The distribution of the likelihood ratio for mixtures of densities from the one-parameter exponential family

We here consider testing the hypothesis ofhomogeneity against the alternative of a two-component mixture of densities. The paper focuses on the asymptotic null distribution of 2 log _n , where _n is the likelihood ratio statistic. The main result, obtained by simulation, is that its limiting distribution appears pivotal (in the sense of constant percentiles over the unknown parameter), but model specific (differs if the model is changed from Poisson to normal, say), and is not at all well approximated by the conventional ₍₂₎ ² -distribution obtained by counting parameters. In Section 3, the binomial with sample size parameter 2 is considered. Via a simple geometric characterization the case for which the likelihood ratio is 1 can easily be identified and the corresponding probability is found. Closed form expressions for the likelihood ratio _n are possible and the asymptotic distribution of 2 log _n is shown to be the mixture giving equal weights to the one point distribution with all its mass equal to zero and the ²-distribution with 1 degree of freedom. A similar result is reached in Section 4 for the Poisson with a small parameter value (0.1), although the geometric characterization is different. In Section 5 we consider the Poisson case in full generality. There is still a positive asymptotic probability that the likelihood ratio is 1. The upper precentiles of the null distribution of 2 log _n are found by simulation for various populations and shown to be nearly independent of the population parameter, and approximately equal to the (1–2)100 percentiles of ₍₁₎ ² . In Sections 6 and 7, we close with a study of two continuous densities, theexponential and thenormal with known variance. In these models the asymptotic distribution of 2 log _n is pivotal. Selected (1–) 100 percentiles are presented and shown to differ between the two models.     

Partition Lattice q-Analogs Related to q-Stirling Numbers

We construct a family of partially ordered sets (posets) that are q-analogs of the set partition lattice. They are different from the q-analogs proposed by Dowling [5]. One of the important features of these posets is that their Whitney numbers of the first and second kind are just the q-Stirling numbers of the first and second kind, respectively. One member of this family [4] can be constructed using an interpretation of Milne [9] for S[n, k] as sequences of lines in a vector space over the Galois field F _q. Another member is constructed so as to mirror the partial order in the subspace lattice.     

A simple formula for mean multiplexing delay for independent regenerative sources

Simple expressions are given for the mean delay, mean waiting time, and mean busy period length in a multiplexer. Data streams with active periods having a general distribution are permitted, and the data rate during the active periods can be random. Data can also arrive in batches. The key restrictions of the model are that the sources are independent, idle periods are exponentially distributed, and a source generates at least enough data during an active period to keep the server busy throughout the period. The exact formulas allow evaluation of the error in approximations such as a heavy traffic diffusion approximation.Both continuous and discrete time models are considered. The discrete-time model includes that studied by Viterbi and subsequently generalized by Neuts. The Pollaczek-Khinchine formula for the mean amount of work in anM/GI/1 queue is retrieved as a limiting case.Preliminary version presented at IEEE INFOCOM, San Francisco, April 1993.     

Laser Based Techniques for Ultra Trace Isotope Production, Spectroscopy and Detection

A variety of research activities in the field of fundamental and applied nuclear physics has evolved in the last years using resonantly tuned radiation from powerful lasers. The technique of resonance ionization spectroscopy has delivered outstanding results and found broad acceptance in the last years as a particularly efficient and highly selective method for rare and exotic radioisotope studies. It is used for production, spectroscopy and detection of these species and provides complete isobaric, high isotopic and even some isomeric selection, which altogether is needed for on-line investigation of short lived species far off stability as well as for ultra trace determination. Good overall efficiency pushes the experimental limits of detection in elemental trace analysis down to below 10⁶ atoms per sample, and additionally isotopic selectivity as high as 3 × 10¹² has been demonstrated. The widespread potential of resonance ionization techniques is discussed, focusing on the experimental arrangements for applications in selective on-line isotope production, spectroscopy of rare radioisotopes and ultra trace determination of radiotoxic isotopes like ²³⁸Pu to ²⁴⁴Pu, ^135,137Cs, ^89,90Sr or ⁴¹Ca in environmental, technical and biomedical samples.     

Simplification and analysis of a model of social interaction in voting

A recently proposed model of social interaction in voting is investigated by simplifyingit down into a version that is more analytically tractable and which allows a mathematicalanalysis to be performed. This analysis clarifies the interplay of the different elementspresent in the system – social influence, heterogeneity and noise – and leads to a betterunderstanding of its properties. The origin of a regime of bistability is identified. Theinsight gained in this way gives further intuition into the behaviour of the originalmodel.     

Analogy between the Lorenz strange attractor and a bistable stochastic oscillator

The relation between the aperiodic solution of the Lorenz model and that of a stochastic anharmonic oscillator is explored. The stochastic oscillator is constructed by replacing (t) in the Lorenz model by a stochastic variable(t) of specified statistics. The resulting system is of course not isomorphic to the Lorenz model, but does share with it a number of statistical properties. Thus, within the confines of these measures the two systems are physically very similar.     

Extension of statistical replacement to systems with time-correlated fluctuations

Nonlinear systems with correlated stochastic parameters are approximated by simpler systems. This method is an extension of an earlier version of statistical replacement and statistical linearization. The extended method is applicable to systems with correlated fluctuations. We show how this general method reduces to the earlier methods in special cases.