TBwei

TBwei computes weight function psi(u)/u for Tukey's biweight

Syntax

Description

example

w =TBwei(u, c) Weight function for Tukey biweight.

Examples

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  • Weight function for Tukey biweight.
  • x=-6:0.01:6;
    weiTB=TBwei(x,2);
    plot(x,weiTB)
    xlabel('x','Interpreter','Latex')
    ylabel('$W (x) =\psi(x)/x$','Interpreter','Latex')

    Related Examples

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  • Compare four different weight functions.
  • Initialize graphical parameters.

    FontSize=14;
    x=-6:0.01:6;
    ylim1=-0.05;
    ylim2=1.05;
    xlim1=min(x);
    xlim2=max(x);
    LineWidth=2;
    subplot(2,2,1)
    ceff095HU=HUeff(0.95,1);
    weiHU=HUwei(x,ceff095HU);
    plot(x,weiHU,'LineWidth',LineWidth)
    xlabel('$u$','Interpreter','Latex','FontSize',FontSize)
    title('Huber','FontSize',FontSize)
    ylim([ylim1 ylim2])
    xlim([xlim1 xlim2])
    subplot(2,2,2)
    ceff095HA=HAeff(0.95,1);
    weiHA=HAwei(x,ceff095HA);
    plot(x,weiHA,'LineWidth',LineWidth)
    xlabel('$u$','Interpreter','Latex','FontSize',FontSize)
    title('Hampel','FontSize',FontSize)
    ylim([ylim1 ylim2])
    xlim([xlim1 xlim2])
    subplot(2,2,3)
    ceff095TB=TBeff(0.95,1);
    weiTB=TBwei(x,ceff095TB);
    plot(x,weiTB,'LineWidth',LineWidth)
    xlabel('$u$','Interpreter','Latex','FontSize',FontSize)
    title('Tukey biweight','FontSize',FontSize)
    ylim([ylim1 ylim2])
    xlim([xlim1 xlim2])
    subplot(2,2,4)
    ceff095HYP=HYPeff(0.95,1);
    ktuning=4.5;
    weiHYP=HYPwei(x,[ceff095HYP,ktuning]);
    plot(x,weiHYP,'LineWidth',LineWidth)
    xlabel('$u$','Interpreter','Latex','FontSize',FontSize)
    title('Hyperbolic','FontSize',FontSize)
    ylim([ylim1 ylim2])
    xlim([xlim1 xlim2])
    Effective tolerance in routine HYPck=1.9842e-08
    
    Click here for the graphical output of this example (link to Ro.S.A. website)

  • Compare two weight 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=TBwei(x,c);
    plot(x,rhoTB,'LineStyle','-','LineWidth',lwd)
    c=TBeff(0.95,1);
    rhoTB=TBwei(x,c);
    plot(x,rhoTB,'LineStyle','-.','LineWidth',lwd)
    xlabel('$x$','Interpreter','Latex','FontSize',16)
    ylabel('TB weight function $\psi_c(x)/x$','Interpreter','Latex','FontSize',20)
    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 parameters. 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 TBwei transforms vector u as follows \[ TBwei(u)= \left\{ \begin{array}{cc} (c^2/6) \psi(u)/u = (c^2/6) \left[ 1-(u/c) \right]^2 & |u/c| \leq 1 \\ 0 & |u/c|>1 \\ \end{array} \right. \]

    See p. 30 of Maronna et al. (2006) Remark: Tukey's biweight psi-function is almost linear around u = 0 in accordance with Winsor's principle that all distributions are normal in the middle.

    This means that $\psi (u)/u$ is approximately constant over the linear region of $\psi$, so the points in that region tend to get equal weight.

    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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