OPTpsix

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

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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.

    vector of length n 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

    Output Arguments

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    psix —Values of the derivative of the optimal psi function multiplied by u, associated to the residuals or Mahalanobis distances for the n units of the sample. vector of length n

    More About

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    Additional Details

    Function OPTpsix transforms vector u as follows \[ \psi(x)x =x\rho' (x) = \begin{cases} \frac{2.7692 x^2}{c^2} \qquad |x| \leq \frac{2}{3} c \\ -\frac{5.3834 x^2}{c^2} +\frac{43.0672 x^4}{c^4} -\frac{69.9840 x^6}{c^6} +\frac{32.3 x^8}{c^8} \qquad \frac{2}{3} c < |x| \leq c \\ 0 & \; \vert x \vert > c. \end{cases} \] 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.

    See Also

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