HYPc

HYPc computes breakdown point and efficiency associated with constant chyperbolic tangent estimator (for a given value of k=sup CVC)

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Syntax

Description

example

bdp =HYPc(c, v) HYPc will all the default options.

example

bdp =HYPc(c, v, Name, Value) HYPc will all the default options (2).

example

[bdp, eff] =HYPc(___) HYPc will all the default options (3).

example

[bdp, eff, A] =HYPc(___) Analysis of breakdown point and asymptotic efficiency at the normal distribution as a function of c in regression.

example

[bdp, eff, A, B] =HYPc(___)

example

[bdp, eff, A, B, d] =HYPc(___)

Examples

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  • HYPc will all the default options.

    • Just find bdp. 

    bdp=HYPc(2.5,1);
disp('bdp associated with c=2.5 in regression')
disp(bdp)

  • HYPc will all the default options (2).

    • Find both bdp and eff. 

    [bdp,eff]=HYPc(2.5,1);
disp('bdp associated with c=2.5 in regression')
disp(bdp)
disp('eff associated with c=2.5 in regression')
disp(eff)

  • HYPc will all the default options (3).

    • Find bdp, eff, A, B and d. 

    [bdp,eff,A,B,d]=HYPc(2.5,1);
disp('bdp associated with c=2.5 in regression')
disp(bdp)
disp('eff associated with c=2.5 in regression')
disp(eff)

  • Analysis of breakdown point and asymptotic efficiency at the normal distribution as a function of c in regression.

    • cc=2:0.1:6;
% BDPEFF = matrix which will contain
% 1st column = value of c
% 2nd column = breakdown point (bdp)
% 3rd column = asympotic nominal efficiency (eff)
% 4th column = value of parameter A
% 5th column = value of parameter B
% 6th column = value of parameter d
BDPEFF=[cc' zeros(length(cc),5)];
% Fixed value of k (sup CVC) which must be used
kdef=4.5;
jk=1;
for c=cc
[bdp,eff,A,B,d]=HYPc(c,1,'k',kdef);
BDPEFF(jk,2:end)=[bdp, eff, A, B, d];
jk=jk+1;
end
nr=2;
nc=2;
subplot(nr,nc,1)
plot(BDPEFF(:,1),BDPEFF(:,2))
xlabel('c','Interpreter','Latex','FontSize',16)
ylabel('Breakdown point','Interpreter','none')
subplot(nr,nc,2)
plot(BDPEFF(:,1),BDPEFF(:,3))
xlabel('c','Interpreter','Latex','FontSize',16)
ylabel('Asymptotic efficiency','Interpreter','none')
subplot(nr,nc,3)
plot(BDPEFF(:,1),BDPEFF(:,4:5))
xlabel('c','Interpreter','Latex','FontSize',16)
ylabel('A and B','Interpreter','none')
subplot(nr,nc,4)
plot(BDPEFF(:,1),BDPEFF(:,6))
xlabel('c','Interpreter','Latex','FontSize',16)
ylabel('d','Interpreter','none')
suplabel(['Constant k=' num2str(kdef)],'t');

    Input Arguments

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    c — tuning constant c. Scalar.

    Scalar greater than 0 which controls the robustness/efficiency of the estimator

    Data Types: single| double

    v — number of response variables. Scalar.

    Number of variables of the dataset (for regression p=1) UP TO NOW v=1 (JUST REGRESSION) TO DO FOR MULTIVARIATE ANALYSIS

    Data Types: single| double

    Name-Value Pair Arguments

    Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

    Example: 'k',4 , 'shapeeff',1 , 'param',[0.0035;0.0317;0.0827]

    k —supremum of the change of variance curve $supCVC(psi,x) x \in R$.default value is k=4.

    5

    Example: 'k',4

    Data Types: double

    shapeeff —location or shape efficiency.scalar.

    If 1, the efficiency is referred to the shape else (default) is referred to the location. TODO:Hac:shapeeff

    Example: 'shapeeff',1

    Data Types: double

    param —parameters A, B and d.vector with 3 elements.

    Vector of length 3 specifying the parameters A, B and d of the weight function of the hyperbolic tangent estimator.

    param(1)=A param(2)=B param(3)=d If these values are not supplied they will be automatically calculated calling routine HYPck

    Example: 'param',[0.0035;0.0317;0.0827]

    Data Types: double

    Output Arguments

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    bdp —bdp. Scalar

    Breakdown point associated to the supplied value of c

    eff —eff. Scalar

    Efficiency associated to the supplied value of c

    A —parameter A of hyperbolic tangent estimator. Scalar

    For more details see the methodological details inside "More About" below

    B —parameter B of hyperbolic tangent estimator. Scalar

    For more details see the methodological details inside "More About" below

    d —parameter d of hyperbolic tangent estimator. Scalar

    For more details see the methodological details inside "More About" below

    More About

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    Additional Details

    \[ HYPpsi(u) = \left\{ \begin{array}{cc} u & |u| \leq d \\ \sqrt{A (k - 1)} \tanh \left( \sqrt{(k - 1) B^2/A} (c -|u|)/2 \right) sign(u) & d \leq |u| < c, \\ 0 & |u| \geq c. \end{array} \right. \] It is necessary to have $0 < A < B < 2 normcdf(c)-1- 2 c \times normpdf(c) <1$

    References

    Hampel, F.R., Rousseeuw, P.J. and Ronchetti E. (1981), The Change-of-Variance Curve and Optimal Redescending M-Estimators, "Journal of the American Statistical Association", Vol. 76, pp. 643-648 [HRR]

    See Also

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