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Multivariate circular observations, i.e. points on a torus arise frequently in fields where instruments such as compass, protractor, weather vane, sextant or theodolite are used. Multivariate wrapped models are often appropriate to describe data points scattered on p-dimensional torus. However, the statistical inference based on such models is quite complicated since each contribution in the log-likelihood function involves an infinite sum of indices in $${\mathbb {Z}}^p$$ , where p is the dimension of the data. To overcome this problem, for moderate dimension p, we propose two estimation procedures based on Expectation-Maximisation and Classification Expectation-Maximisation algorithms. We study the performance of the proposed techniques on a Monte Carlo simulation and further illustrate the advantages of the new procedures on three real-world data sets.

Spherical distributions, including the von Mises-Fisher density, have received a great attention in the literature because of their usefulness to model circular data lying on the unit sphere. However, there is a paucity of research on proposing spherical densities possing multimode tuned with a single parameter. To fill in this gap, we extend von Mises-Fisher distribution to construct a new density. Moreover, some of the important statistical properties of the proposed distribution including the estimation of parameters are highlighted. To evaluate the performance of the proposed distribution, some simulation studies and analyzing three real-life examples are presented.

The Pearson type family densities are among the most important classes of distributions that play a key role in the directional statistics. Their particular structures make them suitable candidates to analysis data on non-Euclidean space, such as sphere. To model data scattered asymmetrically on such spaces, the researchers confined themselves to extend particular distributions from the class of the Pearson type family densities. Those specific distributions are symmetric in nature but their extended versions are usually heavy tailed. In this paper, we introduce some alternative probability density functions in the class of Pearson type distributions on the sphere having the spherical Student’s $t$, Fisher and Chi-square densities as the subfamilies. Via investigating various theoretical properties of this new subclass, we show that it is inheritably asymmetric. To further evaluating this subclass, some simulation studies are conducted. Also, modeling of two real-life data using the proposed densities and then comparing their fitting consequences with those resulted from invoking other common spherical distributions are considered.

One of the new area of research emerging in the field of statistics is the shape analysis. Shape is defined as all the geometrical information of an object whose location, scale and orientation is not of interest. Diffusion in shape analysis can be studied via either perturbation of the key coordi- nates identifying the initial object or random evolution of the shape itself. Reviewing the first case, we mainly consider the second case and particularly define a new family of diffusion processes. It can be used to model diffusion phenomena represented by shape evolution such as cell motion.

SummaryWe consider the application of Markov chain Monte Carlo (MCMC) estimation methods to random-effects models and in particular the family of discrete time survival models. Survival models can be used in many situations in the medical and social sciences and we illustrate their use through two examples that differ in terms of both substantive area and data structure. A multilevel discrete time survival analysis involves expanding the data set so that the model can be cast as a standard multilevel binary response model. For such models it has been shown that MCMC methods have advantages in terms of reducing estimate bias. However, the data expansion results in very large data sets for which MCMC estimation is often slow and can produce chains that exhibit poor mixing. Any way of improving the mixing will result in both speeding up the methods and more confidence in the estimates that are produced. The MCMC methodological literature is full of alternative algorithms designed to improve mixing of chains and we describe three reparameterization techniques that are easy to implement in available software. We consider two examples of multilevel survival analysis: incidence of mastitis in dairy cattle and contraceptive use dynamics in Indonesia. For each application we show where the reparameterization techniques can be used and assess their performance.

Among many algorithms, gradient descent algorithm (GDA) is a simple tool to derive an optimal quantity in dealing with an optimization problem in the linear space. Apart from the initial value, the step size has a great impact on the convergence rate of this algorithm. Its affect on the geometric structure of the consecutive configurations is more crucial if one works with an optimization problem in the statistical shape analysis. In other words, if the step size of the GDA is not properly tuned, the geometry might not be preserved while the algorithm is moving forward to reach an optimal mean shape. In order to improve the performance of the GDA, we introduce a dynamic step size and a new criterion both to check the geometry in each step of the algorithm and to accelerate the convergence rate. These lead to a new robust algorithm on deriving the intrinsic mean on the shape space. We compare the performance of our proposed procedure to the usual GDA using a real shape data accompanied with simulation studies.