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A new distribution on (0, 1), generalized Log-Lindley distribution, is proposed by extending the Log-Lindley distribution. This new distribution is shown to be a weighted Log-Lindley distribution. Important probabilistic and statistical properties have been derived. An interesting characterization of the weighted distribution in terms of Kullback-Liebler distance and weighted entropy has also been obtained. A useful result in insurance for the distorted premium principal is presented and verified with numerical calculations. New regression models for bounded responses based on this distribution and their application is illustrated by considering modeling a real life data on risk management and another data set on outpatient health expenditure in comparison with beta regression and Log-Lindley regression models. Much better fits for both data sets justify the relevance of the new distribution in statistical modeling and analysis. Furthermore this generalization, apart from adding flexibility for modelling, retains the compactness and tractability of statistical quantities required for statistical analysis, which is a feature of the Log-Lindley distribution. Thus, the generalized Log-Lindley distribution should be a useful addition to statistical models for practitioners., Comment: 37 pages; 2 Figures, 3 Table, Pre-print Version-4.0

In this paper a new generalization of the hyper-Poisson distribution is proposed using the Mittag-Leffler function. The hyper-Poisson, displaced Poisson, Poisson and geometric distributions among others are seen as particular cases. This Mittag-Leffler function distribution (MLFD) belongs to the generalized hypergeometric and generalized power series families and also arises as weighted Poison distributions. MLFD is a flexible distribution with varying shapes like non-increasing with unique mode at zero, unimodal with one / two non-zero modes. It can be under, equi or over dispersed. Various distributional properties like recurrence relation for pmf, cumulative distribution function, generating functions, formulae for different type of moments, their recurrence relations, index of dispersion, its classification, log-concavity, reliability properties like survival, increasing failure rate, unimodality, and stochastic ordering with respect to hyper-Poisson distribution have been discussed. The distribution has been found to fare well when compared with the hyper-Poisson distributions in its suitability in empirical modeling of differently dispersed count data. It is therefore expected that proposed MLFD with its interesting features and flexibility, will be a useful addition as a model for count data., Comment: 24 pages, 11 figures, 9 tables

Let X Nv(0, {\Lambda}) be a normal vector in v dimensions, where {\Lambda} is diagonal. With reference to the truncated distribution of X on the interior of a v-dimensional Euclidean ball, we completely prove a variance inequality and a covariance inequality that were recently discussed by F. Palombi and S. Toti [J. Multivariate Anal. 122 (2013) 355-376]. These inequalities ensure the convergence of an algorithm for the reconstruction of {\Lambda} only on the basis of the covariance matrix of X truncated to the Euclidean ball. The concept of monotone likelihood ratio is useful in our proofs. Moreover, we also prove and utilize the fact that the cumulative distribution function of any positive linear combination of independent chi-square variates is log-concave, even though the same may not be true for the corresponding density function.