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.     

Neutron fluences and dose equivalents measured with passive detectors on LDEF

Neutron fluences were measured on LDEF in the low energy (< 1 MeV) and high energy (> 1 MeV) ranges. The low energy detectors used the ⁶Li(n,)T reaction with Gd foil absorbers to separate thermal (< 0.2 eV) and resonance (0.2 eV−1 MeV) neutron response. High energy detectors contained sets of fission foils (¹⁸¹Ta, ²⁰⁹Bi, ²³²Th, ²³⁸U) with different neutron energy thresholds. The measured neutron fluences together with predicted spectral shapes were used to estimate neutron dose equivalents. The detectors were located in the A0015 and P0006 experiments at the west and Earth sides of LDEF under shielding varying from 1 to 19 g/cm².

Dose equivalent rates varied from 0.8 to 3.3 μSv/d for the low energy neutrons and from 160 to 390 μSv/d for the high energy neutrons. This compares with TLD measured absorbed dose rates in the range of 1000–3000 μGy/d near these locations and demonstrates that high energy neutrons contribute a significant fraction of the total dose equivalent in LEO.

Comparisons between measurements and calculations were made for high energy neutrons based on fission fragment tracks generated by fission foils at different shielding depths. A simple 1-D slab geometry was used in the calculations. Agreement between measurements and calculations depended on both shielding depth and threshold energy of the fission foils. Differences increased as both shielding and thereshold energy increased. The modeled proton/neutron spectra appeared deficient at high energies. A 3-D model of the experiments is needed to help resolve the differences.     

Silver deposition from silver rubidium iodide

The halogenide (Cl^?, Br^?, I^?) complexes of indium(III) were investigated polarographically. The potential obtained in the presence of iodide ion (0.0004 M) was taken as the half-wave potential of “free” indium ion. The half-wave potentials at low halogenide concentrations were corrected for the kinetic effect. The approximate values of the stability constants read off from the curve (half-wave potential versus logarithm of ligand concentration) at points corresponding to the mean ligand numbers n=0.5, 1.5, 2.5, and 3.5 were then refined by the trial and error method. Four stability constants were found for each of the ligands investigated. The logarithms of β_1–4 are 2.70, 3.20, 4.20 and 3.30 for chloride ions, 2.10, 2.40, 2.50 and 0.60 for bromide ions and 1.35, 1.40, 1.30 and 0.50 for iodide ions. It was also assumed that the too high values of β₁ found by Bond resulted from neglect of the kinetic character of the waves.     

Enzyme electrokinetics: electrochemical studies of the anaerobic interconversions between active and inactive states of Allochromatium vinosum [NiFe]-hydrogenase

The cycling between active and inactive states of the catalytic center of [NiFe]-hydrogenase from Allochromatium vinosum has been investigated by dynamic electrochemical techniques. Adsorbed on a rotating disk pyrolytic graphite "edge" electrode, the enzyme is highly electroactive: this allows precise manipulations of the complex redox chemistry and facilitates quantitative measurements of the interconversions between active catalytic states and the inactive oxidized form Ni(r) (also called Ni-B or "ready") as functions of pH, H(2) partial pressure, temperature, and electrode potential. Cyclic voltammograms for catalytic H(2) oxidation (current is directly related to turnover rate) are highly asymmetric (except at pH > 8 and high temperature) due to inactivation being much slower than activation. Controlled potential-step experiments show that the rate of oxidative inactivation increases at high pH but is independent of potential, whereas the rate of reductive activation increases as the potential becomes more negative. Indeed, at 45 degrees C, activation takes just a few seconds at -288 mV. The cyclic asymmetry arises because interconversion is a two-stage reaction, as expected if the reduced inactive Ni(r)-S state is an intermediate. The rate of inactivation depends on a chemical process (rearrangement and uptake of a ligand) that is independent of potential, but sensitive to pH, while activation is driven by an electron-transfer process, Ni(III) to Ni(II), that responds directly to the driving force. The potentials at which fast activation occurs under different conditions have been analyzed to yield the potential-pH dependence and the corresponding entropies and enthalpies. The reduced (active) enzyme shows a pK of 7.6; thus, when a one-electron process is assumed, reductive activation at pH < 7 involves a net uptake of one proton (or release of one hydroxide), whereas, at pH > 8, there is no net exchange of protons with solvent. Activation is favored by a large positive entropy, consistent with the release of a ligand and/or relaxation of the structure around the active site.     

Facile synthesis of unsymmetrical benzotetrathiafulvalenes via cleavage of the corresponding hexathioorthooxalates

Unsymmetrical hexathioorthooxalates of types ($\text{1}$) and ($\text{2}$) undergo elimination of dialkyl disulfide on heating in an organic solvent; the reaction, which is catalyzed by acid, proceeds without fission of the central C:C bond and provides the first general, high yield synthesis of unsymmetrical benzotetrathiafulvalenes of types ($\text{3}$) and ($\text{4}$).