HYPbdp

HYPbdp finds constant c which is associated to the requested breakdown point for hyp. tan. estimator

Syntax

Description

example

c =HYPbdp(bdp, v) HYPbdp with all default options.

example

c =HYPbdp(bdp, v, k) Find value of c, A, B, for fixed break down point.

example

c =HYPbdp(bdp, v, k, traceiter) Find value of c, A, B, for fixed break down point.

example

[c, A] =HYPbdp(___) Efficienty monitoring.

example

[c, A, B] =HYPbdp(___) Analysis of efficiency as a function of k =sup CVC.

example

[c, A, B, d] =HYPbdp(___) Example of use of option traceiter.

Examples

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  • HYPbdp with all default options.
  • Find required value of c when k=4.5 and bdp=0.5.

    c=HYPbdp(0.5,1);

  • Find value of c, A, B, for fixed break down point.
  • Find required values when k=4.5 and bdp=0.5.

    [c,A,B,d]=HYPbdp(0.5,1);
    % In this case
    % c = 2.010311082005501
    % A = 0.008931591866092
    % B = 0.051928487236632
    % d=  0.132017481327058

  • Find value of c, A, B, for fixed break down point.
  • Find required values when k=4.5 and bdp=0.5.

    ktuning=4.2;
    [c,A,B,d]=HYPbdp(0.5,1,ktuning);
    % In this case
    % c =  2.093345907330513
    % A = 0.002411382994023
    % B = 0.026962054035832
    % d=  0.066725444793702

  • Efficienty monitoring.
  • Analysis of efficiency and of paramters A, B abd k as function of bdp for a given value of sup CVC=4

    seqi=0.1:0.1:0.5;
    eff=[seqi' zeros(length(seqi),4)];
    iter=0;
    k=4;
    for i=seqi
    [c,A,B,d] = HYPbdp(i,1,k);
    iter=iter+1;
    eff(iter,2:5)=[B^2/A A B d];
    end
    subplot(2,2,1)
    plot(eff(:,1),eff(:,2))
    title('efficiency')
    subplot(2,2,2)
    plot(eff(:,1),eff(:,3))
    title('A')
    subplot(2,2,3)
    plot(eff(:,1),eff(:,4))
    title('B')
    subplot(2,2,4)
    plot(eff(:,1),eff(:,5))
    title('d')

  • Analysis of efficiency as a function of k =sup CVC.
  • ktun=[2; 3; 4; 5];
    eff=zeros(length(ktun),1);
    for i=1:length(ktun)
    [~,A1,B1]=HYPbdp(0.3,1,ktun(i));
    eff(i)=B1^2/A1;
    end
    % Efficiency increases as sup CVC increases.
    table(ktun,eff)

  • Example of use of option traceiter.
  • traceiter=1
    ktuning=4.5;
    [c,A,B,d]=HYPbdp(0.4,1,ktuning,traceiter);

    Input Arguments

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    bdp — requested breakdown point. Scalar.

    Scalar defining breakdown point (i.e a number between 0 and 0.5)

    Data Types: single|double

    v — number of response variables. Scalar.

    (e.g. in regression p=1)

    Data Types: single|double|int32|int64

    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: 'k',5

    Data Types: double

    traceiter — Level of display. Scalar.

    If traceiter = 1 it is possible to monitor how the value of the objective function E(rho)/\rho(\infty) gets closer to the target (bdp) during the iterations

    Example: 'traceiter',0

    Data Types: single|double|int32|int64

    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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