# OPTpsix

OPTpsix computes psi function (derivative of rho function) times x

## Syntax

• psix=OPTpsix(u,c)example

## Description

 psix =OPTpsix(u, c) Plot of psi(x) function (derivative of rho function) times x.

## Examples

expand all

### Plot of psi(x) function (derivative of rho function) times x.

x=-6:0.01:6;
psixOPT=OPTpsix(x,1.2);
plot(x,psixOPT)
xlabel('x','Interpreter','Latex')
ylabel('$\psi (x) \times x$','Interpreter','Latex')

## Input Arguments

### 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 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

## More About

### Additional Details

Function OPTpsix transforms vector u as follows $\psi(x)x =x\rho' (x) = \begin{cases} \frac{2.7692 x^2}{c^2} \qquad |x| \leq \frac{2}{3} c \\ -\frac{5.3834 x^2}{c^2} +\frac{43.0672 x^4}{c^4} -\frac{69.9840 x^6}{c^6} +\frac{32.3 x^8}{c^8} \qquad \frac{2}{3} c < |x| \leq c \\ 0 & \; \vert x \vert > c. \end{cases}$ Remark: function $\psi(x) *x$ has been precalculated because it is used to find the weights in case of Tau estimators

## References

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

## See Also

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