimate of is zero or negative: .

(errno )

Number of iterations required exceeds : .

(errno )

All winsorized residuals are zero.

Notes

The data consists of a sample of size , denoted by , drawn from a random variable .

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

where is a location parameter, and is a scale parameter. -estimators of and are given by the solution to the following system of equations:

where and are given functions, and is a constant, such that is an unbiased estimator when , for has a Normal distribution. Optionally, the second equation can be omitted and the first equation is solved for using an assigned value of .

The values of are known as the Winsorized residuals.

The following functions are available for and in robust_1var_mestim:

1. Null Weights

 ψ(t)=t χ(t)=t22

Use of these null functions leads to the mean and standard deviation of the data.

2. Huber’s Function

 ψ(t)=max(−c,min(c,t)) χ(t)=|t|22 |t|≤d χ(t)=d22 |t|>d
3. Hampel’s Piecewise Linear Function

 ψh1,h2,h3(t)=−ψh1,h2,h3(−t) ψh1,h2,h3(t)=t 0≤t≤h1 ψh1,h2,h3(t)=h1 h1≤t≤h2 ψh1,h2,h3(t)=h1(h3−t)/(h3−h2) h2≤t≤h3 ψh1,h2,h3(t)=0 t>h3 χ(t)=|t|22 |t|≤d χ(t)=d22 |t|>d
4. Andrew’s Sine Wave Function

 ψ(t)=sin(t) −π≤t≤π ψ(t)=0 otherwise χ(t)=|t|22 |t|≤d χ(t)=d22 |t|>d
5. Tukey’s Bi-weight

 ψ(t)=t(1−t2)2 |t|≤1 ψ(t)=0 otherwise χ(t)=|t|22 |t|≤d χ(t)=d22 |t|>d

where , , , and are constants.

Equations (1) and (2) are solved by a simple iterative procedure suggested by Huber:

and

or

The initial values for and may either be user-supplied or calculated within robust_1var_mestim as the sample median and an estimate of based on the median absolute deviation respectively.

robust_1var_mestim 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