OPTpsider

OPTpsider computes derivative of psi function (second derivative of rho function) for optimal weight function

Syntax

Description

example

psider =OPTpsider(u, c) Plot of derivative of psi function.

Examples

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  • Plot of derivative of psi function.
  • x=-6:0.01:6;
    psiOPTder=OPTpsider(x,1.2);
    plot(x,psiOPTder)
    xlabel('x','Interpreter','Latex')
    ylabel('$\psi''(x)$','Interpreter','Latex')
    title('Optimal')

    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 OPTpsider transforms vector x as follows \[ OPTpsider(u,c) = \left\{ \begin{array}{cc} (1/3.25*c^2) & |u| \leq 2c \\ (1/3.25) \left( -1.944 \frac{1}{c^2} + 1.728 \frac{3u^2}{c^4} - 0.312 \frac{5u^4}{c^6} + 0.016 \frac{7u^6}{c^8} \right) & \qquad 2c \leq |u| \leq 3c \\ 0 & |u|>3c \\ \end{array} \right. \]

    Remark: Optimal psi-function is almost linear around u = 0 in accordance with Winsor's principle that all distributions are normal in the middle.

    This means that \psi (u)/u is approximately constant over the linear region of \psi, so the points in that region tend to get equal weight.

    References

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

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