HApsix

HApsix computes psi function using Hampel proposal times x

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Syntax

Description

example

psiHAx =HApsix(u, ctuning) Plot of psi(x) multiplied by x.

Examples

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  • Plot of psi(x) multiplied by x.

    • Obtain bottom panel of Figure 11.10. p. 375 of Hoaglin et al. (1987) 

    x=-9:0.1:9;
psiHAx=HApsix(x,1);
plot(x,psiHAx)
xlabel('x','Interpreter','Latex')
ylabel(' Hampel $\psi(x) \times 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

    ctuning — tuning parameters. Scalar or Vector.

    Scalar or vector of length 4 which specifies the value of the tuning constant c (scalar greater than 0 which controls the robustness/efficiency of the estimator) and the prefixed values of paramters a, b, c ctuning(1) = tuning constant which will multiply parameters a, b and c of Hampel rho (psi) function ctuning(2) = paramter a of Hampel rho (psi) function ctuning(3) = paramter b of Hampel rho (psi) function ctuning(4) = paramter c of Hampel rho (psi) function Remark: if length(ctuning)==1 values of a, b and c will be set to a=2*ctuning b=4*ctuning c=4*ctuning With these choices, if ctuning=1 the resulting influence function is nearly identical to the biweight with parameter 8.

    Data Types: single| double

    Output Arguments

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    psiHAx —Values of Hampel psi(u)*u function 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 HApsix transforms vector u as follows \[ HApsi(u) = \left\{ \begin{array}{cc} u^2 & |u| <= a \\ a \times sign(u)u & a <= |u| < b \\ a \frac{c-|u|}{c-b} \times sign(u)\times u & b <= |u| < c \\ 0 & |u| >= c \end{array} \right. \]

    where $a$= ctuning(2) *ctuning(1).

    $b$= ctuning(3) *ctuning(1).

    $c$= ctuning(4) *ctuning(1).

    The default (if input ctuning is a scalar) is $a$= 2*ctuning.

    $b$= 4*ctuning.

    $c$= 8*ctuning.

    It is necessary to have 0 <= $a$ <= $b$ <= $c$

    References

    Hoaglin, D.C., Mosteller, F., Tukey, J.W. (1982), "Understanding Robust and Exploratory Data Analysis", Wiley, New York.

    See Also

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