# HUpsix

HUpsix computes psi function (derivative of rho function) times x for Huber

## Syntax

• psix=HUpsix(u,c)example

## Description

 psix =HUpsix(u, c) Plot of psi(x) function multiplied by x.

## Examples

expand all

### Plot of psi(x) function multiplied by x.

% Plot of psi(x) function multiplied  by x.
x=-6:0.01:6;
psixHU=HUpsix(x,2);
plot(x,psixHU)
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

### psix —psi(u) function multiplied by u.  Vector

n-by-1 vector which contains the values of HU psi(u)*u function associated to the residuals or Mahalanobis distances for the n units of the sample.

Function HUpsix transforms vector u as follows $HUpsix(u)= \left\{ \begin{array}{cc} u^2 & \mbox{if } |u/c| \leq 1 \\ c \times \mbox{sign}(u) u & |u/c|>1 \\ \end{array} \right.$

See equation (2.38) p. 29 of Maronna et al. (2006) Remark: Tukey's biweight 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.