PDrho

PDrho computes rho function for minimum density power divergence estimator

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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

    Output Arguments

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    rhoPD —Minimum density power divergence rho function associated to the residuals or Mahalanobis distances for the n units of the sample. n -by- 1 vector

    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, Entropy, Vol. 22, 399.

    https://www.mdpi.com/1099-4300/22/4/399

    See Also

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