In this manuscript, we develop a unified joint modelling and estimation framework for zero-inflated count and longitudinal semi-continuous data, with a focus on models structured around the exponential family and two-part hurdle formulations. We first review and synthesize existing longitudinal hurdle models, identifying a common structure across diverse approaches. Motivated by this foundation, we introduce novel joint models that integrate semi-continuous longitudinal outcomes with time-to-event data, and propose new methods for dynamic prediction in the presence of semi-continuous outcomes. To facilitate flexible estimation and inference across this class of models, we propose a Bayesian estimation strategy based on a Markov Chain Monte Carlo (MCMC) algorithm. We have implemented these methods in the R package UHJM (available at https://github.com/tbaghfalaki/UHJM), providing accessible tools for parameter estimation and risk prediction. The utility of our framework is demonstrated through simulation studies and two real-world applications characterized by excess zeros.

A Bayesian update method for exponential family projection filters with non-conjugate likelihoods

A distributed inference algorithm for Dirichlet process mixture models with exponential family components

In medical research, factors such as loss to follow-up can lead to the precise occurrence time of the event of interest for study subjects being unobserved, resulting in right-censored data. The covariates within such data often exhibit complex network structures. Consequently, our study constructs a proportional odds model that incorporates network structures, drawing from both the proportional odds model and the exponential family graphical model. We employ the Sieve maximum likelihood method to estimate unknown parameters while simultaneously identifying the network structure. To validate the effectiveness of the proposed method, we conducted numerical simulations under various settings. The results indicate that the method achieves high accuracy in parameter estimation and network structure identification. Finally, the proposed method was applied to a study on type 2 diabetic retinopathy to explore potential factors influencing the development of retinopathy.

Abstract Recently, a Wasserstein analogue of the Cramer–Rao inequality has been developed using the Wasserstein information matrix (Otto metric). This inequality provides a lower bound on the Wasserstein variance of an estimator, which quantifies its robustness against additive noise. In this study, we investigate conditions for an estimator to attain the Wasserstein–Cramer–Rao lower bound (asymptotically), which we call the (asymptotic) Wasserstein efficiency. We show a condition under which Wasserstein efficient estimators exist for one-parameter statistical models. This condition corresponds to a recently proposed Wasserstein analogue of one-parameter exponential families (e-geodesics). We also show that the Wasserstein estimator, a Wasserstein analogue of the maximum likelihood estimator based on the Wasserstein score function, is asymptotically Wasserstein efficient in location-scale families.

Abstract This article investigates the pseudo phase transition behavior of the six-state clock model on two-dimensional finite-size lattices. By employing the Wang–Landau sampling method and microcanonical inflection-point analysis, we identified two phase transition points and classified both as continuous transitions within the microcanonical framework. Using Metropolis sampling and canonical ensemble analysis, we verified the accuracy of the transition points obtained from the microcanonical approach and further pinpointed the location of a fourth-order dependent transition with high accuracy. Moreover, the fourth-order dependent transition points extracted from two canonical order parameters—the average cluster perimeter and the $M_{\mathbb{Z}_6}$ symmetry—are in excellent agreement and consistently reinforce each other. Through a detailed analysis of the $M_{\mathbb{Z}_6}$ symmetry, we further demonstrate that the position of this fourth-order dependent transition may function as a critical transition warning indicator for both primary phase transitions in the system.

Neurons process sensory stimuli efficiently, showing sparse yet highly variable ensemble spiking activity involving structured higher-order interactions. Notably, while neural populations are mostly silent, they occasionally exhibit highly synchronous activity, resulting in sparse and heavy-tailed spike-count distributions. However, its mechanistic origin-specifically, what types of nonlinear properties in individual neurons induce such population-level patterns-remains unclear. In this study, we derive sufficient conditions under which the joint activity of homogeneous binary neurons generates sparse and widespread population firing rate distributions in infinitely large networks. We then propose a subclass of exponential family distributions that satisfy this condition. This class incorporates structured higher-order interactions with alternating signs and shrinking magnitudes, along with a base-measure function that offsets distributional concentration, giving rise to parameter-dependent sparsity and heavy-tailed population firing rate distributions. Analysis of recurrent neural networks that recapitulate these distributions reveals that individual neurons possess threshold-like nonlinearity, followed by supralinear activation that jointly facilitates sparse and synchronous population activity. These nonlinear features resemble those in modern Hopfield networks, suggesting a connection between widespread population activity and the network's memory capacity. The theory establishes sparse and heavy-tailed distributions for binary patterns, forming a foundation for developing energy-efficient spike-based learning machines.

On the ${q}$-Continuous Natural Exponential Family

Count data are present in many areas of everyday life. Unfortunately, such data are often characterized by over- and under-dispersion. In 1986, Efron introduced the Double Poisson distribution to account for this problem. The aim of this work is to examine the application of this distribution in regression analyses performed in health-related literature by means of a narrative review. The databases Science Direct, PBSC, Pubmed PsycInfo, PsycArticles, CINAHL and Google Scholar were searched for applications. Two independent reviewers extracted data on Double Poisson Regression Models and their applications in the health and life sciences. From a total of 1644 hits, 84 articles were pre-selected and after full-text screening, 13 articles remained. All these articles were published after 2011 and most of them targeted epidemiological research. Both over- and under-dispersion was present and most of the papers used the generalized additive models for location, scale, and shape (GAMLSS) framework. In summary, this narrative review shows that the first steps in applying Efron’s idea of double exponential families for empirical count data have already been successfully taken in a variety of fields in the health and life sciences. Approaches to ease their application in clinical research should be encouraged.

This research is centered on exploring a mathematically tractable and versatile family of probability distributions, specifically focusing on one family member. We have used the exponential distribution as a base distribution to create a novel distribution, which we have aptly named the "New Odd-type Exponential Distribution." In this paper, we provide an overview of the essential characteristics inherent to this innovative distribution. This model showcases a variety of hazard functions, including the reverse-J, decreasing, constant, increasing and constant, bathtub, and S-shaped shapes. The estimation of the distribution's parameters is conducted through both classical and Bayesian methods. We validate the accuracy of the classical estimation procedure through simulation studies. These simulations demonstrate a reduction in biases and mean square errors as sample sizes increase, even for smaller samples. To showcase the practicality of the proposed distribution, we apply it to two sets of real-world data, employing both classical and Bayesian approaches. We evaluate the performance of our suggested distribution model using various model selection criteria and goodness-of-fit test statistics. Empirical evidence from these evaluations confirms that our proposed model surpasses some existing models in the literature. Further, the suggested model was analyzed using the Bayesian approach and the Hamiltonian Monte Carlo method. Its predictive capability was also explored by using the posterior predictive checks, and it can predict the data consistently.

This paper deals with problems of sequential testing of two hypotheses, when the observations follow a distribution from a one-parameter exponential family. It is assumed that the observations come in groups of predefined size (group sequential setting) and that no more than N < ∞ groups will be observed. We suppose that at stage n, a group (sample) containing m observations has a cost of c n ( m ) monetary units, n = 1 , 2 , … , N . We consider as the objective function the expected sampling cost, weighted between some fixed parameter points, which should be minimized over all tests whose error probabilities of the first and the second kind do not exceed some α and β , respectively. We propose a computer-oriented method of construction of the optimal group sequential plans for this problem and develop algorithms for computing them and their performance characteristics. The algorithms are implemented in the R programming language, and are available from a public GitHub repository on the Internet. A series of applications to situations seen in the literature are considered.

We consider the lossless compression bound of any individual data sequence. Conceptually, its Kolmogorov complexity is such a bound yet uncomputable. According to Shannon's source coding theorem, the average compression bound is nH, where n is the number of words and H is the entropy of an oracle probability distribution characterizing the data source. The quantity nH(θ^n) obtained by plugging in the maximum likelihood estimate is an underestimate of the bound. Shtarkov showed that the normalized maximum likelihood (NML) distribution is optimal in a minimax sense for any parametric family. Fitting a data sequence-without any a priori distributional assumption-by a relevant exponential family, we apply the local asymptotic normality to show that the NML code length is nH(θ^n)+d2logn2π+log∫Θ|I(θ)|1/2dθ+o(1), where d is dictionary size, |I(θ)| is the determinant of the Fisher information matrix, and Θ is the parameter space. We demonstrate that sequentially predicting the optimal code length for the next word via a Bayesian mechanism leads to the mixture code whose length is given by nH(θ^n)+d2logn2π+log|I(θ^n)|1/2w(θ^n)+o(1), where w(θ) is a prior. The asymptotics apply to not only discrete symbols but also continuous data if the code length for the former is replaced by the description length for the latter. The analytical result is exemplified by calculating compression bounds of protein-encoding DNA sequences under different parsing models. Typically, compression is maximized when parsing aligns with amino acid codons, while pseudo-random sequences remain incompressible, as predicted by Kolmogorov complexity. Notably, the empirical bound becomes more accurate as the dictionary size increases.

Patients affected by neurological disorders usually present substantial heterogeneity in multi-domain biomarkers and clinical measures. This heterogeneity arises from differences in disease stage, unique characteristics, and membership in distinct latent subtypes. Exploring such complex heterogeneity and identifying disease progression-related markers is crucial for early diagnosis and developing timely and targeted interventions. This paper proposes a mixture exponential family trajectory model to integrate markers from multiple modalities to learn the disease progression. We incorporate continuous neuroimaging and microRNA sequencing biomarkers, categorical clinical symptoms, and ordinal cognitive markers using appropriate exponential family distributions with lower-dimensional latent factors. The mixture model assigns subtype-specific parameters to these distributions for each mixture component, enabling the characterization of patients in heterogeneous latent subgroups. The proposed model can also describe the nonlinear trajectory of disease deterioration and provide a temporal sequence of decline for each marker. We develop a Bayesian estimation procedure coupled with efficient Markov chain Monte Carlo (MCMC) sampling schemes to perform statistical inference for the mixture model. The proposed method is assessed through extensive simulation studies and an application to Parkinson's Progression Markers Initiative (PPMI) to learn the temporal ordering and subtypes of neurodegeneration of Parkinson's disease (PD).

One parameter family of exact solutions in General Relativity with a scalarfield has been found using the Liouville metric. The scalar field potential hasexponential form. The solution corresponding to the naked singularity providessmooth extension of the Friedmann Universe with accelerated expansion throughthe zero of the scale factor back in time. All geodesics are found explicitly.Their analysis shows that the Liouville solution is a global one: everygeodesic is either continued to infinite value of the canonical parameter inboth directions or ends up at the singularity at its finite value. Moreover,analysis of geodesics shows that the naked singularity located outside theFriedmann Universe attracts matter and therefore provides its acceleratingexpansion inside the light cone.

The discussion revolves around the broader concept of exponential family distributions, exploring their applications in survival analysis and reliability engineering. Various distributions within this family, such as the Weibull, Rayleigh, and Gompertz distributions, are examined in terms of their suitability for modelling different phenomena, including instantaneous failure events, independent sums of events, and decreasing processes over time. Additionally, a newly derived distribution, termed the Meenakshi’s-Gompertz distribution, is introduced, with its parameters interchangeably representing scale and shape properties.

Abstract This article considers exponential families of truncated multivariate normal distributions with one-sided truncation for some or all coordinates. We observe that if all components are one-sided truncated, then this family is not full. The family of truncated multivariate normal distributions is extended to a full family, and the extended family is investigated in detail. We identify the canonical parameter space of the extended family and establish that the family is not regular and not even steep. We also consider maximum likelihood estimation for the location vector parameter and the positive definite (symmetric) matrix dispersion parameter of a truncated non-singular multivariate normal distribution. It is shown that if the sample size is sufficiently large then, almost surely, the maximizer of the likelihood function is unique, provided that it exists. It is also shown that each solution to the score equations for the location and dispersion parameters satisfies the method-of-moments equations. Finally, it is observed that similar results arise in the case of an arbitrary number of truncated components.

In this study, we designed a new family of distributions called new odd reparameterized exponential transformed-X family of distributions. This was achieved by utilizing an exponential distribution with a constant scale parameter as the transformer and then the odd function of the baseline distribution as the generalizer. The new family was used to extend the classical Weibull distribution. We further studied the characteristics of the new extended Weibull distribution which include the quantile function, moment, moment generating function, mean residual life function, order statistic, entropy and extropy. Again, the parameters were estimated using both non-Bayesian and Bayesian approaches. A comprehensive Monte Carlo simulation was conducted under four different scenarios. The proposed distribution was fitted to HIV/AIDS and COVID-19 data and then compared with the baseline distribution (Weibull) and other related models. The new distribution commands superior fit with a probability value of 0.9959 and 0.7086 in the two datasets respectively.

Optimised conjugate prior for model structure estimation in the exponential family

TVEG: Model Selection of the Time-Varying Exponential Family Distributions Graphical Models

Abstract The Bernstein–von Mises theorem (BvM) gives conditions under which the posterior distribution of a parameter $\theta \in \varTheta \subseteq \mathbb R^{d}$ based on $n$ independent samples is asymptotically normal. In the high-dimensional regime, a key question is to determine the growth rate of $d$ with $n$ required for the BvM to hold. We show that up to a model-dependent coefficient, $n\gg d^{2}$ suffices for the BvM to hold in two settings: arbitrary generalized linear models, which include exponential families as a special case, and multinomial data, in which the parameter of interest is an unknown probability mass functions on $d+1$ states. Our results improve on the tightest previously known condition for posterior asymptotic normality, $n\gg d^{3}$. Our statements of the BvM are non-asymptotic, taking the form of explicit high-probability bounds. To prove the BvM, we derive a new simple and explicit bound on the total variation distance between a measure $\pi \propto \,\mathrm{e}^{-nf}$ on $\varTheta \subseteq \mathbb R^{d}$ and its Laplace approximation.