TBrho

TBrho computes rho function for Tukey's biweight

Syntax

Description

example

rhoTB =TBrho(u, c) Plot of rho function.

Examples

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  • Plot of rho function.
  • close all
    x=-6:0.01:6;
    rhoTB=TBrho(x,2);
    plot(x,rhoTB,'LineWidth',2)
    xlabel('$u$','Interpreter','Latex')
    ylabel('$\rho (u,2)$','Interpreter','Latex')
    text(x(1)-0.8,rhoTB(1),'c^2/6')
    text(x(end)+0.2,rhoTB(end),'c^2/6')
    title('$\rho (u,c)$','Interpreter','Latex')
    hold('on')
    c=2;
    stem(c,c^2/6,'LineStyle',':','LineWidth',1)
    stem(-c,c^2/6,'LineStyle',':','LineWidth',1)

    Related Examples

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  • Compare two rho functions for 2 different values of c.
  • 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;
    hold('on')
    c=TBbdp(0.5,1);
    rhoTB=TBrho(x,c);
    rhoTB=rhoTB/max(rhoTB);
    plot(x,rhoTB,'LineStyle','-','LineWidth',lwd)
    c=TBeff(0.95,1);
    rhoTB=TBrho(x,c);
    rhoTB=rhoTB/max(rhoTB);
    plot(x,rhoTB,'LineStyle','-.','LineWidth',lwd)
    xlabel('$x$','Interpreter','Latex','FontSize',16)
    ylabel('TB. Normalized $\rho_c(x)$','Interpreter','Latex','FontSize',20)
    legend({'$c_{(bdp=0.5 \mapsto eff=0.29)}$', '$c_{(eff=0.95 \mapsto bdp=0.12)}$'},'Interpreter','Latex','Location','SouthEast','FontSize',16)
    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

    c — tuning parameter. Scalar.

    Scalar greater than 0 which controls the robustness/efficiency of the estimator (beta in regression or mu in the location case ...)

    Data Types: single| double

    More About

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    Additional Details

    function TBrho transforms vector u as follows \[ TBrho(u)= \left\{ \begin{array}{cc} (c^2/6) \left\{ 1-[1-(u/c)^2]^3 \right\} & |u/c| \leq 1 \\ (c^2/6) & |u/c| >1 \\ \end{array} \right. \]

    See equation (2.37) p. 29 of Maronna et al. (2006).

    Remark: equation (2.37) is written in standardized terms in such a way that $\rho(c)=1$, so it is the previous expression divided by $(c^2/6)$

    References

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

    Riani, M., Cerioli, A. and Torti, F. (2014), On consistency factors and efficiency of robust S-estimators, "TEST", Vol. 23, pp. 356-387.

    http://dx.doi.org/10.1007/s11749-014-0357-7

    See Also

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