# TBpsider

TBpsider computes derivative of psi function (second derivative of rho function) for Tukey's biweight

## Syntax

• psider=TBpsider(u,c)example

## Description

 psider =TBpsider(u, c) Plot the derivative of Tukey's biweght psi function.

## Examples

expand all

### Plot the derivative of Tukey's biweght psi function.

x=-6:0.01:6;
c=1.5476;
psiTBder=TBpsider(x,c);
plot(x,psiTBder)
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

### psider —derivative of psi function.  Vector

n x 1 vector which contains the values of the derivative of the Tukey biweight psi function associated to the residuals or Mahalanobis distances for the n units of the sample.

Function TBpsider transforms vector x as follows $TBpsider(x)= \left\{ \begin{array}{cc} 1- (x/c)^2 [6- 5 (x/c)^2] & \mbox{if} |x/c|<=1 \\ 0 & \mbox{if} |x/c|>1 \\ \end{array} \right.$

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.