HYPpsider

HYPpsider computes derivative of psi function for hyperbolic tangent estimator

Syntax

  • psiHYPder=HYPpsider(u, cktuning)example

Description

example

psiHYPder =HYPpsider(u, cktuning) Plot of derivative of hyperbolic psi function.

Examples

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  • Plot of derivative of hyperbolic psi function.
  • % Plot of derivative of hyperbolic psi function.
    x=-9:0.1:9;
    ctuning=6;
    ktuning=4.5;
    psiHYPder=HYPpsider(x,[ctuning,ktuning]);
    plot(x,psiHYPder)
    xlabel('x','Interpreter','Latex')
    ylabel(' Hyperbolic $\psi''(x) $','Interpreter','Latex')

    Related Examples

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  • Comparison among four derivatives of psi function.
  • TB, Optimal, Hampel, Hyperbolic

    bdp=0.5;
    c=TBbdp(bdp,1);
    subplot(2,2,1)
    % 1st panel is Tukey biweight
    x=-4:0.01:4;
    psiTBder=TBpsider(x,c);
    plot(x,psiTBder)
    xlabel('x','Interpreter','Latex')
    ylabel('$\psi''(x)$','Interpreter','Latex')
    title('Tukey biweight')
    subplot(2,2,2)
    % 2nd panel is optimal
    c=OPTbdp(bdp,1);
    c=c/3;
    % Remark: in this case it is necessary to multiply by 3.25*c^2 because the
    % optimal has been standardized in such a way that sup(rho(x))=1
    psiOPTder=(3.25*c^2)*OPTpsider(x,c);
    plot(x,psiOPTder)
    xlim([-4 4])
    xlabel('x','Interpreter','Latex')
    ylabel('$\psi''(x)$','Interpreter','Latex')
    title('Optimal')
    subplot(2,2,3)
    % 3rd panel is Hampel
    % Obtain bottom panel of Figure 11.10 p. 375 of
    % Hoaglin et al. (1987)
    c=HAbdp(bdp,1);
    psiHA=HApsider(x,c);
    plot(x,psiHA)
    xlabel('x','Interpreter','Latex')
    ylabel('$\psi''(x) $','Interpreter','Latex')
    title('Hampel')
    subplot(2,2,4)
    % 4th panel is hyperbolic
    % c=HYPbdp(0.5,1);
    %ktuning=4.5;
    ktuning=4.5;
    % Precalculated values
    c = 2.010311082005501;
    A = 0.008931591866092;
    B = 0.051928487236632;
    d=  0.132017481327058;
    % Alternatively the values can be found using
    %[c,A,B,d]=HYPbdp(0.5,1,ktuning);
    psiHYPder=HYPpsider(x,[c,ktuning,A,B,d]);
    plot(x,psiHYPder)
    xlabel('x','Interpreter','Latex')
    ylabel('$\psi''(x) $','Interpreter','Latex')
    title('Hyperbolic')

    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

    cktuning — tuning parameters. Vector of length 2 or of length 5.

    cktuning specifies the value of the tuning constant c (scalar greater than 0 which controls the robustness/efficiency of the estimator) and the prefixed value k (sup of the change-of-variance sensitivity) and the values of parameters A, B and d.

    cktuning(1) = c;

    cktuning(2) = k = supCVC(psi,x) x \in R;

    cktuning(3)=A;

    cktuning(4)=B;

    cktuning(5)=d;

    Remark - if length(cktuning)==2 values of A, B and d will be computed automatically

    Data Types: single| double

    Output Arguments

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    psiHYPder —Derivative of psi function. Vector

    n x 1 vector which contains the values of hyperbolic psi'(u) function associated to the residuals or Mahalanobis distances for the n units of the sample.

    More About

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    Additional Details

    Function HYPpsi transforms vector u as follows \[ HYPpsider(u)= \left\{ \begin{array}{cc} 1 & |u| \leq d, \\ 0.5 B (1-k) \left( 1/\cosh \left( \sqrt{(k - 1) B^2/A} (c -|u|)/2 \right)^2 \right) & d \leq |u| < c, \\ 0 & |u| \geq c. \end{array} \right. \] It is necessary to have $0 < A < B < 2 normcdf(c)-1- 2 c normpdf(c) <1$

    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]'

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