# RKbdp

RKbdp finds the constants c associated to the supplied breakdown point and asymptotic rejection point

## Description

c =RKbdp(bdp, v) Find c and M given bdp and ARP.

c =RKbdp(bdp, v, ARP) Computation of c and M for a series of values of bdp (v=3).

[c, M] =RKbdp(___) Computation of c and M for a series of values of bdp (v=10).

## Examples

expand all

### Find c and M given bdp and ARP.

bdp=0.5;
% Suppose the dimension is 3
v=3;
% The constants c and M associated to a breakdown point of 50 per cent
% and an ARP of 0.05 when there are three variables are
% c=1.133183024897769 and M= 1.662300458017338
[c,M]=RKbdp(bdp,v)

### Computation of c and M for a series of values of bdp (v=3).

v=3;
bdp=0.01:0.01:0.5;
cM=zeros(length(bdp),2);
for i=1:length(bdp)
[c,M]=RKbdp(bdp(i),v);
cM(i,:)=[c M];
end
subplot(2,1,1)
plot(bdp,cM(:,1))
subplot(2,1,2)
plot(bdp,cM(:,2))

### Computation of c and M for a series of values of bdp (v=10).

ARP fixed to 0.01.

v=10;
bdp=0.01:0.01:0.5;
cM=zeros(length(bdp),2);
for i=1:length(bdp)
[c,M]=RKbdp(bdp(i),v,0.01);
cM(i,:)=[c M];
end
subplot(2,1,1)
plot(bdp,cM(:,1))
subplot(2,1,2)
plot(bdp,cM(:,2))

## Input Arguments

### bdp — breakdown point. Scalar.

Scalar defining breakdown point (i.e a number in the interval [0 0.5). Please notice that the maximum achievable breakdown point is (n-p)/(2*n), and therefore the value 0.5 is reached only when the sample size goes to infinity. However, this routine assume a sample of size infinity and allows you to specify a bdp equal to 0.5.

Data Types: single|double

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

e.g. in regression v=1

Data Types: single|double|int32|int64

### ARP — asymptotic rejection probability. Scalar.

The asymptotic rejection probability of an estimator is defined as the probability in large sample under a reference distribution that a Malanobis distance excees $c_0$, where $c_0=inf \{ u_0 | w(u)=0, \forall u>u_0 \}$.

$w(u)$ is the weight function (defined in RKwei.m). In other words, given $c_0=sup(\rho(u))$,if an estimator is normed to the normal distribution ARP is $1-F_{\chi^2_v}(c_0^2)$.

The default value of ARP is 0.05.

Example: 0.04

Data Types: double

## Output Arguments

### c —Requested tuning constant. Scalar

Tuning constatnt of Rocke rho function (translated Tukey Biweight) associated to requested breakdown point and asymptotic rejection probability

### M —Requested tuning constant. Scalar

Tuning constant of Rocke rho function (translated Tukey Biweight) associated to requested breakdown point and asymptotic rejection probability

## References

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

Rocke D.M. (1996), Robustness properties of S-estimators of multivariate location and shape in high dimension, The Annals of Statistics, Vol. 24, pp. 1327-1345.