# TBeff

Tbeff finds the constant c which is associated to the requested efficiency for Tukey's biweight

## Syntax

• ceff=TBeff(eff, v)example
• ceff=TBeff(eff, v, shapeeff)example
• ceff=TBeff(eff, v, shapeeff, approxsheff)example

## Description

 ceff =TBeff(eff, v) Find c for a given efficiency.

 ceff =TBeff(eff, v, shapeeff)

 ceff =TBeff(eff, v, shapeeff, approxsheff) Find shape efficiency.

## Examples

expand all

### Find c for a given efficiency.

The constant c associated to a nominal location efficiency of 95% in regression is c = 4.685064948559557

c=TBeff(0.95,1)

c=TBeff(0.95,1,1)

### Find shape efficiency.

Compare constant c for the range of values of p from 1 to 200 using approximate and exact formula for fixed shape efficiency Number of variables

p=200;
pp=1:1:p;
% Initialize the matrix which stores the values of c for the two
% methods
cc=[pp' zeros(p,2)];
eff=0.75;
% eff=0.99;
for i=pp
% Use exact formula for finding the value of c associated to a fixed
% level of shape efficiency
cc(i,2)=TBeff(eff,i,1);
% Use approximate formula for finding the value of c associated to a fixed
% level of shape efficiency
cc(i,3)=TBeff(eff,i,1,1);
end
figure
plot(cc(:,1),cc(:,2),'LineStyle','-','LineWidth',2)
hold('on')
plot(cc(:,1),cc(:,3),'LineStyle','-.','LineWidth',2)
legend('True value of c', 'Approximate value of c','Location','best')
xlabel('Number of variables')
ylabel('Value of c')

## Input Arguments

### eff — required efficienty. Scalar.

Scalar which contains the required efficiency (of location or scale estimator).

Data Types: single|double Generally eff=0.85, 0.9 or 0.95

### v — Number of response variables. Scalar.

e.g. in regression p=1

Data Types: single|double|int32|int64

### shapeeff — Location or shape efficiency. Scalar.

If 1, the efficiency is referred to shape else (default) is referred to location

Example: 'shapeeff',1 

Data Types: double

### approxsheff — approximate or exact calculations. Dummy scalar.

If 1, when p > 1 the approximate formula for scale efficiency is used else (default) the exact formula of the variance of the robust estimator of the scale is used

Example: 'approxsheff',1 

Data Types: double

## Output Arguments

### ceff —Requested tuning constant. Scalar

Tuning constatnt of Tukey Biweigh rho function associated to requested value of efficiency

## References

Maronna, R.A., Martin D. and Yohai V.J. (2006), "Robust Statistics, Theory and Methods", Wiley, New York.