HYPeff

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

Syntax

Description

example

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

example

c =HYPeff(eff, v, k) Example of use of option Find parameters for fixed efficiency and k.

example

c =HYPeff(eff, v, k, traceiter)

example

[c, A] =HYPeff(___)

example

[c, A, B] =HYPeff(___)

example

[c, A, B, d] =HYPeff(___)

Examples

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  • 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

    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: '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 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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