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.     

Dynamic fluctuations in optical bistability

We present results on the time correlation function of a system undergoing absorptive optical bistability. We first use the Zwanzig-Mori formalism to calculate time correlation functions both near marginal stability points and in the coexistence region. Near marginal stability the theory predicts large deviations from a single exponential form of the correlation function. The truncated continued fraction expansion is shown to become inapplicable close to the coexistence point. The difficulties are due to the presence of long time scales, viz. the very large mean first passage times between the two metastable steady states. When these scales are important we show that the memory kernel relaxation is no longer faster than that of a field fluctuation. An increase in the size of the system increases the disparity of the time scales and thus exacerbates the problems of the projection operator formalism as used here following reference [14].We next present an ansatz for the correlation function incorporating the four major time scales important near coexistence, the two single branch relaxation times and the two mean first passage times for transitions between the stable states. This form of the correlation function avoids the difficulties cited in connection with the use of the projection operator method.     

UV-holographic gratings in TS-diacetylene single crystals

Holographic gratings with grating periods between 0.37 m and 4.4 m, respectively, were recorded on the surface of TS-diacetylene single crystals by two interfering 257 nm laser beams. Diffraction efficiencies of up to 35 % were obtained for readout with red light of = 633 nm. The main modulation mechanism was phase modulation. The spatial resolution was better than 1600 lines/mm. The upper limit of the polymer chain length was 0.6 m. The holographic sensitivity was 4.5 cm²/J for immediate read-out at 633 nm without processing after exposure. — A weakly exposed latent hologram may be developed simply by gentle annealing the crystal which increases the sensitivity by one order of magnitude.     

First experimental evidence for quantum echoes in scattering systems

A self-pulsing effect termed quantum echoes has been observed in experiments with an open superconducting and a normal conducting microwave billiard whose geometry provides soft chaos, i.e., a mixed phase space portrait with a large stable island. For such systems a periodic response to an incoming pulse has been predicted. Its period has been associated with the degree of development of a horseshoe describing the topology of the classical dynamics. The experiments confirm this picture and reveal the topological information.     

TERAHERTZ QUALITY CONTROL OF POLYMERIC PRODUCTS

We report on experiments that evaluate the potential of terahertz (THz) time-domain spectroscopy (TDS) for quality control of polymeric compounds. We investigate specimens out of a polyethylene compound with silver-coated titanium dioxide nanospheres and a glass-fiber reinforced epoxy composite. We further examine an industrial polymer product produced by injection molding. Our data demonstrates that THz imaging is a powerful tool for contactless quality control in the polymer industry.