OPTpsix

OPTpsix computes psi function (derivative of rho function) times x

Syntax

Description

example

psix =OPTpsix(u, c) Plot of psi(x) function (derivative of rho function) times x.

Examples

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  • Plot of psi(x) function (derivative of rho function) times x.
  • x=-6:0.01:6;
    psixOPT=OPTpsix(x,1.2);
    plot(x,psixOPT)
    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

    c — tuning parameters. Scalar.

    Scalar greater than 0 which controls the robustness/efficiency of the estimator (beta in regression or mu in the location case ...)

    Data Types: single| double

    More About

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    Additional Details

    Function OPTpsix transforms vector u as follows \[ OPTpsix(u,c) = \left\{ \begin{array}{cc} \frac{1}{3.25*c^2} u^2 & |u| \leq 2c \\ (1/3.25) * (-1.944 * \frac{u^2}{c^2} + 1.728 \frac{u^4}{c^4} - 0.312 \frac{x^6}{c^6} + 0.016 \frac{x^8}{c^8} & \qquad 2c \leq |u| \leq 3c \\ 0 & |u|>3c \\ \end{array} \right. \] Remark: function $\psi(x) *x$ has been precalculated because it is used to find the weights in case of Tau estimators

    References

    Maronna, R.A., Martin D. and Yohai V.J. (2006), "Robust Statistics, Theory and Methods", Wiley, New York.

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