Kwadratiese vorme in stogastiese veranderlikes.

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-Wilk32 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 paper10, "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 this paper it is also shown how these quadratic statistics are related to degenerate Ustatistics and correlation-type test statistics, leading to the same limiting distribution. A number of extensions of other researchers are also discussed, as well as more recent developments based on the original work of Venter. To be more specific, at the time of Venter's research, the general approach to deriving the limiting distribution of such quadratic statistics was by means of a socalled stochastic process approach. In that case the statistic was written as a quadratic functional of the empirical process or of the quantile process, for which the limiting distribution was known in terms of the Brownian Bridge process.… [ABSTRACT FROM AUTHOR]