ASrho

ASrho computes rho function for Andrew's sine function

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Syntax

Description

example

rhoAS =ASrho(u, c) Plot of rho function.

Examples

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  • Plot of rho function.

    • close all
x=-7:0.01:7;
c=2;
rhoAS=ASrho(x,c);
plot(x,rhoAS,'LineWidth',2)
xlabel('$u$','Interpreter','Latex')
ylabel('$\rho (u,2)$','Interpreter','Latex')
text(-c*pi-1,2*c-0.1,'2*c')
text(+c*pi+0.5,2*c-0.1,'2*c')
title('$\rho (u,c)$','Interpreter','Latex')
hold('on')
stem(c*pi,2*c,'LineStyle',':','LineWidth',1)
stem(-c*pi,2*c,'LineStyle',':','LineWidth',1)
text(c*pi-0.8,0.1,'c \pi')
text(-c*pi+0.2,0.1,'-c \pi')
    Click here for the graphical output of this example (link to Ro.S.A. website).

    Related Examples

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  • Compare two rho functions for 2 different values of c.

    • In the first we fix the bdp (value of efficiency is automatically given), while in the second we find the efficiency (the value of bdp is automatically given) 

    close all
x=-6:0.01:6;
lwd=2;
hold('on')
c=ASbdp(0.5,1);
rhoAS=ASrho(x,c);
rhoAS=rhoAS/max(rhoAS);
plot(x,rhoAS,'LineStyle','-','LineWidth',lwd)
c=ASeff(0.95,1);
rhoAS=ASrho(x,c);
rhoAS=rhoAS/max(rhoAS);
plot(x,rhoAS,'LineStyle','-.','LineWidth',lwd)
xlabel('$x$','Interpreter','Latex','FontSize',16)
ylabel('AS. Normalized $\rho_c(x)$','Interpreter','Latex','FontSize',20)
legend({'$c_{(bdp=0.5 \mapsto eff=0.2856)}$', '$c_{(eff=0.95 \mapsto bdp= 0.1217)}$'},'Interpreter','Latex','Location','SouthEast','FontSize',16)
    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.

    Vector 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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    rhoAS —Andrew's sine function rho associated to the residuals or Mahalanobis distances for the n units of the sample. vector of length n

    More About

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    Additional Details

    function ASrho transforms vector u as follows \[ ASrho(u)= \left\{ \begin{array}{cc} c (1-\cos (u / c)) & |u/c| \leq \pi \\ 2c & |u/c| > \pi \\ \end{array} \right. \]

    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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