The main intent of this paper is to introduce a new statistical procedure for testing a complete sample for normality. The test statistic is obtained by dividing the square of an appropriate linear combination of the sample order statistics by the usual symmetric estimate of variance. This ratio is both scale and origin invariant and hence the statistic is appropriate for a test of the composite hypothesis of normality. Testing for distributional assumptions in general and for normality in particular has been a major area of continuing statistical research-both theoretically and practically. A possible cause of such sustained interest is that many statistical procedures have been derived based on particular distributional assumptions-especially that of normality. Although in many cases the techniques are more robust than the assumptions underlying them, still a knowledge that the underlying assumption is incorrect may temper the use and application of the methods. Moreover, the study of a body of data with the stimulus of a distributional test may encourage consideration of, for example, normalizing transformations and the use of alternate methods such as distribution-free techniques, as well as detection of gross peculiarities such as outliers or errors. The test procedure developed in this paper is defined and some of its analytical properties described in ? 2. Operational information and tables useful in employing the test are detailed in ? 3 (which may be read independently of the rest of the paper). Some examples are given in ? 4. Section 5 consists of an extract from an empirical sampling study of the comparison of the effectiveness of various alternative tests. Discussion and concluding remarks are given in ?6. 2. THE W TEST FOR NORMALITY (COMPLETE SAMPLES) 2 1. Motivation and early work This study was initiated, in part, in an attempt to summarize formally certain indications of probability plots. In particular, could one condense departures from statistical linearity of probability plots into one or a few 'degrees of freedom' in the manner of the application of analysis of variance in regression analysis? In a probability plot, one can consider the regression of the ordered observations on the expected values of the order statistics from a standardized version of the hypothesized distribution-the plot tending to be linear if the hypothesis is true. Hence a possible method of testing the distributional assumptionis by means of an analysis of variance type procedure. Using generalized least squares (the ordered variates are correlated) linear and higher-order

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An Analysis of Variance Test for Normality (Complete Samples)

Unbiased plotting positions — A review

We consider a class of nonlinear filters whose output is given by a linear combination of the order statistics of the input sequence. Assuming a constant signal in white noise, the coefficients in the linear combination are chosen to minimize the output MSE for several noise distributions. It is shown that the optimal order statistic filter (OSF) tends toward the median filter as the noise becomes more impulsive. The optimal OSF is applied to an actual noisy image and is shown to perform well, combining properties of both the averaging and median filters. A more general design scheme for applications involving nonconstant signals is also given.

A generalization of median filtering using linear combinations of order statistics

A family of nonlinear filters based on order statistics is presented. A mathematical tool derived through robust estimation theory, order statistics has allowed engineers to develop nonlinear filters with excellent robustness properties. These filters are well suited to digital image processing because they preserve the edges and the fine details of an image much better than conventional linear filters. The probabilistic and deterministic properties of the best known and most widely used filter in this family, the median filter, are discussed. In addition, the authors consider filters that, while not based on order statistics, are related to them through robust estimation theory. A table that ranks nonlinear filters under a variety of performance criteria is included. Most of the topics treated are very active research areas, and the applications are varied, including HDTV, multichannel signal processing of geophysical and ECG/EEG data, and a variety of telecommunications applications. >

/pdf/order-statistics-in-digital-image-processing-1rhdihn5hi.pdf

Order statistics in digital image processing

L'analyse d'image touche a l'heure actuelle de nombreux domaines, avec des objectifs aussi varies que l'aide au diagnostic pour les images medicales, la vision artificielle en robotique ou l'analyse des ressources terrestres a partir des images prises par satellite. Le but du traitement de ces images est a la fois simple dans son concept et difficile dans sa realisation. Simple en effet, puisqu'il s'agit de reconnaitre des objets que notre systeme visuel percoit rapidement, du moins pour la majorite d'entre eux. Difficile cependant, car dans la grande quantite d'informations contenues dans l'image, il faut extraire des elements pertinents pour l'application visee et ceci independamment de la qualite de l'image. L'analyse d'image s'est donc dotee d'outils et de methodes puissants issus de domaines aussi varies que les mathematiques, le traitement du signal, ou l'informatique. Cet ouvrage presente un des aspects les plus importants du traitement des images : la " segmentation ". Il recapitule d'abord les grandeurs observables et calculables sur une image et les algorithmes de manipulation des structures de donnees associees. Il detaille ensuite les traitements preliminaires, tels le filtrage du bruit et les deux types d'approche de la segmentation, l'extraction des contours et celle des regions. Chacune fait l'objet d'une etude theorique et de nombreux resultats illustrent les performances. Une des originalites de l'ouvrage est l'etude comparative des differentes techniques appliquees sur un meme corpus d'images reelles.

https://hal.archives-ouvertes.fr/hal-00706168/document

E. H. Lloyd

Papers

Least-squares estimation of location and scale parameters using order statistics