OPTwei

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

Syntax

Description

example

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

Examples

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  • Plot of weight function.
  • x=-6:0.1:6;
    weiOPT=OPTwei(x,2);
    plot(x,weiOPT)
    xlabel('x','Interpreter','Latex')
    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 \[ OPTwei(u,c) = \left\{ \begin{array}{cc} 1/(3.25*c^2) & |u| \leq 2 \\ (1/3.25) \left( -1.944 * 1 / c^2 + 1.728 \frac{u^2}{c^4} - 0.312\frac{u^2}{c^6} + 0.016 \frac{u.^6}{c^8} \right) & \qquad 2c\leq |u|\leq 3c \\ 0 \end{array} \right. \]

    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.

    References

    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.

    See Also

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