TBpsi

TBpsi computes psi function (derivative of rho function) for Tukey's biweight

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Syntax

Description

example

psi =TBpsi(u, c) Plot of psi function.

Examples

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  • Plot of psi function.

    • close all
x=-6:0.01:6;
c=2;
psiTB=TBpsi(x,c);
plot(x,psiTB,'LineWidth',2)
xlabel('$u$','Interpreter','Latex')
ylabel('$\psi(u,2)$','Interpreter','Latex','FontSize',14)
hold('on')
ax=axis;
line([-c;-c],[ax(3);0],'LineStyle',':','LineWidth',1)
line([c;c],[ax(3);0],'LineStyle',':','LineWidth',1)

    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

    Output Arguments

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    psi —Tukey's psi associated to the residuals or Mahalanobis distances for the n units of the sample. n -by- 1 vector

    More About

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    Additional Details

    Function TBpsi transforms vector u as follows \[ TBpsi(u)= \left\{ \begin{array}{cc} (c^2/6) u[1-(u/c)^2]^2 & |u/c| \leq 1 \\ 0 & |u/c|>1 \\ \end{array} \right. \]

    See equation (2.38) p. 29 of Maronna et al. (2006) Remark: Tukey's biweight 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.

    Riani, M., Cerioli, A. and Torti, F. (2014), On consistency factors and efficiency of robust S-estimators, "TEST", Vol. 23, pp. 356-387.

    http://dx.doi.org/10.1007/s11749-014-0357-7

    See Also

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