A Bayesian solution to the Behrens–Fisher problem

A simple solution to the Behrens–Fisher problem based on Bayes factors is presented, and its relation with the Behrens–Fisher distribution is explored. The construction of the Bayes factor is based on a simple hierarchical model, and has a closed form based on the densities of general Behrens–Fisher distributions. Simple asymptotic approximations of the Bayes factor, which are functions of the Kullback–Leibler divergence between normal distributions, are given, and it is also proved to be consistent. Some examples and comparisons are also presented.

APPLICATIONS OF TWO NON-CENTRAL HYPERGEOMETRIC DISTRIBUTIONS OF BIASED SAMPLING STATISTICAL MODELS

Statistical models of biased sampling of two non-central hypergeometric distributions Wallenius' and Fisher's distribution has been extensively used in the literature, however, not many of the logic of hypergeometric distribution have been investigated by different techniques. This research work examined the procedure of the two non-central hypergeometric distributions and investigates the statistical properties which includes the mean and variance that were obtained. The parameters of the distribution were estimated using the direct inversion method of hyper simulation of biased urn model in the environment of R statistical software, with varying odd ratios (w) and group sizes (mi). It was discovered that the two non - central hypergeometric are approximately equal in mean, variance and coefficient of variation and differ as odds ratios (w) becomes higher and differ from the central hypergeometric distribution with ω = 1. Furthermore, in univariate situation we observed that Fisher distribution at (ω = 0.2, 0.5, 0.7, 0.9) is more consistent than Wallenius distribution, although central hypergeometric is more consistent than any of them. Also, in multinomial situation, it was observed that Fisher distribution is more consistent at (ω = 0.2, 0.5), Wallenius distribution at (ω = 0.7, 0.9) and central hypergeometric at (ω = 0.2)

Progressive von Mises–Fisher Filtering Using Isotropic Sample Sets for Nonlinear Hyperspherical Estimation

In this work, we present a novel scheme for nonlinear hyperspherical estimation using the von Mises–Fisher distribution. Deterministic sample sets with an isotropic layout are exploited for the efficient and informative representation of the underlying distribution in a geometrically adaptive manner. The proposed deterministic sampling approach allows manually configurable sample sizes, considerably enhancing the filtering performance under strong nonlinearity. Furthermore, the progressive paradigm is applied to the fusing of measurements of non-identity models in conjunction with the isotropic sample sets. We evaluate the proposed filtering scheme in a nonlinear spherical tracking scenario based on simulations. Numerical results show the evidently superior performance of the proposed scheme over state-of-the-art von Mises–Fisher filters and the particle filter.