OPTrho computes rho function for optimal weight function




rhoOPT =OPTrho(u, c) Plot of rho function.


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  • Plot of rho function.
  • x=-6:0.01:6;
    ylabel('$\rho (x)$','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 parameter. 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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    rhoOPT —residuals after rho filter. Vector

    Vector of length n which contains optima rho values associated to the residuals or Mahalanobis distances for the n units of the sample.

    Function OPTrho transforms vector u as follows

    More About

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    Additional Details

    Yohai and Zamar (1997) showed that the $\rho$ function given above 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.

    \[ \label{opt} \rho(x) = \begin{cases} 1.3846 \left( \frac{x}{c} \right)^2 \qquad |x| \leq \frac{2}{3} c \\ 0.5514-2.6917\left( \frac{x}{c} \right)^2+10.7668\left( \frac{x}{c} \right)^4-11.6640\left( \frac{x}{c} \right)^6+4.0375\left( \frac{x}{c} \right)^8 \qquad \frac{2}{3} c < |x| \leq c \\ 1 \qquad |x| >c \end{cases} \]


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

    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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