# OPTpsi

OPTpsi computes psi function (derivative of rho function) for optimal weight function

## Syntax

• psiOPT=OPTpsi(u,c)example

## Description

 psiOPT =OPTpsi(u, c) Plot of psi function (derivative of rho function) for optimal weight function.

## Examples

expand all

### Plot of psi function (derivative of rho function) for optimal weight function.

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

## Input Arguments

### 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

### psiOPT —Values of optimal psi function associated to the residuals or Mahalanobis distances for the n units of the sample. n -by- 1 vector

function OPTpsi transforms vector u as follows $OPTpsi(u,c) = \left\{ \begin{array}{cc} \frac{u}{3.25*c^2} & |u| \leq 2c \\ = (1/3.25) \left( -1.944 \frac{u}{c^2} + 1.728 \frac{u^3}{c^4} - 0.312 \frac{u^5}{c^6} + 0.016 \frac{u^7}{c^8} \right) & \qquad 2c \leq |u| \leq 3c \\ 0 & |u|>3c \\ \end{array} \right.$

Remark: Optimal 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.