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1. In iteroparous species, intermittent breeding is an important life-history tactic that can greatly affect animal population growth and viability. Despite its importance, few studies have quantified the consequences of breeding pauses on lifetime reproductive output, principally because calculating lifetime reproductive output requires knowledge of each individual's entire reproductive history. This information is extremely difficult to obtain in wild populations. 2. We applied novel statistical approaches that account for uncertainty in state assessment and individual heterogeneity to an 18-year capture-recapture dataset of 6,631 female southern elephant seals from Macquarie Island. We estimated survival and breeding probabilities, and investigated the consequences of intermittent breeding on lifetime reproductive output. 3. We found consistent differences in females' demographic performance between two heterogeneity classes. In particular, breeding imbued a high cost on survival in the females from the heterogeneity class 2, assumed to be females of lower quality. Individual quality also appeared to play a major role in a female's decision to skip reproduction with females of poorer quality more likely to skip breeding events than females of higher quality. 4. Skipping some breeding events allowed females from both heterogeneity classes to increase lifetime reproductive output over females that bred annually. However, females of lower quality produced less offspring over their lifetime. 5. Intermittent breeding seems to be used by female southern elephant seals as a tactic to offset reproductive costs on survival and enhance lifetime reproductive output but remains unavoidable and driven by individual-specific constraints in some other females.

We analyse the identifiability of the number of components in k‐variate, M‐component finite mixture models in which each component distribution has independent marginals, including models in latent class analysis. Without making parametric assumptions on the component distributions, we investigate how one can identify the number of components from the distribution function of the observed data. When k≥2, a lower bound on the number of components (M) is non‐parametrically identifiable from the rank of a matrix constructed from the distribution function of the observed variables. Building on this identification condition, we develop a procedure to estimate a lower bound on the number of components consistently.

Various techniques have been proposed in the literature to account for the observed and unobserved heterogeneity in the crash dataset. Those include techniques such as the finite mixture model (FMM), or hierarchical techniques. The FMM could provide a flexible framework by providing various distributions for various individual observations. However, the shortcoming of the standard FMM is that it cannot account for the heterogeneity in a single model’s structure, and the data needs to be disaggregated to its resultant subsamples. That would result in a loss of information. On the other hand, a second plausible approach is to use a hierarchical technique to account for the data heterogeneities, being based on various explanatory variables, and based on engineering intuition. In the context of traffic safety, while some researchers, for instance, considered the seasonality, some others considered highway systems or even genders. However, a question might arise: are the same observations within a same hierarchy homogenous? Are all the observations within different clusters heterogeneous? Additionally, how about other variables? Although the results in the literature highlighted accounting for the structure of the dataset would result in an acceptable interclass correlation (ICC), and also result in a significant improvement in terms of reduction in the deviance information criteria (DIC), there is no justification why to use those specific hierarchies and reject others. A more reasonable approach is to let the algorithm come up with the best distributions based on the provided parameters and accommodate observations to the related mixtures. In that approach those observations that belong to various subjective hierarchies, e.g., winter versus summer, but found to be similar would be set in a similar cluster. That is why we proposed this methodology to implement an objective hierarchy of the FMM to be used for the hierarchical technique. Here, due to the label switching problem of the FMM in the context of Bayesian, the FMM first conducted in the context of maximum likelihood estimates, and then assigned observations were used for the final analysis. The results of the DIC highlighted a significant improvement in the model fit compared with a subjective assigned hierarchy based on highway system. Additionally, although the subjective model resulted in a very low ICC due to so much heterogeneity in the dataset, the implemented methodology resulted in an acceptable ICC (0.3), justifying the use of hierarchy. The Bayesian hierarchical finite mixture model (BHFMM) is one of earliest application in traffic safety studies. The findings of this study have important implications for the future studies to account for a higher heterogeneity of the crash dataset based on the distance of observations to each cluster.

Common diseases including cancer are heterogeneous. It is important to discover disease subtypes and identify both shared and unique risk factors for different disease subtypes. The advent of high-throughput technologies enriches the data to achieve this goal, if necessary statistical methods are developed. Existing methods can accommodate both heterogeneity identification and variable selection under parametric models, but for survival analysis, the commonly used Cox model is semiparametric. Although finite-mixture Cox model has been proposed to address heterogeneity in survival analysis, variable selection has not been incorporated into such semiparametric models. Using regularization regression, we propose a variable selection method for the finite-mixture Cox model and select important, subtype-specific risk factors from high-dimensional predictors. Our estimators have oracle properties with proper choices of penalty parameters under the regularization regression. An expectation-maximization algorithm is developed for numerical calculation. Simulations demonstrate that our proposed method performs well in revealing the heterogeneity and selecting important risk factors for each subtype, and its performance is compared to alternatives with other regularizers. Finally, we apply our method to analyze a gene expression dataset for ovarian cancer DNA repair pathways. Based on our selected risk factors, the prognosis model accounting for heterogeneity consistently improves the prediction for the survival probability in both training and test datasets.

The risk-balancing hypothesis (RBH) suggests that farms will take less business risk as their financial risk increases, but existing literature provides empirical evidence that the RBH might be invalid under certain circumstances. We present a unified model that explains the conditions under which the RBH holds or is invalidated by recognizing the role of latent heterogeneity among farms. We generalize the RBH idea and trace the source of credit risk back to latent heterogeneity among farms. We then apply recent literature to longitudinal data from a panel of Dutch farms and classify segments using a finite mixture regression fixed-effects model and find that the RBH may not apply to all groups in the same way.

We present a novel methodology for estimating the parameters of a finite mixture model (FMM) based on partially rank-ordered set (PROS) sampling and use it in a fishery application. A PROS sampling design first selects a simple random sample of fish and creates partially rank-ordered judgement subsets by dividing units into subsets of prespecified sizes. The final measurements are then obtained from these partially ordered judgement subsets. The traditional expectation–maximization algorithm is not directly applicable for these observations. We propose a suitable expectation–maximization algorithm to estimate the parameters of the FMMs based on PROS samples. We also study the problem of classification of the PROS sample into the components of the FMM. We show that the maximum likelihood estimators based on PROS samples perform substantially better than their simple random sample counterparts even with small samples. The results are used to classify a fish population using the length-frequency data.

Background This study addresses a recurrent biological problem, that is to define a formal clustering structure for a set of tissues on the basis of the relative abundance of multiple alternatively spliced isoforms mRNAs generated by the same gene. To this aim, we have used a model-based clustering approach, based on a finite mixture of multivariate Gaussian densities. However, given we had more technical replicates from the same tissue for each quantitative measurement, we also employed a finite mixture of linear mixed models, with tissue-specific random effects. Results A panel of human tissues was analysed through quantitative real-time PCR methods, to quantify the relative amount of mRNA encoding different IGF-1 alternative splicing variants. After an appropriate, preliminary, equalization of the quantitative data, we provided an estimate of the distribution of the observed concentrations for the different IGF-1 mRNA splice variants in the cohort of tissues by employing suitable kernel density estimators. We observed that the analysed IGF-1 mRNA splice variants were characterized by multimodal distributions, which could be interpreted as describing the presence of several sub-population, i.e. potential tissue clusters. In this context, a formal clustering approach based on a finite mixture model (FMM) with Gaussian components is proposed. Due to the presence of potential dependence between the technical replicates (originated by repeated quantitative measurements of the same mRNA splice isoform in the same tissue) we have also employed the finite mixture of linear mixed models (FMLMM), which allowed to take into account this kind of within-tissue dependence. Conclusions The FMM and the FMLMM provided a convenient yet formal setting for a model-based clustering of the human tissues in sub-populations, characterized by homogeneous values of concentrations of the mRNAs for one or multiple IGF-1 alternative splicing isoforms. The proposed approaches can be applied to any cohort of tissues expressing several alternatively spliced mRNAs generated by the same gene, and can overcome the limitations of clustering methods based on simple comparisons between splice isoform expression levels.

In recent years, customer value has become a major focus among strategy researchers and practitioners as an essential element of a firm's competitive strategy. Many firms have been interested in Customer Value Analysis (CVA) which involves a structural analysis of the antecedent factors of perceived value (i.e., perceived quality and perceived price) to assess their relative importance in the perceptions of their buyers. We develop a statistical approach for performing CVA utilizing a recursive simultaneous equation model that is formulated to accommodate buyer heterogeneity. In particular, the proposed finite-mixture methodology allows one to estimate the relative effects and integration rules of perceived value drivers at the market segment level, as well as to simultaneously determine the (unknown) segments themselves. We demonstrate the utility of the proposed methodology via an actual commercial application involving a large electric utility company. Finally, we discuss the contributions of our research from the perspective of firm strategy and how it may be extended in the future.

This research introduces a new tool for analyzing both what customers buy and how often they shop. Unlike traditional models that focus only on in-store purchases, the MVL-Poisson Model captures shopping frequency, basket composition, and consumer response to prices and promotions. It segments customers by preferences and visit-frequency, reveals cross-category demand relationships, and highlights how promotions influence not just purchases but also store visits. It is computationally practical and can be implemented with standard retail data and analytics software. In an application to convenience store data, the model had high predictive accuracy and generated insights aligned with managerial intuition. We found that shoppers with similar preferences may visit at very different rates—a critical finding for targeting promotions effectively. Focusing only on in-store behavior underestimates the impact of promotions, as promotions also drive store traffic. Using insights on consumer preferences and cross-category relationships, we show how our model can be used to create optimal bundle promotions customized to particular segments.

The method of moments (Philos. Trans. R. Soc. Lond. Ser. A 185 (1894) 71–110) is one of the most widely used methods in statistics for parameter estimation, by means of solving the system of equations that match the population and estimated moments. However, in practice and especially for the important case of mixture models, one frequently needs to contend with the difficulties of non-existence or nonuniqueness of statistically meaningful solutions, as well as the high computational cost of solving large polynomial systems. Moreover, theoretical analyses of the method of moments are mainly confined to asymptotic normality style of results established under strong assumptions. This paper considers estimating a k-component Gaussian location mixture with a common (possibly unknown) variance parameter. To overcome the aforementioned theoretic and algorithmic hurdles, a crucial step is to denoise the moment estimates by projecting to the truncated moment space (via semidefinite programming) before solving the method of moments equations. Not only does this regularization ensure existence and uniqueness of solutions, it also yields fast solvers by means of Gauss quadrature. Furthermore, by proving new moment comparison theorems in the Wasserstein distance via polynomial interpolation and majorization techniques, we establish the statistical guarantees and adaptive optimality of the proposed procedure, as well as oracle inequality in misspecified models. These results can also be viewed as provable algorithms for generalized method of moments (Econometrica 50 (1982) 1029–1054) which involves nonconvex optimization and lacks theoretical guarantees.