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A nonparametric goodness-of-fit test for random variables with values in a separable Hilbert space is investigated. To verify the null hypothesis that the data come from a specific distribution, an integral type test based on a Cramér-von-Mises statistic is suggested. The convergence in distribution of the test statistic under the null hypothesis is proved and the test's consistency is concluded. Moreover, properties under local alternatives are discussed. Applications are given for data of huge but finite dimension and for functional data in infinite dimensional spaces. A general approach enables the treatment of incomplete data. In simulation studies the test competes with alternative proposals.
The readout of gamma detectors is considerably simplified when the event intensity is encoded as a pulse width (Pulse Width Modulation, PWM). Time-to-Digital-Converters (TDC) replace the conventional ADCs and multiple TDCs can be realized easily in one PLD chip (Programmable Logic Device). The output of a PWM stage is only one digital signal per channel which is well suited for transport so that further processing can be performed apart from the detector. This is particularly interesting for large systems with high channel density (e.g. high resolution scanners). In this work we present a circuit with a linear transfer function that requires a minimum of components by performing the PWM already in the preamp stage. This allows a very compact and also cost-efficient implementation of the front-end electronics.