# MMregeda

MMregeda computes MM estimator in linear regression for a series of values of efficiency

## Syntax

• out =MMregeda(y,X)example
• out =MMregeda(y,X,Name,Value)example
• [out , varargout]=MMregeda(___)example

## Description

 out =MMregeda(y, X) MMregeda with all default options.

 out =MMregeda(y, X, Name, Value) MMregeda with optional input arguments.

 [out , varargout] =MMregeda(___) Comparing the output of different MMreg runs.

## Examples

expand all

### MMregeda with all default options.

n=200;
p=3;
randn('state', 123456);
X=randn(n,p);
% Uncontaminated data
y=randn(n,1);
% Contaminated data
ycont=y;
ycont(1:5)=ycont(1:5)+6;
[out]=MMregeda(ycont,X);

### MMregeda with optional input arguments.

MMregeda using the optimal rho function

n=200;
p=3;
randn('state', 123456);
X=randn(n,p);
% Uncontaminated data
y=randn(n,1);
% Contaminated data
ycont=y;
ycont(1:5)=ycont(1:5)+6;
[out]=MMregeda(ycont,X,'Srhofunc','optimal','rhofunc','optimal');

### Comparing the output of different MMreg runs.

state=100;
randn('state', state);
n=100;
X=randn(n,3);
bet=[3;4;5];
y=3*randn(n,1)+X*bet;
y(1:20)=y(1:20)+13;
%For outlier detection we consider both the nominal individual 1%
%significance level and the simultaneous Bonferroni confidence level.
% Define nominal confidence level
conflev=[0.99,1-0.01/length(y)];
% Define number of subsets
nsamp=3000;
% MM  estimators
[out]=MMregeda(y,X,'conflev',conflev(1));
laby='Scaled MM residuals';
resfwdplot(out)
Total estimated time to complete S estimate:  0.05 seconds


## Input Arguments

### y — Response variable. Vector.

A vector with n elements that contains the response variable. y can be either a row or a column vector.

Data Types: single| double

### X — Data matrix of explanatory variables (also called 'regressors') of dimension (n x p-1). Rows of X represent observations, and columns represent variables.

Missing values (NaN's) and infinite values (Inf's) are allowed, since observations (rows) with missing or infinite values will automatically be excluded from the computations.

Data Types: single| double

### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as  Name1,Value1,...,NameN,ValueN.

Example:  'conflev',0.99 , 'eff',[0.85 0.90 0.95 0.99] , 'effshape',1 , 'InitialEst',[] , 'intercept',false , 'nocheck',true , 'refsteps',10 , 'rhofunc','optimal' , 'rhofuncparam',5 , 'Snsamp',1000 , 'tol',1e-10 , 'plots',0 

### conflev —Confidence level which is used to declare units as outliers.scalar.

Usually conflev=0.95, 0.975 0.99 (individual alpha) or 1-0.05/n, 1-0.025/n, 1-0.01/n (simultaneous alpha).

Default value is 0.975

Example:  'conflev',0.99 

Data Types: double

### eff —nominal efficiency.scalar | vector.

Vector defining nominal efficiency (i.e. a series of numbers between 0.5 and 0.99). The default value is the sequence 0.5:0.01:0.99 Asymptotic nominal efficiency is:

$(\int \psi' d\Phi)^2 / (\psi^2 d\Phi)$

Example:  'eff',[0.85 0.90 0.95 0.99] 

Data Types: double

### effshape —location or scale effiicency.dummy scalar.

If effshape=1 efficiency refers to shape efficiency else (default) efficiency refers to location

Example:  'effshape',1 

Data Types: double

### InitialEst —starting values of the MM-estimator.[] (default) | structure.

InitialEst must contain the following fields

Value Description
beta

v x 1 vector (estimate of the initial regression coefficients)

scale

scalar (estimate of the scale parameter).

If InitialEst is empty (default) or InitialEst.beta continas NaN values, program uses S estimators. In this last case it is possible to specify the options given in function Sreg.

Example:  'InitialEst',[] 

Data Types: struct or empty value

### intercept —Indicator for constant term.true (default) | false.

Indicator for the constant term (intercept) in the fit, specified as the comma-separated pair consisting of 'Intercept' and either true to include or false to remove the constant term from the model.

Example:  'intercept',false 

Data Types: boolean

### nocheck —Check input arguments.boolean.

If nocheck is equal to true no check is performed on matrix y and matrix X. Notice that y and X are left unchanged. In other words the additional column of ones for the intercept is not added.

As default nocheck=false.

Example:  'nocheck',true 

Data Types: boolean

### refsteps —Maximum iterations.scalar.

Scalar defining maximum number of iterations in the MM loop. Default value is 100.

Example:  'refsteps',10 

Data Types: double

### rhofunc —rho function.string.

String which specifies the rho function which must be used to weight the residuals in MM step.

Possible values are 'bisquare';

'optimal';

'hyperbolic';

'hampel';

'mdpd'.

'bisquare' uses Tukey's $\rho$ and $\psi$ functions.

See TBrho and TBpsi.

'optimal' uses optimal $\rho$ and $\psi$ functions.

See OPTrho and OPTpsi.

'hyperbolic' uses hyperbolic $\rho$ and $\psi$ functions.

See HYPrho and HYPpsi.

'hampel' uses Hampel $\rho$ and $\psi$ functions.

See HArho and HApsi.

'mdpd' uses Minimum Density Power Divergence $\rho$ and $\psi$ functions.

See PDrho and PDpsi.

The default is bisquare

Example:  'rhofunc','optimal' 

Data Types: char

### rhofuncparam —Additional parameters for the specified rho function in the MM step.scalar | vector.

For hyperbolic rho function it is possible to set up the value of k = sup CVC (the default value of k is 4.5).

For Hampel rho function it is possible to define parameters a, b and c (the default values are a=2, b=4, c=8)

Example:  'rhofuncparam',5 

Data Types: single | double

### Soptions —options if initial estimator is S and InitialEst is empty.srhofunc,Snsamp,Srefsteps, Sreftol, Srefstepsbestr, Sreftolbestr, Sminsctol, Sbestr.

See function Sreg for more details on these options.

It is necessary to add to the S options the letter S at the beginning. For example, if you want to use the optimal rho function the supplied option is 'Srhofunc','optimal'. For example, if you want to use 3000 subsets, the supplied option is 'Snsamp',3000.

Note that the rho function which is used in the MMstep is the same as the one used in the S step.

Example:  'Snsamp',1000 

Data Types: single | double

### tol —Tolerance.scalar.

Scalar controlling tolerance in the MM loop.

Default value is 1e-7

Example:  'tol',1e-10 

Data Types: double

### plots —Plot on the screen.scalar | structure.

If plots = 1, generates a plot with the robust residuals against index number. The confidence level used to draw the confidence bands for the residuals is given by the input option conflev. If conflev is not specified a nominal 0.975 confidence interval will be used.

Example:  'plots',0 

Data Types: single | double

## Output Arguments

### out — description Structure

A structure containing the following fields

Value Description
auxscale

scalar, S estimate of the scale (or supplied external estimate of scale, if option InitialEst is not empty)

Beta

p x length(eff) matrix containing MM estimate of regression coefficients for each value of eff

RES

n x length(eff) matrix containing scaled MM residuals for each value of eff out.RES(:,jj)=(y-X*out.Beta(:,jj))/out.auxscale

Weights

n x length(eff) matrix. Weights assigned to each observation for each value of eff out.Outliers : n x length(eff) Boolean matrix containing the outliers which have been found for each value of eff.

out.Sbeta : p x 1 vector containing S estimate of regression coefficients (or supplied initial external estimate of regression coefficients, if option InitialEst is not empty)

Ssingsub

Number of subsets without full rank in the S preliminary part. Notice that out.singsub > 0.1*(number of subsamples) produces a warning

conflev

Confidence level that was used to declare outliers

rhofunc

string identifying the rho function which has been used

rhofuncparam

vector which contains the additional parameters for the specified rho function which have been used. For hyperbolic rho function the value of k =sup CVC. For Hampel rho function the parameters a, b and c

Outliers

Boolean matrix containing the list of the units declared as outliers for each value of eff using confidence level specified in input scalar conflev

eff

vector containing the value of eff which have been used.

Sbeta

vector. S initial estimate of regression coefficients.

y

response vector y.

X

data matrix X.

class

'MMregeda'.

## References

Riani, M., Cerioli, A., Atkinson, A.C. and Perrotta, D. (2014), Monitoring Robust Regression, "Electronic Journal of Statistics", Vol. 8, pp. 646-677.

Maronna, R.A., Martin D. and Yohai V.J. (2006), "Robust Statistics, Theory and Methods", Wiley, New York.