ASpsider

ASpsider computes derivative of psi function (second derivative of rho function) for Andrew's sine function

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Syntax

Description

example

psiderAS =ASpsider(u, c) Plot the derivative of Andrew's psi function (when bdp=0.5).

Examples

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  • Plot the derivative of Andrew's psi function (when bdp=0.5).

    • x=-6:0.01:6;
c=ASbdp(0.5,1);
psiASder=ASpsider(x,c);
plot(x,psiASder)
xlabel('x','Interpreter','Latex')
ylabel('$\psi''(x)$','Interpreter','Latex')
    Click here for the graphical output of this example (link to Ro.S.A. website).

    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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    psiderAS —derivative of psi function. Vector

    n x 1 vector which contains the values of the derivative of the Andrew's psi function associated to the residuals or Mahalanobis distances for the n units of the sample.

    More About

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    Additional Details

    Function ASpsider transforms vector x as follows \[ ASpsi(u)= \left\{ \begin{array}{cc} (1/c) \cos(u/c) & |u/c| \leq \pi \\ 0 & |u/c|> \pi \\ \end{array} \right. \]

    Remark: Andrew's sine functionon 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

    Andrews, D.F., Bickel, P.J., Hampel, F.R., Huber, P.J., Rogers, W.H., and Tukey, J.W. (1972), "Robust Estimates of Location: Survey and Advances", Princeton Univ. Press, Princeton, NJ. [p. 203]

    Andrews, D. F. (1974). A Robust Method for Multiple Linear Regression, "Technometrics", V. 16, pp. 523-531, https://doi.org/10.1080/00401706.1974.10489233

    See Also

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