# HArho

HArho computes rho function using Hampel proposal

## Syntax

• rhoHA=HArho(u, ctuning)example

## Description

 rhoHA =HArho(u, ctuning) Plot of rho function.

## Examples

expand all

### Plot of rho function.

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

x=-9:0.1:9;
rhoHA=HArho(x,1);
plot(x,rhoHA,'LineWidth',2)
xlabel('$u$','Interpreter','Latex')
ylabel(' Hampel $\rho(u,[2, 4, 8])$','Interpreter','Latex','FontSize',14)

## Related Examples

expand all

### Hampel rho function using a redescending slope of -1/3.

% Hampel rho function using a redescending slope of -1/3.
x=-9:0.1:9;
rhoHA=HArho(x,[1,1.5,3.5,8]);
plot(x,rhoHA)
xlabel('x','Interpreter','Latex')
ylabel(' Hampel $\rho(x)$','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

### rhoHA —Hampel rho associated to the residuals or Mahalanobis distances for the n units of the sample. n -by- 1 vector

Function HArho transforms vector u as follows $HArho(u) = \left\{ \begin{array}{cc} \frac{u^2}{2} & |u| \leq a \\ a \times |u| -0.5 a^2 & a \leq |u| < b \\ ab-0.5a^2+0.5(c-b)a \left[ 1- \left( \frac{c-|u|}{c-b}\right)^2 \right] & b \leq |u| < c \\ ab-0.5a^2+0.5(c-b)a & |u| \geq c \end{array} \right.$

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

$b$= ctun *ctuning(3).

$c$= ctun *ctuning(4).

The default is $a$= 2*ctun.

$b$= 4*ctun.

$c$= 8*ctun.

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.