Topic
Natural exponential family
About: Natural exponential family is a research topic. Over the lifetime, 1973 publications have been published within this topic receiving 60189 citations. The topic is also known as: NEF.
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TL;DR: In this article, a novel approach to data analysis using fractional order calculus is presented, which can be applied to any distribution and shows remarkable improvement even if the parameters of a particular distribution have been optimised to achieve the best fit to data.
12 citations
TL;DR: The moment distributions of a nonnegative random variable are defined and their applications in life length studies are indicated in this paper, where some properties of the moment distributions are employed to characterize the discrete distributions in the class of modified power series distributions introduced by the author.
Abstract: The moment distributions of a nonnegative random variable are defined and their applications in life length studies are indicated. Some properties of the moment distributions are employed to characterize the discrete distributions in the class of modified power series distributions introduced by the author (1974). In particular, characterisations of Poisson, binomial and negative binomial distributions are obtained.
12 citations
TL;DR: In this article, an exponential family of distributions which generalises the exponential distribution for censored failure time data is analyzed, analogous to the way in which the class of generalised linear models generalizes the normal distribution.
Abstract: We analyse an exponential family of distributions which generalises the exponential distribution for censored failure time data, analogous to the way in which the class of generalised linear models generalises the normal distribution. The parameter of the distribution depends on a linear combination of covariates via a possibly nonlinear link function, and we allow another level of heterogeneity: the data may contain "immune" individuals who are not subject to failure. Thus the data is modelled by a mixture of a distribution from the exponential family and a "mass at infinity" representing individuals who never fail. Our results include large sample distributions for parameter estimators and for hypothesis test statistics obtained by maximising the likelihood of a sample. The asymptotic distribution of the likelihood ratio test statistic for the hypothesis that there are no immunes present in the population is shown to be "non-standard"; it is a 50-50 mixture of a chi-squared distribution on 1 degree of freedom and a point mass at 0. Our analysis clearly shows how "negligibility" of individual covariate values and "sufficient followup" conditions are required for the asymptotic properties.
12 citations
TL;DR: In this article, the Marshall-Olkin generalized Erlang-truncated exponential (MOGETE) distribution was introduced as a generalization of the ETE distribution.
Abstract: This article introduces the Marshall–Olkin generalized Erlang-truncated exponential (MOGETE) distribution as a generalization of the Erlang-truncated exponential (ETE) distribution. The hazard rate of the new distribution could be increasing, decreasing or constant. Explicit-closed form mathematical expressions of some of the statistical and reliability properties of the new distribution were given and the method of maximum likelihood estimation was used to estimate the model parameters. The usefulness and flexibility of the new distribution was illustrated with two real and uncensored lifetime data-sets. The MOGETE distribution with a smaller goodness of fit statistics always emerged as a better candidate for the data-sets than the ETE, Exp Frechet and Exp Burr XII distributions.
12 citations
TL;DR: In this article, the authors introduce a univariate family of distributions, called the beta-log-normal family, motivated by a mixture representation of some of the more typical distributions and which generalises naturally to the multivariate case.
Abstract: In the context of parametric survival analysis, it is necessary to specify probability distributions on (0, ∞). Typically, the exponential, Weibull, gamma, Pareto or log-normal is used. However, attempts to generalise these distributions to a multivariate setting have proved problematic. This paper introduces a univariate family of distributions, called the beta-log-normal family, motivated by a mixture representation of some of the more typical distributions and which generalises naturally to the multivariate case.
12 citations