HYPpsix

HYPpsix computes psi function for hyperbolic tangent estimator times x

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Syntax

  • psiHYPx=HYPpsix(u, cktuning)example

Description

example

psiHYPx =HYPpsix(u, cktuning) plot of psi(x)*x for Hyperbolic estimator.

Examples

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  • plot of psi(x)*x for Hyperbolic estimator.

    • % plot of psi(x)*x for Hyperbolic estimator.
x=-9:0.1:9;
ctuning=6;
ktuning=4.5;
psiHYPx=HYPpsix(x,[ctuning,ktuning]);
plot(x,psiHYPx)
xlabel('x','Interpreter','Latex')
ylabel('$\psi (x) \times x$','Interpreter','Latex')

    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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    psiHYPx —psi(u)*u function. Vector

    n x 1 vector which contains the values of hyperbolic psi(u)*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 HYPpsix transforms vector $u$ as follows \[ HYPpsix(u) = \left\{ \begin{array}{cc} u^2 & |u| \leq d \\ \sqrt{A (k - 1)} \tanh \left( \sqrt{(k - 1) B^2/A} (c -|u|)/2 \right) sign(u) u & d \leq |u| < c, \\ 0 & |u| \geq c. \end{array} \right. \] It is necessary to have $0 < A < B < 2 normcdf(c)-1- 2 c normpdf(c) <1$

    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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