Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie, Sep 1, 2008
Quadratic forms in random variables In the late 1960s J.H. Venter started to investigate the use ... more Quadratic forms in random variables 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 vroeere 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.
In this paper a partial review is given of results on asymptotic theory and includes distribution... more In this paper a partial review is given of results on asymptotic theory and includes distribution theory, invariance principles, finding the limiting distribution, the effect of estimation of unknown parameters on the limit distribution and relationships to other classes of statistics. A number of examples are discussed.
ABSTRACT We develop and study in the framework of Pareto-type distributions a class of nonparamet... more ABSTRACT We develop and study in the framework of Pareto-type distributions a class of nonparametric kernel estimators for the conditional second order tail parameter. The estimators are obtained by local estimation of the conditional second order parameter using a moving window approach. Asymptotic normality of the proposed class of kernel estimators is proven under some suitable conditions on the kernel function and the conditional tail quantile function. The nonparametric estimators for the second order parameter are subsequently used to obtain a class of bias-corrected kernel estimators for the conditional tail index. In particular it is shown how for a given kernel function one obtains a bias-corrected kernel function, and that replacing the second order parameter in the latter with a consistent estimator does not change the limiting distribution of the bias-corrected estimator for the conditional tail index. The finite sample behavior of some specific estimators is illustrated with a simulation experiment. The developed methodology is also illustrated on fire insurance claim data.
Studies in Economics and Econometrics, Apr 1, 2005
Abstract LULU smoothers is a class of non-linear smoothers introduced by Rohwer (1989) and has si... more Abstract LULU smoothers is a class of non-linear smoothers introduced by Rohwer (1989) and has since been studied extensively by him, from a mathematical point of view, culminating in the publishing of Rohwer (2005). It has also been successfully applied in image processing, engineering and the earth sciences. The purpose of this paper is to discuss linear and non-linear smoothers very briefly and to introduce LULU smoothers to the econometrical and statistical literature as an alternative to the existing linear and non-linear smoothers. An overview of LULU smoothers will be given and their most important properties will be discussed. Their attractive way of dealing with impulsive noise in the form of blockpulses and of decomposing the variation in a series will be highlighted and illustrated by applying it to the Standard and Poor 500 series.
... A natural candidate is s (a ), viz the Winsorized nn variance with random Winsorizing proport... more ... A natural candidate is s (a ), viz the Winsorized nn variance with random Winsorizing proportion a . In De ... Tukey, JW and Mc Laughlin, DH (1963 ... Less vulnerable confi-dence and sipnificance ~rocedures for location based on a - single sample: Triming/Winsorization 1. SankhyZ ...
Minimum distance parameter estimation using weighted Cramer-von Mises statistics is considered fo... more Minimum distance parameter estimation using weighted Cramer-von Mises statistics is considered for the general one-dimensional case. Under rather general conditions, the derived estimators are asymptotically normal. Consideration is given to appropriate weights to produce Fisher-efficient estimators. In fact, estimators can be obtained with influence curves proportional to any desired smooth function, and hence prescribed first-order robustness properties. Many such curves
Two proposals are made for constructing adaptive estimators of the parameters in a linear regress... more Two proposals are made for constructing adaptive estimators of the parameters in a linear regression model. These estimators are based on regression trimmed means and use an idea of Jaeckel [(1971) Ann Math Statist 42, 1540-1552] and the bootstrap respectively. These adaptive trimmed means as well as some nonadaptive trimmed means are studied by Monte Carlo. A one-step biweight is
Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie, Sep 1, 2008
Quadratic forms in random variables In the late 1960s J.H. Venter started to investigate the use ... more Quadratic forms in random variables 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 vroeere 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.
In this paper a partial review is given of results on asymptotic theory and includes distribution... more In this paper a partial review is given of results on asymptotic theory and includes distribution theory, invariance principles, finding the limiting distribution, the effect of estimation of unknown parameters on the limit distribution and relationships to other classes of statistics. A number of examples are discussed.
ABSTRACT We develop and study in the framework of Pareto-type distributions a class of nonparamet... more ABSTRACT We develop and study in the framework of Pareto-type distributions a class of nonparametric kernel estimators for the conditional second order tail parameter. The estimators are obtained by local estimation of the conditional second order parameter using a moving window approach. Asymptotic normality of the proposed class of kernel estimators is proven under some suitable conditions on the kernel function and the conditional tail quantile function. The nonparametric estimators for the second order parameter are subsequently used to obtain a class of bias-corrected kernel estimators for the conditional tail index. In particular it is shown how for a given kernel function one obtains a bias-corrected kernel function, and that replacing the second order parameter in the latter with a consistent estimator does not change the limiting distribution of the bias-corrected estimator for the conditional tail index. The finite sample behavior of some specific estimators is illustrated with a simulation experiment. The developed methodology is also illustrated on fire insurance claim data.
Studies in Economics and Econometrics, Apr 1, 2005
Abstract LULU smoothers is a class of non-linear smoothers introduced by Rohwer (1989) and has si... more Abstract LULU smoothers is a class of non-linear smoothers introduced by Rohwer (1989) and has since been studied extensively by him, from a mathematical point of view, culminating in the publishing of Rohwer (2005). It has also been successfully applied in image processing, engineering and the earth sciences. The purpose of this paper is to discuss linear and non-linear smoothers very briefly and to introduce LULU smoothers to the econometrical and statistical literature as an alternative to the existing linear and non-linear smoothers. An overview of LULU smoothers will be given and their most important properties will be discussed. Their attractive way of dealing with impulsive noise in the form of blockpulses and of decomposing the variation in a series will be highlighted and illustrated by applying it to the Standard and Poor 500 series.
... A natural candidate is s (a ), viz the Winsorized nn variance with random Winsorizing proport... more ... A natural candidate is s (a ), viz the Winsorized nn variance with random Winsorizing proportion a . In De ... Tukey, JW and Mc Laughlin, DH (1963 ... Less vulnerable confi-dence and sipnificance ~rocedures for location based on a - single sample: Triming/Winsorization 1. SankhyZ ...
Minimum distance parameter estimation using weighted Cramer-von Mises statistics is considered fo... more Minimum distance parameter estimation using weighted Cramer-von Mises statistics is considered for the general one-dimensional case. Under rather general conditions, the derived estimators are asymptotically normal. Consideration is given to appropriate weights to produce Fisher-efficient estimators. In fact, estimators can be obtained with influence curves proportional to any desired smooth function, and hence prescribed first-order robustness properties. Many such curves
Two proposals are made for constructing adaptive estimators of the parameters in a linear regress... more Two proposals are made for constructing adaptive estimators of the parameters in a linear regression model. These estimators are based on regression trimmed means and use an idea of Jaeckel [(1971) Ann Math Statist 42, 1540-1552] and the bootstrap respectively. These adaptive trimmed means as well as some nonadaptive trimmed means are studied by Monte Carlo. A one-step biweight is
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