510 Mathematik
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- 2014 (2) (remove)
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- Dichte <Stochastik> (1)
- Doppelt nichtzentrale F-Verteilung (1)
- Doppelt nichtzentrale t-Verteilung (1)
- Doubly noncentral F-distribution (1)
- Doubly noncentral t-distribution (1)
- Extended sign regular (1)
- Funktionalanalysis (1)
- Modified Bessel function (1)
- Modifizierte Besselfunktion (1)
- Parameterabhängigkeit (1)
Zu den klassischen Verteilungen der mathematischen Statistik zählen die zentralen F- und t-Verteilungen. Die vorliegende Arbeit untersucht Verallgemeinerungen dieser Verteilungen, die sogenannten doppelt nichtzentralen F- und t-Verteilungen, welche in der statistischen Testtheorie von Bedeutung sind. Die Tatsache, dass die zugehörigen Wahrscheinlichkeitsdichten nur in Form von Parameterintegral- bzw. Doppelreihendarstellungen gegeben sind, stellt eine große Herausforderung bei der Untersuchung analytischer Eigenschaften dar. Unter Verwendung von Techniken aus der Theorie der vorzeichenregulären Funktionen gelingt es, die bisher vermutete, jedoch lediglich aus Approximationen abgeleitete, strikt unimodale Gestalt der Dichtefunktion für eine große Klasse doppelt nichtzentraler Verteilungen zu zeigen. Dieses Resultat gestattet die Untersuchung des eindeutig bestimmten Modus als Funktion gewisser Nichtzentralitätsparameter. Hier erweist sich die Theorie der vorzeichenregulären Funktionen als wichtiges Hilfsmittel, um monotone Abhängigkeiten nachzuweisen.
In splitting theory of locally convex spaces we investigate evaluable characterizations of the pairs (E, X) of locally convex spaces such that each exact sequence 0 -> X -> G -> E -> 0 of locally convex spaces splits, i.e. either X -> G has a continuous linear left inverse or G -> E has a continuous linear right inverse. In the thesis at hand we deal with splitting of short exact sequences of so-called PLH spaces, which are defined as projective limits of strongly reduced spectra of strong duals of Fréchet-Hilbert spaces. This class of locally convex spaces contains most of the spaces of interest for application in the theory of partial differential operators as the space of Schwartz distributions , the space of real analytic functions and various spaces of ultradifferentiable functions and ultradistributions. It also contains non-Schwartz spaces as B(2,k,loc)(Ω) and spaces of smooth and square integrable functions that are not covered by the current theory for PLS spaces. We prove a complete characterizations of the above problem in the case of X being a PLH space and E either being a Fréchet-Hilbert space or a strong dual of one by conditions of type (T ). To this end, we establish the full homological toolbox of Yoneda Ext functors in exact categories for the category of PLH spaces including the long exact sequence, which in particular involves a thorough discussion of the proper concept of exactness. Furthermore, we exhibit the connection to the parameter dependence problem via the Hilbert tensor product for hilbertizable locally convex spaces. We show that the Hilbert tensor product of two PLH spaces is again a PLH space which in particular proves the positive answer to Grothendieck- problème des topologies. In addition to that we give a complete characterization of the vanishing of the first derivative of the functor proj for tensorized PLH spectra if one of the PLH spaces E and X meets some nuclearity assumptions. To apply our results to concrete cases we establish sufficient conditions of (DN)-(Ω) type and apply them to the parameter dependence problem for partial differential operators with constant coefficients on B(2,k,loc)(Ω) spaces as well as to the smooth and square integrable parameter dependence problem. Concluding we give a complete solution of all the problems under consideration for PLH spaces of Köthe type.