A martingale-transform goodness-of-fit test for the form of the conditional variance

In the common nonparametric regression model the problem of testing for a specific parametric form of the variance function is considered. Recently Dette and Hetzler (2008) proposed a test statistic, which is based on an empirical process of pseudo residuals. The process converges weakly to a Gaussian process with a complicated covariance kernel depending on the data generating process. In the present paper we consider a standardized version of this process and propose a martingale transform to obtain asymptotically distribution free tests for the corresponding Kolmogorov-Smirnov and Cramer-von-Mises functionals. The finite sample properties of the proposed tests are investigated by means of a simulation study. --nonparametric regression,goodness-of-it test,martingale transform,conditional variance

A martingale-transform goodness-of-fit test for the form of the conditional variance

In the common nonparametric regression model the problem of testing for a specific parametric form of the variance function is considered. Recently Dette and Hetzler (2008) proposed a test statistic, which is based on an empirical process of pseudo residuals. The process converges weakly to a Gaussian process with a complicated covariance kernel depending on the data generating process. In the present paper we consider a standardized version of this process and propose a martingale transform to obtain asymptotically distribution free tests for the corresponding Kolmogorov-Smirnov and Cram\'{e}r-von-Mises functionals. The finite sample properties of the proposed tests are investigated by means of a simulation study.Comment: 24 pages

Specification Tests Indexed by Bandwidths

In this note we consider several goodness-of-fit tests for model specification in nonparametric regression models which are based on kernel methods. In order to circumvent the problem of choosing a bandwidth for the corresponding test statistic we propose to consider the statistics as stochastic processes indexed with bandwidths proportional to the asymptotically optimal bandwidth for the estimation of the regression function. We prove weak convergence of these processes to centered Gaussian processes and suggest to use functionals of these processes as test statistics for the problem of model specification. A bootstrap test is proposed to obtain a good approximation of the nominal level. The results are illustrated by means of a simulation study and the new test is compared with some of the currently available procedures

Pacing and Decision Making in Sport and Exercise: The Roles of Perception and Action in the Regulation of Exercise Intensity

In pursuit of optimal performance, athletes and physical exercisers alike have to make decisions about how and when to invest their energy. The process of pacing has been associated with the goal-directed regulation of exercise intensity across an exercise bout. The current review explores divergent views on understanding underlying mechanisms of decision making in pacing. Current pacing literature provides a wide range of aspects that might be involved in the determination of an athlete's pacing strategy, but lacks in explaining how perception and action are coupled in establishing behaviour. In contrast, decision-making literature rooted in the understanding that perception and action are coupled provides refreshing perspectives on explaining the mechanisms that underlie natural interactive behaviour. Contrary to the assumption of behaviour that is managed by a higher-order governor that passively constructs internal representations of the world, an ecological approach is considered. According to this approach, knowledge is rooted in the direct experience of meaningful environmental objects and events in individual environmental processes. To assist a neuropsychological explanation of decision making in exercise regulation, the relevance of the affordance competition hypothesis is explored. By considering pacing as a behavioural expression of continuous decision making, new insights on underlying mechanisms in pacing and optimal performance can be developed. © 2014 Springer International Publishing Switzerland

Tests auf parametrische Struktur der Varianzfunktion in der nichtparametrischen Regression

In der vorliegenden Arbeit werden Testverfahren auf eine parametrische Struktur der Varianzfunktion im nichtparametrischen Regressionsmodell entwickelt. Dies umfasst insbesondere Tests für die in vielen Anwendungen getroffene Annahme der Homoskedastizität. Zunächst wird eine Testprozedur im Fall univariater Einflussgrößen vorgestellt, die auf einem mit Pseudoresiduen konstruierten empirischen Prozess basiert, für den die schwache Konvergenz gegen einen Gaußprozess nachgewiesen wird. Da die asymptotische Verteilung von unbekannten Größen abhängt, wird danach die schwache Konvergenz des geeignet transformierten Prozesses gegen eine Brownsche Bewegung mit transformierter Zeit nachgewiesen. Schließlich wird ein Test für die Nullhypothese der Homoskedastizität im multivariaten Fall konstruiert, der auf einer nichtparametrischen Schätzung der Regressionsfunktion basiert. Das Verhalten der vorgeschlagenen Testverfahren für endliche Stichproben wird mittels einer Simulationsstudie untersucht

Specification tests indexed by bandwidths

Abstract In this note we consider several goodness-of-fit tests for model specification in nonparametric regression models which are based on kernel methods. In order to circumvent the problem of choosing a bandwidth for the corresponding test statistic we propose to consider the statistics as stochastic processes indexed with bandwidths proportional to the asymptotically optimal bandwidth for the estimation of the regression function. We prove weak convergence of these processes to centered Gaussian processes and suggest to use functionals of these processes as test statistics for the problem of model specification. A bootstrap test is proposed to obtain a good approximation of the nominal level. The results are illustrated by means of a simulation study and the new test is compared with some of the currently available procedures

A simple test for the parametric form of the variance function in nonparametric regression

Homoscedasticity, Nonparametric regression, Pseudo residuals, Empirical process, Goodness-of-fit testing, Bootstrap,