Search Results (100)

In this article, we propose a first-order integer-valued autoregressive (INAR(1)) model based on binomial thinning, in which the innovation sequence follows a zero–one–two-modified Poisson (ZOTMPo) distribution. The proposed model accommodates overdispersion and allows for inflation or deflation at low-count values, particularly at zero, one, and two, which are commonly observed in public health count time series. We derive the main probabilistic properties of the model and estimate the unknown parameters using the conditional maximum likelihood (CML) method. A Monte Carlo simulation study is conducted to evaluate the finite-sample performance of the estimators. Furthermore, we establish the information-theoretic properties of the model, specifically deriving the Shannon entropy and conditional entropy bounds to quantify the dynamical complexity and predictability of the stochastic process. The practical utility of the model is illustrated using two real-world datasets on dengue fever incidence and Escherichia coli (E. coli) enteritis. Model performance is assessed using standard information criteria and forecast accuracy measures, as well as the Euclidean distance between observed and fitted probabilities for zero, one, and two. Diagnostic checks, including analysis of residual autocorrelation, cumulative periodograms, and jump process behavior, provide further confirmation of the fitted model’s adequacy. The results indicate that the proposed ZOTMPo–INAR(1) model provides an effective framework for modeling overdispersed count time series with a modified low-count structure. Full article

Modeling count data using generalized linear mixed models (GLMMs) is a common practice in ecological research. However, datasets such as bird vocalization counts often display overdispersion, where the variance exceeds the mean, violating Poisson assumptions and potentially leading to biased inference. This study investigates the call counts of owl nestlings by comparing Poisson and Negative Binomial (NB) GLMMs to assess the impact of overdispersion on model performance. The models incorporate a random intercept for nest identity, thereby accounting for the hierarchical structure of the data and the correlation among observations within the same nest. Parameter estimation was performed using the Laplace approximation and the Adaptive Gauss–Hermite Quadrature (AGHQ) method to evaluate likelihood-based inference under different estimation approaches. Although the overdispersion diagnostic indicated extra-Poisson variation, model comparison based on the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and log-likelihood showed that the Poisson GLMM provided a better overall fit than the NB-GLMM. These findings demonstrate that the presence of overdispersion alone does not necessarily justify replacing a Poisson GLMM with a NB-GLMM and highlight the importance of combining distributional diagnostics with likelihood-based model selection when analyzing hierarchical ecological count data. Full article

The Poisson–Three-Parameter Lindley (PTPL) distribution constitutes a flexible Poisson mixture model for overdispersed count data, encompassing several classical count distributions as special or limiting cases. Despite its growing use in applied contexts, no formal goodness-of-fit test specifically designed for this distribution is currently available. In this paper, we propose and study a new goodness-of-fit test for the PTPL model based on a Cramér–von Mises type distance between the empirical and theoretical probability generating functions (PGFs). For polynomial weight functions, the test statistic admits an explicit closed-form representation; in practice, it is computed efficiently via numerical quadrature. The null distribution of the statistic is approximated via parametric bootstrap. We establish theoretical properties of the proposed procedure, including consistency against fixed alternatives and the validity of the bootstrap approximation. Monte Carlo simulations with sample sizes $n\in \{50, 100, 150, 200, 500\}$ for size evaluation and $n\in \{100, 250, 500\}$ for power comparisons, as well as weight exponents $a\in \{0, 1, 2\}$, show that the empirical size is well controlled at both the 5% and 10% nominal levels, and that the test exhibits competitive power against Poisson, Negative Binomial, COM-Poisson, and Zero-Inflated Poisson alternatives. A real data application to five overdispersed count datasets further illustrates the practical utility of the method. The empirical size is further verified across twelve parameter configurations spanning dispersion indices from 1.37 to 59.33, confirming bootstrap validity under strong overdispersion. Full article

Background and Objectives: Macroeconomic instability, particularly income loss, inflation and unemployment, is increasingly recognized as a psychosocial stressor that may influence both symptom burden and healthcare-seeking behavior. This single-center study investigated the association of income, inflation and unemployment with private-sector hospital visits for primary headache disorders and assessed whether economic stressors were associated with different patterns across demographic groups. Materials and Methods: We conducted a single-center, retrospective, ecological quarterly time-series analysis of hospital visits for primary headache disorders between 2016 and 2024 in a private tertiary care hospital in Turkey. After exclusions, 18,522 eligible hospital-visit records were included and categorized by sex and age (<18, 18–64, and ≥65 years). National data on real gross domestic product (GDP), consumer price index (CPI), unemployment and a COVID-19 period indicator were used. Counts were modeled with log-linked Poisson or negative binomial generalized linear models selected through overdispersion diagnostics, with seasonal controls and HAC-robust inference. Results: In most groups, higher GDP was associated with more primary headache visits, whereas higher inflation was consistently associated with fewer visits. The association with unemployment was heterogeneous: visits decreased significantly among the working-age population but increased among older adults. Contemporaneous models outperformed one-quarter lagged alternatives, suggesting that private-sector healthcare seeking may change within the same quarter as macroeconomic shocks. Conclusions: In this private hospital setting, macroeconomic deterioration was associated with reduced primary headache visits, particularly among working-age patients. These findings suggest that financial constraints may suppress private-sector healthcare utilization despite possible increases in stress-related symptoms, and that private hospital data may underestimate headache-related healthcare need during economic crises. Full article

Modeling over-dispersed count data is a common challenge in applied statistics, especially in engineering applications where repeated events, system faults, and clustered observations often produce variability beyond that allowed by the classical Poisson model. In this paper, we introduce and study the Poisson–QGamma distribution, a new compound discrete model obtained by mixing the Poisson distribution with the QGamma distribution. The proposed distribution is analytically tractable and flexible enough to capture over-dispersion, skewness, and excess kurtosis, which are frequently observed in real count data. Several statistical properties of the distribution are derived, including the probability mass function, cumulative distribution function, survival and hazard rate functions, moments, dispersion index, skewness, kurtosis, entropy, and generating functions. Parameter estimation is considered using maximum likelihood, method of moments, least squares, and weighted least squares methods. The finite-sample behavior of these estimators is examined through Monte Carlo simulation. A regression model based on the Poisson–QGamma distribution is also developed for count responses with covariates. The proposed model is compared with classical and competing count models using simulation and real-data applications. Three engineering-related datasets, involving power grid failure counts, environmental sensor event counts, and packet loss counts in communication networks, are analyzed to illustrate the practical value of the model. The results show that the Poisson–QGamma model provides a better fit than several standard alternatives, including the Poisson, negative binomial, Poisson–Lindley, generalized Poisson, and COM–Poisson models, particularly in the presence of over-dispersion and heavy-tailed behavior. Overall, the proposed distribution offers a parsimonious and effective tool for modeling over-dispersed count data, while also contributing to the broader class of compound discrete distributions. Full article

The Poisson exponentially weighted moving average (PEWMA) control chart rests upon the equidispersion assumption of the pure Poisson distribution, a structural symmetry condition stipulating that the process mean and variance are equal. In manufacturing environments characterized by latent process heterogeneity, this assumption is systematically violated: the resulting distributions are inherently asymmetric, heavily right-skewed, and overdispersed. This structural asymmetry renders standard PEWMA control limits artificially narrow, inducing a substantial inflation of false alarm rates. This paper introduces the Poisson mixture EWMA (PM-EWMA) control chart, which models the latent heterogeneous structure of count data as a finite Poisson mixture distribution, with parameters estimated via the Expectation–Maximization (EM) algorithm without requiring prior labeling of process states. The optimal number of components is determined via the Bayesian Information Criterion (BIC) as the primary criterion, supplemented by the Akaike Information Criterion (AIC), its bias-corrected variant (AICc), and the log-likelihood ratio diagnostic. The PM-EWMA chart incorporates the exact mixture variance, accounting for both within-component and between-component variability, into the EWMA control limit structure, thereby providing a theoretically justified correction under the fitted Poisson mixture assumption. A Monte Carlo simulation study comprising 495 factorial configurations benchmarks the PM-EWMA chart against both the standard PEWMA chart and the negative binomial EWMA (NB-EWMA) chart with oracle dispersion calibration, confirming stable in-control ARL performance and demonstrating improved discrimination relative to the misspecified PEWMA baseline. Empirical validation using fabric defect count data from two textile manufacturers in Türkiye, with Overdispersion Indices of 6.01 and 2.74, respectively, demonstrates false alarm reductions ranging from 40.9% to 89.2% relative to the standard PEWMA chart, depending on the smoothing parameter and degree of overdispersion. Full article

This paper introduces the Touchard process, a flexible two-parameter stochastic framework for modeling count data that depart from the classical Poisson assumptions. In contrast to standard Poisson processes, the proposed model allows for both nonstationary and dependent increments, enabling the representation of overdispersion, underdispersion, and temporal dependence within a unified structure. The main contribution lies in extending weighted Poisson models to a stochastic-process setting through recursively defined transition probabilities associated with Touchard marginal distributions. We derive key theoretical properties, including admissibility conditions and a recursive formulation for the transition probabilities, and propose an efficient simulation algorithm. Maximum likelihood estimation is developed for parameter inference, and a likelihood ratio framework is used for model comparison. An empirical application to motor vehicle crash data illustrates the ability of the model to capture dynamic patterns that are not adequately described by classical Poisson-based approaches. Full article

Count outcomes are commonly analyzed using Poisson regression, but empirical data often exhibit overdispersion, excess ties, heaping, or other departures from the Poisson distribution. This paper evaluates a rank-Poisson transformation, denoted poisrank, designed to map observed counts onto Poisson quantiles before fitting a Poisson regression model. Our goal is to test whether a rank-Poisson transformation offers a useful general-purpose strategy when count data do not satisfy Poisson assumptions. Using an empirical example and a Monte Carlo simulation study with Poisson, overdispersed, rounded, and gapped count distributions, we compared Poisson regression on raw counts, Poisson regression after the poisrank transformation, quasi-Poisson regression, and additional comparison approaches. Although the transformation made the marginal distribution more similar to a Poisson distribution, it generally did not outperform standard alternatives for inference. In particular, quasi-Poisson regression more consistently maintained appropriate rejection rates with overdispersion whereas poisrank tended to be conservative and often reduced power. These findings suggest that the rank-Poisson transformation is better understood as an exploratory robustness device than as a preferred replacement for established count-data methods. Full article

The rational-power shifted Lagrangian distribution and its corresponding regression model are discrete Lagrange probability distributions. The proposed model is constructed from a shifted Lagrangian framework with a rational-power component that introduces an additional shape parameter and provides greater flexibility in modeling dispersion and tail behavior. The derivation of the distribution is presented, and its main statistical properties are discussed, together with maximum likelihood estimation based on the expected Fisher information matrix. Using this distribution, a rational-power shifted Lagrangian regression model is formulated for analyzing count data. Simulation results are used to examine the performance of the parameter estimators and to compare the proposed model with the Poisson and modified Sunil models. A real data application using domestic violence data is also provided to illustrate its practical usefulness. The proposed model has a better fit and lower information criteria than the competing models, suggesting it could be used to model overdispersed count data. Full article

We develop a semi-parametric framework for representing discrete probability mass functions through orthogonal polynomial representations. Classical count models, such as the Poisson and negative binomial distributions, impose restrictive structural assumptions that often fail to accommodate empirical features including heavy overdispersion, multimodality, and nonstandard tail behavior. To address these limitations, we introduce a linear-tilt model constructed from orthonormal polynomial systems associated with Poisson and negative binomial baselines, namely the Charlier and Meixner families. The proposed representation improves the baseline distribution using additional information from empirical moments. This allows the distribution to flexibly adjust its shape, capturing differences in skewness and kurtosis. We establish theoretical properties of the expansion within a weighted Hilbert space formulation, where the coefficients arise as orthogonal projections that can be expressed as expectations of the corresponding polynomial basis functions. In addition, we analyze approximation behavior and provide numerical bounds on the resulting numerical error and convergence properties of truncated approximations. The practical relevance of the proposed methodology is illustrated through applications to several empirical datasets, demonstrating its ability to capture complex distributional structures while preserving a tractable semi-parametric form. Full article

Extended reality (XR), encompassing augmented reality (AR), virtual reality (VR), and mixed reality (MR), is a key enabling technology for virtual worlds, and XR-related patents continue to grow rapidly. However, patent-based XR technology analysis faces a fundamental challenge: document–keyword matrix (DKM) built from patent titles and abstracts are typically high dimensional, sparse, and often exhibit excess zeros, which can distort inference when conventional text mining pipelines are applied without a generative count perspective. In this study, we propose a statistically grounded XR technology analysis framework that combines likelihood-based count modeling with interpretable structure mining to map XR sub-technologies from a patent DKM. Using an XR patent–keyword matrix, we fit Poisson regression (PR), negative binomial regression (NBR), and zero-inflated negative binomial regression (ZINBR) models via maximum likelihood estimation (MLE), controlling for document-length effects. Model selection by Akaike information criterion (AIC) consistently favored NBR for both target keywords, indicating substantial overdispersion in XR patent counts. We interpret exponentiated coefficients as incidence rate ratios (IRRs) and construct a technology relatedness network from significant IRR edges, revealing a dual-axis XR structure: reality is anchored in an AR or VR experience and content axis such as virtual and augment, whereas extend is embedded in a structure and integration axis for example, surface, edge, layer, and connectivity-related terms. To show how the proposed method can be applied to real domains, we searched the XR patent documents, and analyzed them for XR technology analysis. Full article

This study evaluates the performance of a semi-dynamic negative binomial model with cubic spline smoothing to characterize the spatiotemporal dynamics of COVID-19 mortality in Peru, a setting marked by significant data inconsistency and reporting delays. Using nationwide weekly mortality data, we compared a Poisson regression against a semi-dynamic NB model with a population offset and cubic splines (df = 6). The models were evaluated using Akaike Information Criterion and log-likelihood to handle overdispersion and temporal non-stationarity. The NB model demonstrated a superior fit, reducing the AIC from 136,596.4 to 75,668.25 and improving log-likelihood by over 30,000 points. Demographic analysis revealed an 81.6% higher risk of death in males (IRR = 1.816; 95% CI: 1.753–1.881) and an exponential gradient with age, peaking at an IRR of 4.717 (95% CI: 4.499–4.945) for individuals ≥80 years. Departmental fixed effects identified significant spatial heterogeneity, with higher diffusion in coastal regions. The semi-dynamic NB model with splines provides a robust, parsimonious, and scalable framework for epidemiological surveillance in resource-limited settings. By effectively correcting for overdispersion and stabilizing weekly reporting fluctuations, this approach offers a reliable tool for public health decision making in environments with fragmented data quality. Full article

Background/Objectives: Publicly available open databases offer advantages in terms of accessibility and transparency. However, their application in intensive care research remains limited. Therefore, in this study, we examined whether simple nationwide comparisons of intensive care unit (ICU) practice patterns are feasible using an open database. Methods: A multicenter, cross-sectional study was conducted using data from the Bed Function Report. ICU wards reimbursed under ICU admission fee types were included and classified as high-acuity or standard ICUs. The ward-level procedure frequencies of procedures, including mechanical ventilation, were calculated. Comparisons were performed according to ICU admission fee type and geographic region. Quasi-Poisson regression models with offsets for annual ICU admissions were applied, accounting for overdispersion. Results: A total of 602 ICUs were included in the study. Non-metropolitan ICUs demonstrated higher procedural rates for mechanical ventilation compared with metropolitan ICUs (rate ratio [RR], 1.11; 95% confidence interval [CI], 1.02–1.21). Standard ICUs consistently had lower procedural rates for mechanical ventilation than high-acuity ICUs (RR, 0.74; 95% CI, 0.68–0.81). Group analyses indicated that regional differences in procedure frequencies were evident in standard ICUs, but not in high-acuity ICUs. Conclusions: This study demonstrated the feasibility of comparing ICU practice patterns across different regions and facility types in Japan using a nationwide open public database. This approach may serve as an initial step in a stepwise research framework that links open-database profiling to patient-level analysis using more detailed data sources. Full article

We propose the Bivariate Poisson–X–Exponential Distribution (BPXED), a flexible bivariate count model obtained by compounding Poisson variables with a shared X–Exponential latent mixing distribution. The model extends the Poisson–X–Exponential (PXED) distribution and includes several bivariate Poisson-type models as special or limiting cases. Closed-form expressions are derived for the joint probability mass function, probability generating function, moments, and covariance structure, showing that dependence arises from shared latent heterogeneity and is restricted to positive correlation. Parameter estimation is developed using maximum likelihood, regression-based, and Bayesian approaches, and a Monte Carlo simulation study demonstrates a good finite-sample performance. Applications to soccer scores, reliability failures, and correlated photon counts illustrate improved goodness-of-fit over classical and recent competing models. Overall, BPXED provides an analytically tractable and interpretable framework for modeling positively dependent and overdispersed bivariate count data. Full article

The New Polynomial Exponential Distribution (NPED), introduced by Beghriche et al. (2022), provides a flexible one-parameter family capable of representing diverse hazard shapes and heavy-tailed behavior. Regression frameworks based on the NPED, however, have not yet been established. This paper introduces two methodological extensions: (i) a generalized linear model (NPED-GLM) in which the distribution parameter depends on covariates, and (ii) a Poisson–NPED count regression model suitable for overdispersed and heavy-tailed count data. Likelihood-based inference, asymptotic properties, and simulation studies are developed to investigate the performance of the estimators. Applications to engineering failure-count data and insurance claim frequencies illustrate the advantages of the proposed models relative to classical Poisson, negative binomial, and Poisson–Lindley regressions. These developments substantially broaden the applicability of the NPED in actuarial science, reliability engineering, and applied statistics. Full article