1887

n Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie - Kwadratiese vorme in stogastiese veranderlikes

Supplement 1
  • ISSN : 0254-3486
  • E-ISSN: 2222-4173

Abstract


In the late 1960s J.H. Venter started to investigate the use of test statistics for normality based on quadratic distances between the order statistics of the sample and the corresponding hypothetical quantiles. These types of statistics are closely related to the (already known at that time) Shapiro-Wilk statistics, for which the limiting distribution was not yet known. The behaviour of the statistics investigated by Venter was such that the standard approach of that time, the so-called stochastic process approach, was insufficient to derive their limiting distribution. However, by writing the statistic in terms of order statistics from a uniform distribution and employing the representation of such order statistics in terms of independent, identically distributed exponential random variables, he was able to approximate the statistic by a quadratic form in independent, identically distributed random variables. This led him to the study of the limiting behaviour of the latter, for which results were not available to handle the statistics he was interested in. The results needed were derived and applied to the statistics of interest, constituting pioneering research that in later years led to the derivation of the limiting distribution of, inter alia, the Shapiro-Wilk statistic and many other statistics of a quadratic type. The words of Del Barrio, Cuesta-Albertos, Matran and Rodriguez-Rodriguez in a recent paper, "All the proofs of the asymptotic behaviour of these statistics ... rely on the results in ...", emphasize the fundamental contributions of Venter's earlier work.
In the current paper the above-mentioned contribution and the work flowing from it, are discussed and placed in a historical context. In particular, it is shown that by using an expression for the distribution of uniform order statistics in terms of ratios of sums of independent, identically distributed exponential random variables, the test statistic can be shown to be asymptotically equivalent to a quadratic form in independent, identically distributed random variables. For the latter the results known at that time were insufficient and stronger results had to be developed in order to obtain a limiting distribution. This limiting distribution was obtained as that of a linear combination of independent chi-squared random variables with one degree of freedom each. The latter's characteristic function could be found quite easily and inverted numerically to obtain critical values for the test. The constants in the linear combination are closely related to the eigenvalues of the matrix whose entries are the constants in the quadratic form. It was shown how these constants can be found by transforming the required integral equation into a differential equation for which the classical orthogonal polynomials provide solutions.

In die laat sestigerjare van die vorige eeu het J.H.Venter begin ondersoek instel na die gebruik van toetsstatistieke vir normaliteit gebaseer op kwadratiese afstande tussen die rangordestatistieke van die steekproef en die ooreenkomstige hipotetiese kwantiele. Hierdie tipe statistieke is nou verwant aan die (op daardie stadium reeds bekende) Shapiro-Wilk-statistieke, waarvan die limietverdeling nog nie bekend was nie. Die gedrag van statistieke wat deur Venter ondersoek is, was sodanig dat die standaardbenadering van daardie tyd, die sogenaamde stogastiese prosesbenadering, nie voldoende was om hul limietverdeling af te lei nie. Deur egter die statistiek te herlei na 'n uitdrukking in terme van rangordestatistieke uit 'n uniforme verdeling, en gebruik te maak van die voorstelling van sodanige rangordestatistieke in terme van onafhanklike, eksponensiaalverdeelde stogastiese veranderlikes, kon hy die statistiek by benadering skryf as 'n kwadratiese vorm in onafhanklike, identiesverdeelde stogastiese veranderlikes. Dit het hom gelei tot die studie van die limietgedrag van laasgenoemde, 'n onderwerp waaroor daar op die betrokke stadium nie voldoende resultate beskikbaar was om sy tipe statistieke te hanteer nie. Daardie resultate is ontwikkel en toegepas op die tersaaklike statistieke waarmee hy baanbrekerswerk verrig het en wat later sou lei tot die herleiding van die limietverdeling van onder andere die Shapiro-Wilk-statistiek en vele ander statistieke van 'n kwadratiese aard. Die woorde van Del Barrio, Cuesta-Albertos, Matran en Rodriguez-Rodriguez in 'n onlangse artikel, te wete "All the proofs of the asymptotic behaviour of these statistics .... rely on the results in ... ", benadruk die fundamentele bydrae van die vroeëre werk van Venter.


In hierdie artikel word die bogenoemde bydrae en uitvloeisels daarvan bespreek en binne 'n historiese konteks geplaas. Enkele uitbreidings van die resultate deur ander navorsers word ookbespreek, sowel as meer onlangse ontwikkelinge wat op die oorspronklike werk van Venter gebaseer is.

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/content/aknat/09/sup-1/EJC20284
2008-09-01
2019-10-20

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