OPTwei computes weight function psi(u)/u for optimal weight function




w =OPTwei(u, c) Plot of weight function.


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  • Plot of weight function.
  • x=-6:0.1:6;
    ylabel('$W (x) =\psi(x)/x$','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 OPTwei transforms vector u as follows Remark: Yohai and Zamar (1997) showed that the optimal $\rho$ function is optimal in the following highly desirable sense: the final M estimate has a breakdown point of one-half and minimizes the maximum bias under contamination distributions (locally for small fraction of contamination), subject to achieving a desidered nominal asymptotic efficiency when the data are Gaussian.

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


    Yohai V.J., Zamar R.H. (1997) Optimal locally robust M-estimates of regression. "Journal of Planning and Statistical Inference", Vol. 64, pp. 309-323.

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