Matrix‐Exponential Distributions: Calculus and Interpretations via Flows

By considering randomly stopped deterministic flow models, we develop an intuitively appealing way to generate probability distributions with rational Laplace–Stieltjes transforms on [0,∞). That approach includes and generalizes the formalism of PH-distributions. That generalization results in the class of matrix-exponential probability distributions. To illustrate the novel way of thinking that is required to use these in stochastic models, we retrace the derivations of some results from matrix-exponential renewal theory and prove a new extension of a result from risk theory. Essentially the flow models allows for keeping track of the dynamics of a mechanism that generates matrix-exponential distributions in a similar way to the probabilistic arguments used for phase-type distributions involving transition rates. We also sketch a generalization of the Markovian arrival process (MAP) to the setting of matrix-exponential distribution. That process is known as the Rational arrival process (RAP).     

A new class of probability distributions via cosine and sine functions with applications

ABSTRACT

In this paper, we introduce a new class of (probability) distributions, based on a cosine-sine transformation, obtained by compounding a baseline distribution with cosine and sine functions. Some of its properties are explored. A special focus is given to a particular cosine-sine transformation using the exponential distribution as baseline. Estimations of parameters of a particular cosine-sine exponential distribution are performed via the maximum likelihood estimation method. A simulation study investigates the performances of these estimates. Applications are given for four real data sets, showing a better fit in comparison to some existing distributions based on some goodness-of-fit tests.     

Characterizing parameters of multivariate elliptical distributions*

This paper defines new parameters characterizing multivariate elliptical distributions. Mardia's coefficient of multivariate kurtosis is shown to be essentially one of these parameters. A simple relation is established between centered multivariate product moments and second moments of the variables. The general results are verified on the contaminated normal distribution as an example.     

A new discrete distrlbution,with applications to survival,dispersal and dispersion

We describe a new discrete probability distribution with several useful properties for the analysis and modelling of survival processes and dispersion. First, the model can be used to describe survival processes with monotonically decreasing, constant, or increasing hazard functions, simply by tuning one parameter. Also, the model can describe counts that are overdispersed (contagious) or underdispersed, since the variance can exceed, equal, or be less than the mean. All of these properties are demonstrated both theoretically and with ecological examples, using ad-hoc parameter estimation techniques. Finally, the equations are tractable compared with, say, the negative binomial, and easily incorporated into larger models.     

My life as a statistician * **

This article was written for a keynote speech to present a few highlights of my life as a statistician. It has seven sections beginning with an introduction in Section 17 some prospects of biostatistics in Section 2, and a tribute to J. Neyman in Section 3. In Section 4 is a sample of my work including motivation for my studies on the life table and competing risks, contributions and comments in survival analyses, a solution for Kolmogorov differential equations, and an explicit formula for the n-step transition probability in discrete time Markov chains.     

A test for the composite hypothesis that a population has a gamma distribution

A test of the composite hypothesis that a population has a gamma distribution is presented. The test is conducted by using a rank test of bivariate independence, such as the one .based on Kendallfs sample tau coefficient. The performance of the test is examined by means of a Monte Carlo study.     

Properties of Entropies of Record Values in Reliability and Life Testing Context

In this article, the entropies of record value distributions from some continuous probability models are computed and their properties are investigated, both analytically when feasible, and numerically. In an attempt to establish relationships between entropies of the parent and the corresponding record value distributions, the entropies of record value distributions associated with the uniform, exponential, Weibull, classical Pareto, normal, gamma, beta, and Cauchy distributions are considered in this article. The entropy of record value distributions associated with the uniform, exponential, Weibull, and Pareto distributions, have tractable closed forms. The entropies of record value distributions associated with the normal, gamma, beta, and Cauchy distributions do not have tractable closed forms and need further investigations. Some general conclusions are drawn in the final section.     

On the Ratio of Two Folded Normal Distributions

A class of distributions associated with the ratio of two folded normal random variables is introduced which strictly includes the half standard Cauchy distribution. The properties of this class of distributions are studied, along with a graph of the possible shapes of its density functions. The salient features of this class are mathematical tractability and statistical applicability. Utility of the class of distributions is demonstrated by presenting four applications.     

Full Bayesian Significance Test for Zero-Inflated Distributions

ABSTRACT

In this article we propose a new Kolmogorov-Smirnov type test for testing the nonlinearity of time series, based on the one which was originally introduced by An and Cheng. Our simulation study shows that the suggested test performs better than the original test. An associated bootstrap test is also found to outperform remarkably the non-bootstrap test.