Consistency of the penalized MLE for two-parameter gamma mixture models

Two-parameter gamma distributions are widely used in liability theory, lifetime data analysis, financial statistics, and other areas. Finite mixtures of gamma distributions are their natural extensions, and they are particularly useful when the population is suspected of heterogeneity. These distributions are successfully employed in various applications, but many researchers falsely believe that the maximum likelihood estimator of the mixing distribution is consistent. Similarly to finite mixtures of normal distributions, the likelihood function under finite gamma mixtures is unbounded. Because of this, each observed value leads to a global maximum that is irrelevant to the true distribution. We apply a seemingly negligible penalty to the likelihood according to the shape parameters in the fitted model. We show that this penalty restores the consistency of the likelihoodbased estimator of the mixing distribution under finite gamma mixture models. We present simulation results to validate the consistency conclusion, and we give an example to illustrate the key points.     

One Pot Synthesis of Octahydroquinazolinone Derivatives Using (Me (Im)<Superscript>12</Superscript>) H<Subscript>4</Subscript>CuPW<Subscript>11</Subscript>O<Subscript>39</Subscript> as a Surfactant Type Catalyst

This study is aimed at proposing a practical green procedure for the synthesis of octahydroquinazolinone derivatives using benzaldehyde, dimedone and urea under microwave irradiation in water. A surfactant type polyoxometalate-based organic–inorganic hybrid was able to efficiently catalyze this synthesis. The catalyst was prepared and characterized by Fourier transform infrared, UV–Vis, X-ray diffraction, and thermogravimetric analysis. The employed catalyst exerted a synergistic effect; the anion part acted as a catalyst while the cation part acted as a surfactant in order to increase the concentration of organic reactants in water. The main advantage of this method is its remarkable yield in short reaction periods, which results in the rendering of products with high purity. Moreover, this heterocatalyst is capable of being easily recovered and reused several times. The influences of reaction conditions were studied systematically, and a possible catalysis mechanism was proposed for understanding the highly efficient heterogeneous catalytic behavior.     

Characteristics of in-out intermittency in delay-coupled FitzHugh–Nagumo oscillators

We analyze a pair of delay-coupled FitzHugh–Nagumo oscillators exhibiting in-out intermittency as a part of the generating mechanism of extreme events. We study in detail the characteristics of in-out intermittency and identify the invariant subsets involved – a saddle fixed point and a saddle periodic orbit – neither of which are chaotic as in the previously reported cases of in-out intermittency. Based on the analysis of a periodic attractor possessing in-out dynamics, we can characterize the approach to the invariant synchronization manifold and the spiralling out to the saddle periodic orbit with subsequent ejection from the manifold. Due to the striking similarities, this analysis of in-out dynamics also explains in-out intermittency     

Robust optimization in countably infinite linear programs

A robust optimization framework for countably infinite linear programs (CILPs) is developed. It is shown that a particular robust counterpart of a nominal CILP can be reformulated as another CILP. A bound on the probability of constraint violation is derived. A convergent algorithm for solving this robust CILP is proposed.