Weighted Order Statistic Research Articles

Adaptive Stack Filtering with Application to Image Processing

With the aid of threshold decomposition, it is shown that optimal stack filters under the mean absolute error (MAE) criterion are equal to optimal (or Bayesian) classifiers subject to stacking constraints under the mean classification error (MCE) criterion. Nonadaptive and adaptive constrained least mean absolute (LMA) algorithms are developed for the esti- mation of stack filters through the linearization of the unit step function in the objective function. The convergence of the al- gorithms is proven under certain conditions. Although the methods do not generally give optimal stack filters under the MAE criterion, these algorithms have several distinct merits compared to other stack filter optimization methods: 1) the op- timization problem has a unique solution that approximates the optimal stack filters in the least mean square sense; 2) the meth- ods can be implemented in the binary and real domains; and 3) for stack filters defined by linearly separable positive Boolean functions (PBF's) or weighted order statistic (WOS) filters, the number of the parameters to be estimated is reduced to the window width of the filter. A comparison between images re- stored by the new algorithms and the stack filtering algorithm optimal under the MAE criterion confirms the effectiveness of the proposed algorithms.

Application of the power-shift transformation†

Some of the practical consequences of entertaining the power-shift transformation are examined for both analysis of variance and simple random sampling data. The appropriateness of different procedures for estimating the transformation parameters is also considered. Standard maximum likelihood (ml) estimation is shown not to be feasible for the joint estimation of the power- shift parameters. However, it is possible to compute local ml estimates for some problems, but special estimation methods may be necessary. An alternative weighted order statistic (wos) estimator is shown to be useful in some situations. Characteristics of ml and wos estimation for the power-shift transformation are examined by a Monte Carlo experiment. A Box and Cox (1964) example and several other data sets are used to illustrate the estimation procedures

Standard and goodness-of-fit parameter estimation methods for the three-parameter lognormal distribution

A class of goodness-of-fit estimators is found to provide a useful alternative in certain situations to the standard maximum likelihood method which has some undesirable estimation characteristics for estimation from the three-parameter lognormal distribution. The class of goodness-of-fit tests considered include the Shapiro-Wilk and Filliben tests which reduce to a weighted linear combination of the order statistics that can be maximized in estimation problems. The weighted order statistic estimators are compared to the standard procedures in Monte Carlo simulations. Robustness of the procedures are examined and example data sets analyzed.