OPTc

OPTc computes breakdown point and efficiency associated with constant c for Optimal rho function

Syntax

Description

example

bdp =OPTc(c, v) bdp associated with a particular c.

example

bdp =OPTc(c, v, shapeeff) bdp and eff associated with a particular c.

example

[bdp, eff] =OPTc(___) Breakdown vs efficiency.

example

[bdp, eff, approxsheff] =OPTc(___) An example with 3 output arguments.

Examples

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  • Breakdown vs efficiency.
  • %% Breakdown vs efficiency.
    %Analysis of breakdown point and asymptotic efficiency
    %at the normal distribution as a function of c in regression.
    c=1:0.01:4;
    CC=[c' zeros(length(c),1)];
    jk=0;
    for j=c
    jk=jk+1;
    [bdp,eff]=OPTc(j,1);
    CC(jk,2:3)=[bdp,eff];
    end
    subplot(2,1,1)
    plot(c',CC(:,2))
    xlabel('c','Interpreter','Latex','FontSize',16)
    ylabel('Breakdown point','Interpreter','none')
    subplot(2,1,2)
    plot(c',CC(:,3))
    xlabel('c','Interpreter','Latex','FontSize',16)
    ylabel('Asymptotic efficiency','Interpreter','none')
    Click here for the graphical output of this example (link to Ro.S.A. website). Graphical output could not be included in the installation file because toolboxes cannot be greater than 20MB. To load locally the image files, download zip file http://rosa.unipr.it/fsda/images.zip and unzip it to <tt>(docroot)/FSDA/images</tt> or simply run routine <tt>downloadGraphicalOutput.m</tt>

  • An example with 3 output arguments.
  • c=5;
    % third input argument is 1, that is shape efficiency
    [bdp, eff, approxeff] = OPTc(c,2,1);

    Input Arguments

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    c — tuning constant c. Scalar.

    Scalar greater than 0 which controls the robustness/efficiency of the estimator

    Data Types: single| double

    v — number of response variables. Scalar.

    Number of variables of the dataset (for regression v=1)

    Data Types: single| double

    Optional Arguments

    shapeeff — location or shape efficiency. Scalar.

    If shapeeff=1, the efficiency is referred to the shape else (default) is referred to the location estimator

    Example: 1

    Data Types: double

    Output Arguments

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    bdp —bdp. Scalar

    Breakdown point associated to the supplied value of c

    eff —eff. Scalar

    Efficiency associated to the supplied value of c

    approxsheff —Approximate value of efficiency. Scalar

    Approximate value of efficiency in case input option shapeeff=1 and v>1.

    This output is left for comparability with the value which comes out from R library robustbase.

    More About

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    Additional Details

    $\rho$ ($\psi$) function which is considered is standardized using intervals 0---(2/3)c , (2/3)c---c, >c.

    $\rho$ function is

    \[ OPTrho(u)= \left\{ \begin{array}{lr} 1.3846 \left(\frac{u}{c}\right)^2 & |\frac{u}{c}| \leq \frac{2}{3} \\ 0.5514-2.6917 \left(\frac{u}{c}\right)^2 +10.7668\left(\frac{u}{c}\right)^4-11.6640\left(\frac{u}{c}\right)^6+4.0375\left(\frac{u}{c}\right)^8 & \qquad \frac{2}{3} \leq |\frac{u}{c}|\leq 1 \\ 1 & |\frac{u}{c}|>1 \\ \end{array} \right. \] |t/c|>1 Therefore, the input c for the (rho) psi function above corresponds to c/3 in the rho (psi) function defined in the interval 0---2c, 2c---3c, >3c

    References

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

    See Also

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