Normal Exponential Distributions Research Articles

On the Multivariate Extended Skew-Normal, Normal-Exponential, and Normal-Gamma Distributions

This paper presents expressions for the multivariate normal-exponential and normal-gamma distributions. It then presents properties of these distributions. These include conditional distributions and a new extension to Stein's lemma. It is also shown that the multivariate normal-gamma and normal-exponential distribution are not in general closed under conditioning, although they are closed under linear transformations. The paper also demonstrates that there are relationships between the extended skew-normal distribution and the normal-gamma and normal-exponential distributions. Specifically, it is shown that certain limiting cases of the extended skew-normal distribution are normal-gamma and normal-exponential. One interpretation of these results is that the normal-exponential distribution may be considered to be an alternative model to the extended skew-normal in some situations. An alternative point of view, however, is that the normal-exponential distribution is redundant since it may be replicated by ...

Heteroscedastic normal–exponential mixture models: Bayesian and classical approaches

In this paper, we introduce a Bayesian analysis for mixture of distributions belonging to the exponential family. As a special case we consider a mixture of normal exponential distributions including joint modeling of the mean and variance. We also consider joint modeling of the mean and variance heterogeneity. Markov Chain Monte Carlo (MCMC) methods are used to obtain the posterior summaries of interest. We also introduce and apply an EM algorithm, where the maximization is obtained applying the Fisher scoring algorithm. Finally, we also include analysis of real data sets to illustrate the proposed methodology.

Photochromism of Diarylethene Single Molecules in Polymer Matrices

Robust fluorescent photoswitching molecules, having perylene bisimide as the fluorescent unit and diarylethene as the switching unit, were prepared, and their photochromic reactions were measured at the single-molecule level in various polymer matrices. The histograms of the fluorescent on and off times were found to deviate from normal exponential distribution and showed a peak when the molecules are embedded in rigid polymer matrices, such as Zeonex or poly(methyl methacrylate) (PMMA). In soft polymer matrices, such as poly(n-buthyl methacrylate) (PnBMA), exponential distribution was observed for the on and off times. The abnormal distribution suggests that the quantum yields of the photoreactions are not constant and the molecules undergo the reactions after absorbing a certain number of photons. A multilocal minima model was proposed to explain the environmental effect.