# TBpsix

TBpsix computes psi function (derivative of rho function) times x for Tukey's biweight

## Syntax

• psix=TBpsix(u,c)example

## Description

 psix =TBpsix(u, c) Plot of psi function (derivative of rho function) times x for Tukey's biweight.

## Examples

expand all

### Plot of psi function (derivative of rho function) times x for Tukey's biweight.

x=-6:0.01:6;
psixTB=TBpsix(x,2);
plot(x,psixTB)
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 —Values of TB psi(u)*u function associated to the residuals or Mahalanobis distances for the n units of the sample. n -by- 1 vector

Function TBpsix transforms vector u as follows $TBpsix(u)= \left\{ \begin{array}{cc} u^2[1-(u/c)^2]^2 if |u/c| \leq 1 \\ 0 & |u/c|>1 \\ \end{array} \right.$

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.