HYPwei

HYPwei computes weight function psi(u)/u for hyperbolic tangent estimator

Description

w =HYPwei(u, cktuning) Weight function for hyperbolic tangent estimator.

Examples

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Weight function for hyperbolic tangent estimator.

x=-6:0.01:6;
ctuning=4;
ktuning=4.5;
weiHYP=HYPwei(x,[ctuning,ktuning]);
plot(x,weiHYP)
xlabel('x','Interpreter','Latex')
ylabel('$W (x) =\psi(x)/x$','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

cktuning — tuning parameters. Vector of length 2 or of length 5.

cktuning specifies specifies the value of the tuning constant c (scalar greater than 0 which controls the robustness/efficiency of the estimator) and the prefixed value k (sup of the change-of-variance sensitivity) and the values of parameters A, B and d:

cktuning(1) = c;

cktuning(2) = k = supCVC(psi,x) x \in R;

cktuning(3)=A;

cktuning(4)=B;

cktuning(5)=d;

Remark: if length(cktuning)==2 values of A, B and d will be computed automatically

Data Types: single| double

Output Arguments

w —hyperbolic weights. Vector

n x 1 vector contains the hyperbolic weights associated to the scaled residuals or Mahalanobis distances for the n units of the sample

Function HYPwei transforms vector u as follows $HYPwei(u) = \left\{ \begin{array}{cc} 1 & |u| \leq d, \\ \sqrt(A * (k - 1)) * tanh(sqrt((k - 1) * B^2/A)*(c-|u|)/2) .* sign(u)/u & d \leq |u| < c, \\ 0 & |u| \geq c. \\ \end{array} \right.$ where $0 < d < c$ is such that $d = \sqrt{[A(k-1)]}\tanh [\frac{1}{2}\sqrt{\frac{(k-1)B^2}{A}}(c - d)],$

$A$ and $B$ satisfy suitable conditions, and $k$ is related to the bound in the Change of Variance Curve.

More precisely, it is necessary to have $0 < A < B < 2 *normcdf(c)-1- 2*c*normpdf(c) <1$ Remark: hyperbolic psi-function is linear around u = 0 in accordance with Winsor's principle that all distributions are normal in the middle.

This means that \psi (u)/u is approximately constant over the linear region of \psi, so the points in that region tend to get equal weight.

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]