This paper provides a mixture modeling framework using the bivariate generalized exponential distribution. We study different properties of this mixture distribution. Hierarchical EM algorithm is developed for finding the estimates of the parameters. The algorithm takes very large sample size to work as it contains many stages of approximation. Numerical Results are provided for more illustration.
We study properties and parameter estimation of finite-state homogeneous continuous-time bivariate Markov chains. Only one of the two processes of the bivariate Markov chain is observable. The general form of the bivariate Markov chain studied here makes no assumptions on the structure of the generator of the chain, and hence, neither the underlying process nor the observable process is necessarily Markov. The bivariate Markov chain allows for simultaneous jumps of the underlying and observable processes. Furthermore, the inter-arrival time of observed events is phase-type. The bivariate Marko…
Researchers collecting intensive longitudinal data (ILD) are increasingly looking to model psychological processes, such as emotional dynamics, that organize and adapt across time in complex and meaningful ways. This is also the case for researchers looking to characterize the impact of an intervention on individual behavior. To be useful, statistical models must be capable of characterizing these processes as complex, time-dependent phenomenon, otherwise only a fraction of the system dynamics will be recovered. In this paper we introduce a Square-Root Second-Order Extended Kalman Filtering ap…
The correlated binomial (CB) distribution was proposed by Luceño (Computational Statistics $\&$ Data Analysis, 20, 1995, 511-520) as an alternative to the binomial distribution for the analysis of the data in the presence of correlations among events. Due to the complexity of the mixture likelihood of the model, it may be impossible to derive analytical expressions of the maximum likelihood estimators (MLEs) of the unknown parameters. To overcome this difficulty, we develop an expectation-maximization algorithm for computing the MLEs of the CB parameters. Numerical results from simulation stud…
We introduce a new class of Sequential Monte Carlo (SMC) methods, which we call free energy SMC. This class is inspired by free energy methods, which originate from Physics, and where one samples from a biased distribution such that a given function $ξ(θ)$ of the state $θ$ is forced to be uniformly distributed over a given interval. From an initial sequence of distributions $(π_t)$ of interest, and a particular choice of $ξ(θ)$, a free energy SMC sampler computes sequentially a sequence of biased distributions $(\tildeπ_{t})$ with the following properties: (a) the marginal distribution of $ξ(θ…
Statistical analyses of directional or angular data have applications in a variety of fields, such as geology, meteorology and bioinformatics. There is substantial literature on descriptive and inferential techniques for univariate angular data, with the bivariate (or more generally, multivariate) cases receiving more attention in recent years. More specifically, the bivariate wrapped normal, von Mises sine and von Mises cosine distributions, and mixtures thereof, have been proposed for practical use. However, there is a lack of software implementing these distributions and the associated infe…
We give convergence guarantees for estimating the coefficients of a symmetric mixture of two linear regressions by expectation maximization (EM). In particular, we show that the empirical EM iterates converge to the target parameter vector at the parametric rate, provided the algorithm is initialized in an unbounded cone. In particular, if the initial guess has a sufficiently large cosine angle with the target parameter vector, a sample-splitting version of the EM algorithm converges to the true coefficient vector with high probability. Interestingly, our analysis borrows from tools used in th…
The cluster-weighted model (CWM) is a mixture model with random covariates which allows for flexible clustering and density estimation of a random vector composed by a response variable and by a set of covariates. In this class of models, the generalized linear exponential CWM is here introduced especially for modeling bivariate data of mixed-type. Its natural counterpart, in the family of latent class models, is also defined. Maximum likelihood parameter estimates are derived using the EM algorithm and model selection is carried out using the Bayesian information criterion (BIC). Artificial a…
Estimation of finite mixture models when the mixing distribution support is unknown is an important problem. This paper gives a new approach based on a marginal likelihood for the unknown support. Motivated by a Bayesian Dirichlet prior model, a computationally efficient stochastic approximation version of the marginal likelihood is proposed and large-sample theory is presented. By restricting the support to a finite grid, a simulated annealing method is employed to maximize the marginal likelihood and estimate the support. Real and simulated data examples show that this novel stochastic appro…
This paper provides a framework for estimating the mean and variance of a high-dimensional normal density. The main setting considered is a fixed number of vector following a high-dimensional normal distribution with unknown mean and diagonal covariance matrix. The diagonal covariance matrix can be known or unknown. If the covariance matrix is unknown, the sample size can be as small as $2$. The proposed estimator is based on the idea that the unobserved pairs of mean and variance for each dimension are drawn from an unknown bivariate distribution, which we model as a mixture of normal-inverse…
In the mixture modeling frame, this paper presents the polynomial Gaussian cluster-weighted model (CWM). It extends the linear Gaussian CWM, for bivariate data, in a twofold way. Firstly, it allows for possible nonlinear dependencies in the mixture components by considering a polynomial regression. Secondly, it is not restricted to be used for model-based clustering only being contextualized in the most general model-based classification framework. Maximum likelihood parameter estimates are derived using the EM algorithm and model selection is carried out using the Bayesian information criteri…
There are three equivalent ways of representing two jointly observed real-valued signals: as a bivariate vector signal, as a single complex-valued signal, or as two analytic signals known as the rotary components. Each representation has unique advantages depending on the system of interest and the application goals. In this paper we provide a joint framework for all three representations in the context of frequency-domain stochastic modeling. This framework allows us to extend many established statistical procedures for bivariate vector time series to complex-valued and rotary representations…
Mixture models are commonly used when data show signs of heterogeneity and, often, it is important to estimate the distribution of the latent variable responsible for that heterogeneity. This is a common problem for data taking values in a Euclidean space, but the work on mixing distribution estimation based on directional data taking values on the unit sphere is limited. In this paper, we propose using the predictive recursion (PR) algorithm to solve for a mixture on a sphere. One key feature of PR is its computational efficiency. Moreover, compared to likelihood-based methods that only suppo…
A bivariate distribution with continuous margins can be uniquely decomposed via a copula and its marginal distributions. We consider the problem of estimating the copula function and adopt a Bayesian approach. On the space of copula functions, we construct a finite-dimensional approximation subspace that is parametrized by a doubly stochastic matrix. A major problem here is the selection of a prior distribution on the space of doubly stochastic matrices also known as the Birkhoff polytope. The main contributions of this paper are the derivation of a simple formula for the Jeffreys prior and sh…
The integration and transfer of information from multiple sources to multiple targets is a core motive of neural systems. The emerging field of partial information decomposition (PID) provides a novel information-theoretic lens into these mechanisms by identifying synergistic, redundant, and unique contributions to the mutual information between one and several variables. While many works have studied aspects of PID for Gaussian and discrete distributions, the case of general continuous distributions is still uncharted territory. In this work we present a method for estimating the unique infor…
