# HApsix

HApsix computes psi function using Hampel proposal times x

## Syntax

• psiHAx=HApsix(u, ctuning)example

## Description

 psiHAx =HApsix(u, ctuning) Plot of psi(x) multiplied by x.

## Examples

expand all

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

Obtain bottom panel of Figure 11.10. p. 375 of Hoaglin et al. (1987)

x=-9:0.1:9;
psiHAx=HApsix(x,1);
plot(x,psiHAx)
xlabel('x','Interpreter','Latex')
ylabel(' Hampel $\psi(x) \times x$','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

### ctuning — tuning parameters. Scalar or Vector.

Scalar or vector of length 4 which specifies the value of the tuning constant c (scalar greater than 0 which controls the robustness/efficiency of the estimator) and the prefixed values of paramters a, b, c ctuning(1) = tuning constant which will multiply parameters a, b and c of Hampel rho (psi) function ctuning(2) = paramter a of Hampel rho (psi) function ctuning(3) = paramter b of Hampel rho (psi) function ctuning(4) = paramter c of Hampel rho (psi) function Remark: if length(ctuning)==1 values of a, b and c will be set to a=2*ctuning b=4*ctuning c=4*ctuning With these choices, if ctuning=1 the resulting influence function is nearly identical to the biweight with parameter 8.

Data Types: single| double

## Output Arguments

### psiHAx —Values of Hampel psi(u)*u function associated to the residuals or Mahalanobis distances for the n units of the sample. n -by- 1 vector

Function HApsix transforms vector u as follows $HApsi(u) = \left\{ \begin{array}{cc} u^2 & |u| <= a \\ a \times sign(u)u & a <= |u| < b \\ a \frac{c-|u|}{c-b} \times sign(u)\times u & b <= |u| < c \\ 0 & |u| >= c \end{array} \right.$

where $a$= ctuning(2) *ctuning(1).

$b$= ctuning(3) *ctuning(1).

$c$= ctuning(4) *ctuning(1).

The default (if input ctuning is a scalar) is $a$= 2*ctuning.

$b$= 4*ctuning.

$c$= 8*ctuning.

It is necessary to have 0 <= $a$ <= $b$ <= $c$

## References

Hoaglin, D.C., Mosteller, F., Tukey, J.W. (1982), "Understanding Robust and Exploratory Data Analysis", Wiley, New York.