# HUpsi

HUpsi computes psi function (derivative of rho function) for Huber

## Syntax

• psi=HUpsi(u,c)example

## Description

 psi =HUpsi(u, c) Plot of psi function.

## Examples

expand all

### Plot of psi function.

close all
x=-6:0.01:6;
c=1.345;
psiHU=HUpsi(x,c);
plot(x,psiHU,'LineWidth',2)
xlabel('$u$','Interpreter','Latex')
ylabel('$\psi (u,1.345)$','Interpreter','Latex','FontSize',14)
text(-c,-c,'-c=-1.345')
text(c,c+0.1,'c=1.345')
hold('on')
stem(c,c,'LineStyle',':','LineWidth',2)
stem(-c,-c,'LineStyle',':','LineWidth',2)

## 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

### psi —Huber's psi associated to the residuals or Mahalanobis distances for the n units of the sample. n -by- 1 vector

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

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

Riani, M., Cerioli, A. and Torti, F. (2014), On consistency factors and efficiency of robust S-estimators, "TEST", Vol. 23, pp. 356-387.