HYPeff

HYPeff finds constant c which is associated to the requested efficiency for hyperbolic estimator

Syntax

Description

example

c =HYPeff(eff, v) Find parameter c for fixed efficiency.

example

c =HYPeff(eff, v, k) Example of specifying k.

example

c =HYPeff(eff, v, k, traceiter) Find parameters for fixed efficiency and k.

example

[c, A] =HYPeff(___) Example of use of option Find parameters for fixed efficiency and k.

example

[c, A, B] =HYPeff(___) Example to show the issue of multiple solutions problem.

example

[c, A, B, d] =HYPeff(___) Compare the weight function for 6 different links.

Examples

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  • Find parameter c for fixed efficiency.
  • Find value of c for a nominal efficiency of 0.9 when k=4.5 (default value).

    c=HYPeff(0.9,1);
    % In this case
    % c = 3.3139

  • Example of specifying k.
  • Find value of c for a nominal efficiency of 0.9 when k=4

    ktuning=4;
    c=HYPeff(0.9,1,ktuning);
    % In this case
    % c =  3.5443

  • Find parameters for fixed efficiency and k.
  • Find value of c, A, B, for a nominal efficiency of 0.8427 when k=4.5

    ktuning=4.5;
    [c,A,B,d]=HYPeff(0.8427,1,ktuning);
    % In this case
    % c = 3.000130564905703
    % A = 0.604298601602487
    % B = 0.713612241773758
    % d= 1.304379168746527
    % See also Table 2 of HRR p. 645

  • Example of use of option Find parameters for fixed efficiency and k.
  • Find value of c, A, B, for a nominal efficiency of 0.8427 when k=4.5

    ktuning=4.5;
    traceiter =true;
    [c,A,B,d]=HYPeff(0.8427,1,ktuning,traceiter);
    % In this case
    % c = 3.000130564905703
    % A = 0.604298601602487
    % B = 0.713612241773758
    % d= 1.304379168746527
    % See also Table 2 of HRR p. 645

  • Example to show the issue of multiple solutions problem.
  • 5 redescending psi functions are used and the Huber psi.

    load Income1;
    y=Income1{:,"HTOTVAL"};
    % Use contaminated income data 
    y=[y(1:20); 600000; 575000; 590000];
    y=y';
    mady=mad(y,1)/0.675;
    eff=0.95;
    TBc=TBeff(eff,1);
    HUc=HUeff(eff,1);
    HAc=HAeff(eff,1);
    HYPc=HYPeff(eff,1);
    OPTc=OPTeff(eff,1);
    PDc=PDeff(eff);
    mu=0:1000:700000;
    avePSI=zeros(length(mu),6);
    for i=1:length(mu)
    % aveTB(i,2)=mean(TBrho((y-mu(i))./mady,c));
    avePSI(i,1)=mean(HUpsi((y-mu(i))./mady,HUc));
    avePSI(i,2)=mean(HApsi((y-mu(i))./mady,HAc));
    avePSI(i,3)=mean(TBpsi((y-mu(i))./mady,TBc));
    avePSI(i,4)=mean(HYPpsi((y-mu(i))./mady,[HYPc,5]));
    avePSI(i,5)=mean(OPTpsi((y-mu(i))./mady,OPTc));
    avePSI(i,6)=mean(PDpsi((y-mu(i))./mady,PDc));
    end
    % Plotting part
    close
    Link={'Huber', 'Hampel', 'Tukey', 'Hyperbolic' 'Optimal' 'Power divergence'} ;
    for i=1:6
    subplot(2,3,i)
    plot(mu',avePSI(:,i),'LineWidth',2,'Color','k')
    hold('on')
    yline(0) %  line([min(mu);max(mu)],[0;0],'LineStyle',':')
    title(Link(i),'FontSize',14)
    xlabel('$\mu$','FontSize',14,'Interpreter','Latex')
    ylabel('$\overline \psi \left( \frac{ y -\mu}{\hat \sigma} \right)$','FontSize',14,'Interpreter','Latex')
    end

  • Compare the weight function for 6 different links.
  • FontSize=14;
    FontSizetitl=12;
    x=-6:0.01:6;
    ylim1=-0.05;
    ylim2=1.05;
    xlim1=min(x);
    xlim2=max(x);
    LineWidth=2;
    subplot(2,3,1)
    ceff05HU=HUeff(0.95,1);
    weiHU=HUwei(x,ceff05HU);
    plot(x,weiHU,'LineWidth',LineWidth)
    xlabel('$u$','Interpreter','Latex','FontSize',FontSize)
    title('Huber','FontSize',FontSizetitl)
    ylim([ylim1 ylim2])
    xlim([xlim1 xlim2])
    subplot(2,3,2)
    ceff095HA=HAeff(0.95,1);
    weiHA=HAwei(x,ceff095HA);
    plot(x,weiHA,'LineWidth',LineWidth)
    xlabel('$u$','Interpreter','Latex','FontSize',FontSize)
    title('Hampel','FontSize',FontSizetitl)
    ylim([ylim1 ylim2])
    xlim([xlim1 xlim2])
    subplot(2,3,3)
    ceff095TB=TBeff(0.95,1);
    weiTB=TBwei(x,ceff095TB);
    plot(x,weiTB,'LineWidth',LineWidth)
    xlabel('$u$','Interpreter','Latex','FontSize',FontSize)
    title('Tukey biweight','FontSize',FontSizetitl)
    ylim([ylim1 ylim2])
    xlim([xlim1 xlim2])
    subplot(2,3,4)
    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',FontSizetitl)
    ylim([ylim1 ylim2])
    xlim([xlim1 xlim2])
    subplot(2,3,5)
    ceff095OPT=OPTeff(0.95,1);
    % ceff095OPT=ceff095OPT/3;
    weiOPT=OPTwei(x,ceff095OPT);
    weiOPT=weiOPT/max(weiOPT);
    plot(x,weiOPT,'LineWidth',LineWidth)
    xlabel('$u$','Interpreter','Latex','FontSize',FontSize)
    title('Optimal','FontSize',FontSizetitl)
    ylim([ylim1 ylim2])
    xlim([xlim1 xlim2])
    subplot(2,3,6)
    ceff095PD=PDeff(0.95);
    weiPD=PDwei(x,ceff095PD);
    weiPD=weiPD/max(weiPD);
    plot(x,weiPD,'LineWidth',LineWidth)
    xlabel('$u$','Interpreter','Latex','FontSize',FontSize)
    title('Power divergence','FontSize',FontSizetitl)
    ylim([ylim1 ylim2])
    xlim([xlim1 xlim2])

    Input Arguments

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    eff — efficiency. Scalar.

    Scalar which contains the required efficiency (of location or scale estimator).

    Generally eff=0.85, 0.9 or 0.95

    Data Types: single| double

    v — number of response variables. Scalar.

    Number of variables of the dataset (for regression v=1) UP TO NOW v=1 (JUST REGRESSION) TO DO FOR MULTIVARIATE ANALYSIS

    Data Types: single| double

    Optional Arguments

    k — supremum of the change of variance curve. Scalar.

    $\sup CVC(psi,x) x \in R$ Default value is k=4.5.

    Example: 5

    Data Types: double

    traceiter — Level of display. Scalar.

    If traceiter = 1 it is possible to monitor how the value of the objective function B^2/A gets closer to the target (eff) during the iterations

    Example: 'traceiter',0

    Data Types: double

    Output Arguments

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    c —parameter c of hyperbolic tangent estimator. Scalar

    For more details see the methodological details inside "More About" below

    A —parameter A of hyperbolic tangent estimator. Scalar

    For more details see the methodological details inside "More About" below

    B —parameter B of hyperbolic tangent estimator. Scalar

    For more details see the methodological details inside "More About" below

    d —parameter d of hyperbolic tangent estimator. Scalar

    For more details see the methodological details inside "More About" below

    More About

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    Additional Details

    \[ HYPpsi(u) = \left\{ \begin{array}{cc} u & |u| \leq d \\ \sqrt{A (k - 1)} \tanh \left( \sqrt{(k - 1) B^2/A} (c -|u|)/2 \right) sign(u) & d \leq |u| < c, \\ 0 & |u| \geq c. \end{array} \right. \] It is necessary to have $0 < A < B < 2 normcdf(c)-1- 2 c \times normpdf(c) <1$

    References

    Hampel, F.R., Rousseeuw, P.J. and Ronchetti E. (1981), The Change-of-Variance Curve and Optimal Redescending M-Estimators, "Journal of the American Statistical Association", Vol. 76, pp. 643-648. [HRR]

    See Also

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