To model count data with excess zeros, ones and twos, for the first time we introduce a so-called zero-one-two-inflated Poisson (ZOTIP) distribution, including the zero-inflated Poisson (ZIP) and the zero-and-one-inflated Poisson (ZOIP) distributions as two special cases. We establish three equivalent stochastic representations for the ZOTIP random variable to develop important distributional properties of the ZOTIP distribution. The Fisher scoring and expectation–maximization (EM) algorithms are derived to obtain the maximum likelihood estimates of parameters of interest. Bootstrap confidence intervals are also provided. Testing hypotheses are considered, simulation studies are conducted, and two real data sets are used to illustrate the proposed methods.

In this paper, we propose a new multivariate Laplace distribution from normal variance mixture models, called as Type II multivariate Laplace distribution. Unlike the original multivariate Laplace distribution whose all components must have the same value for the mixing variate, the random components in the new distribution could have different value for its own mixing variate and are correlated only through the dependence structure of the normal random vector. Thus, it contains the multiplication of independently identical distribution univariate Laplace distributions as a special case if the normal covariance matrix is diagonal. A tractable stochastic representation is used to derive the probability density function and other statistical properties. The maximum likelihood estimates of parameters via an ECM (expectation/conditional maximization) algorithm and the Bayesian methods are derived. Some simulation studies are conducted to evaluate the performance of the proposed methods. Applications in two real data sets indicate that the Type II multivariate Laplace distribution could have a better performance and is distinct from the original one.

I have a confession: although I have read biographies of crystallographers, this is the first book on crystallography that I have read from cover to cover. I am one of those crystallographers who h...

Bivariate continuous negatively correlated proportional data defined in the unit square ${(0,1)}^2$ often appear in many different disciplines, such as medical studies, clinical trials and so on. To model this type of data, the paper proposes two new bivariate continuous distributions (i.e., negatively correlated proportional inverse Gaussian (NPIG) and negatively correlated proportional gamma (NPGA) distributions) for the first time and provides corresponding distributional properties. Two mean regression models are further developed for data with covariates. The normalized expectation–maximization (N-EM) algorithm and the gradient descent algorithm are combined to obtain the maximum likelihood estimates of parameters of interest. Simulations studies are conducted, and a data set of cortical thickness for schizophrenia is used to illustrate the proposed methods. According to our analysis between patients and controls of cortical thickness in typical mutual inhibitory brain regions, we verified the compensatory of cortical thickness in patients with schizophrenia and found its negative correlation with age.