PDrho

PDrho computes rho function for minimum density power divergence estimator

Syntax

Description

example

rhoPD =PDrho(u, alpha) Plot of rho function.

Examples

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  • Plot of rho function.
  • close all
    x=-6:0.01:6;
    alpha=1;
    rhoPD=PDrho(x,alpha);
    plot(x,rhoPD,'LineWidth',2)
    xlabel('$u$','Interpreter','Latex')
    ylabel('$\rho (u,1)$','Interpreter','Latex')

    Related Examples

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  • Compare two rho functions for 2 different values of alpha.
  • In the first we fix the bdp (value of efficiency is automatically given), while in the second we find the efficiency (the value of bdp is automatically given)

    close all
    x=-6:0.01:6;
    lwd=2;
    alpha1=1;
    bdp1=1-1/sqrt(1+alpha1);
    eff1=(sqrt(1+2*alpha1)/(1+alpha1))^3;
    hold('on')
    rhoPD=PDrho(x,alpha1);
    plot(x,rhoPD,'LineStyle','-','LineWidth',lwd)
    alpha2=0.5;
    bdp2=1-1/sqrt(1+alpha2);
    eff2=(sqrt(1+2*alpha2)/(1+alpha2))^3;
    rhoPD=PDrho(x,alpha2);
    plot(x,rhoPD,'LineStyle','-.','LineWidth',lwd)
    xlabel('$x$','Interpreter','Latex','FontSize',16)
    ylabel('MDPD. Normalized $\rho_\alpha(x)$','Interpreter','Latex','FontSize',20)
    legend({['$\alpha=' num2str(alpha1) '\mapsto bdp=' num2str(bdp1,2) '\;  eff=' num2str(eff1,2) '$'], ...
    ['$\alpha=' num2str(alpha2) '\mapsto bdp=' num2str(bdp2,2) '\;  eff=' num2str(eff2,2) '$']},...
    'Interpreter','Latex','Location','SouthEast','FontSize',12)
    Click here for the graphical output of this example (link to Ro.S.A. website)

    Input Arguments

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    u — scaled residuals or Mahalanobis distances. Vector.

    n x 1 vector containing residuals or Mahalanobis distances for the n units of the sample

    Data Types: single| double

    alpha — tuning parameter. Scalar.

    Scalar in the interval (0,1] which controls the robustness/efficiency of the estimator (beta in regression or mu in the location case ...). The greater is alpha the greater is the bdp and smaller is the efficiency.

    Data Types: single| double

    More About

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    Additional Details

    function PDrho transforms vector u as follows \[ PDrho(u,alpha)= 1-\exp(-\alpha (u^2/2)); \]

    References

    Riani, M. Atkinson, A.C., Corbellini A. and Perrotta A. (2020), Robust Regression with Density Power Divergence: Theory, Comparisons and Data Analysis, submitted.

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