Exponential Dispersion Models Research Articles

Optimization design method for multi-stress accelerated degradation test based on Tweedie exponential dispersion process

In the operational environment of electronic products, simultaneous multiple stresses contribute to product degradation, with their interactions often being intricate. To effectively analyze and optimize multi-stress accelerated degradation tests, we introduce a Multi-Stress Generalized Coupling accelerated degradation model based on the Tweedie exponential dispersion process, coupled with a multi-decision optimization approach for test design. The model integrates stress interactions elucidated through fuzzy correlation analysis, reflecting the accelerated degradation behavior consistent with the Tweedie exponential dispersion process. Addressing the complexities of multi-parameter estimation, we propose a parameter estimation method underpinned by the Sine Cosine Algorithm for multi-objective optimization within the proposed framework. Considering budgetary constraints, D-optimization and V-optimization strategies are deployed to finetune test scheme allocations, optimizing resource utilization. Validation through a three-stress accelerated degradation test on LED chips confirms the model's efficacy in reliability assessment. Our findings indicate the superior descriptive power of the Tweedie exponential dispersion model for complex degradation phenomena and the proposed model's closer approximation to stress conditions in real-world settings, with a methodological error maintained below 2 %. Furthermore, the test optimization method enhances the D-optimization and V-optimization objective values by 21.20 % and reduces them by 4.08 %, respectively, marking a substantial improvement over traditional schemes and endorsing the holistic advancement of multi-stress accelerated degradation testing methodologies.

Generalized Linear Model (GLM) Applications for the Exponential Dispersion Model Generated by the Landau Distribution

The exponential dispersion model (EDM) generated by the Landau distribution, denoted by EDM-EVF (exponential variance function), belongs to the Tweedie scale with power infinity. Its density function does not have an explicit form and, as of yet, has not been used for statistical aspects. Out of all EDMs belonging to the Tweedie scale, only two EDMs are steep and supported on the whole real line: the normal EDM with constant variance function and the EDM-EVF. All other absolutely continuous steep EDMs in the Tweedie scale are supported on the positive real line. This paper aims to accomplish an overall picture of all generalized linear model (GLM) applications belonging to the Tweedie scale by including the EDM-EVF. This paper introduces all GLM ingredients needed for its analysis, including the respective link function and total and scaled deviance. We study its analysis of deviance, derive the asymptotic properties of the maximum likelihood estimation (MLE) of the covariate parameters, and obtain the asymptotic distribution of deviance, using saddlepoint approximation. We provide numerical studies, which include estimation algorithm, simulation studies, and applications to three real datasets, and demonstrate that GLM using the EDM-EVF performs better than the linear model based on the normal EDM. An R package accompanies all of these.

A delineation of new classes of exponential dispersion models supported on the set of nonnegative integers

A delineation of new classes of exponential dispersion models supported on the set of nonnegative integers

The Influence of the COVID-19 Pandemic on Baseline Concussion Symptom Assessments Among Adolescents.

Introduction The COVID-19 pandemic resulted in the cancellation of high school sports in spring 2020, a modified resumption of sports in the 2020-2021 academic year, and a return to pre-pandemic sports in 2021-2022. This cancellation had a major impact on the quality of life of adolescent athletes, but it is unknown exactly how these pandemic-driven sports disruptions on athlete baseline (preseason) symptoms affected quality of life. Therefore, the current study retrospectively evaluated symptom inventories from Immediate Post-Concussion Assessment and Cognitive Testing (ImPACT) assessments to determine whether the cancellation of sports during the COVID-19 pandemic affected baseline (preseason) self-reported symptoms among adolescent athletes. Methods Our study used a retrospective cohort design to evaluate high school athletes with complete ImPACT assessments in the academic years before (2018-2019 and 2019-2020), during (2020-2021), and after (2021-2022) the pandemic. Specifically, data from a 22-item symptom report called the Post-Concussion Symptom Scale (PCSS) assessed during ImPACT was collected and analyzed using generalized linear models with a Tweedie exponential dispersion model and post hoc Tukey's honestly significant difference tests. The main outcomes were the total symptom severity score and the affective cluster score. Secondary outcomes were the analysis of the vestibular-somatic, cognitive-sensory, and sleep-arousal symptom clusters. Results Of the 104,274 ImPACT assessments, the total symptom severity score on the PCSS was different across years (p<0.001). There were lower symptom scores in 2020-2021 (5.33, 95% CI = 5.13-5.54) than in 2018-2019 (6.82, 95% CI = 6.63-7.01), 2019-2020 (6.94, 95% CI = 6.75-7.14), and 2021-2022 (6.44, 95% CI = 6.25-6.64). The cluster scores on the PCSS for affective, cognitive-sensory, sleep-arousal, and vestibular-somatic were also lower (p<0.001) in 2020-2021 than in 2018-2019, 2019-2020, and 2021-2022. Conclusion Contrary to our expectations, total symptom severity and cluster scores on the PCSS during the pandemic (2020-2021) were significantly lower than during the years before and after the pandemic-driven sports disruptions, suggesting the pandemic did not negatively affect these athletes as expected. These results also suggested that self-reported symptoms utilized in the PCSS component of ImPACT may not be as sensitive to sports disruption among adolescent athletes as other quality-of-life measures, especially during the COVID-19 pandemic.

A comprehensive review of synthesis kinetics and formation mechanism of geopolymers.

Geopolymers are synthesized by alkali or acid activation of aluminosilicate materials. This paper critically reviews the synthesis kinetics and formation mechanism of geopolymers. A variety of mechanistic tools such as Environmental Scanning Electron Microscopy (ESEM) and in situ Energy Dispersive X-ray diffractometry (EDXRD), in situ Isothermal Conduction Calorimetry (ICC), in situ Attenuated Total Reflectance Fourier Transform Infrared Spectroscopy (ATR-FTIR), 1H low-field Nuclear Magnetic Resonance (NMR) and Isothermal Conduction Calorimetry (ISC), and others and phenomenological models such as the John-Mehl-Avrami-Kolmogorov (JMAK) model, modified Jandar model, and exponential and Knudson linear dispersion models were used to study the geopolymerization kinetics and many mechanisms were proposed for the synthesis of geopolymers. The mechanistic tools and phenomenological models provided new insights about geopolymerization kinetics and formation mechanisms but each of the techniques used possesses some limitations. These limitations need to be removed and new methods or techniques must be developed to overcome these challenges and get more detailed information about all types of geopolymers. The formation mechanism consists of three to four stages such as dissolution of raw materials, polymerization of silica and alumina, condensation, and reorganization. The Si/Al ratio above the Si/Al ratio of reactants is more suitable and it increases the rate or degree of reaction and produces a higher compressive strength geopolymer. The Na/Al ratio of 1, water-to-solid (W/S) ratio of 0.30-0.45, a temperature in the range of 30 °C to 85 °C, and a curing time of 24 hours are the best for the synthesis of geopolymers. The growing demand for geopolymers in various fields requires the development of new advanced techniques for further understanding of kinetics and mechanisms for tailoring the properties of geopolymers for specific applications.

The Exponential Dispersion Model Generated by the Landau Distribution—A Comprehensive Review and Further Developments

The paper comprehensively studies the natural exponential family and its associated exponential dispersion model generated by the Landau distribution. These families exhibit probabilistic and statistical properties and are suitable for modeling skewed continuous data sets on the whole real line. The study explores and further develops various probabilistic properties, including reciprocity, self-decomposability, reproducibility, unimodality, and characterizations. It delves into statistical aspects such as maximum-likelihood estimation, hypothesis testing, and generalized linear models.

A Software Tool For Sparse Estimation Of A General Class Of High-dimensional GLMs

Generalized linear models are the workhorse of many inferential problems. Also in the modern era with high-dimensional settings, such models have been proven to be effective exploratory tools. Most attention has been paid to Gaussian, binomial and Poisson settings, which have efficient computational implementations and where either the dispersion parameter is largely irrelevant or absent. However, general GLMs have dispersion parameters $\phi$ that affect the value of the log-likelihood. This in turn, affects the value of various information criteria such as AIC and BIC, and has a considerable impact on the computation and selection of the optimal model. The R-package [dglars](https://CRAN.R-project.org/package=dglars) is one of the standard packages to perform high-dimensional analyses for GLMs. Being based on fundamental likelihood considerations, rather than arbitrary penalization, it naturally extends to the general GLM setting. In this paper, we present an improved predictor-corrector (IPC) algorithm for computing the differential geometric least angle regression (dgLARS) solution curve, proposed in [@Augug13] and [@pazira]. We describe the implementation of a stable estimator of the dispersion parameter proposed in [@pazira] for high-dimensional exponential dispersion models. A simulation study is conducted to test the performance of the proposed methods and algorithms. We illustrate the methods using an example. The described improvements have been implemented in a new version of the R-package [dglars](https://CRAN.R-project.org/package=dglars).

Tweedie hidden Markov random field and the expectation-method of moments and maximisation algorithm for brain MR image segmentation

ABSTRACT In this paper, a segmentation algorithm of brain magnetic resonance imaging is elaborated. The proposed method rests on Tweedie hidden Markov random field processing and the Expectation Method of moments and Maximisation algorithm. This new type of stochastic process stands for a Markov Random Field whose state sequence cannot be observed directly and whose spatial information is encoded through the mutual influences of neighbouring sites. Tweedie models are a special case of exponential dispersion models with power mean variance. In image processing problems as medical image segmentation no work has been recorded in terms of demonstrating the applicability of the Tweedie models with hidden Markov random field and different Tweedie parameters values p. Hence, our research stands for the pioneering work aiming at proving the efficiency of the Tweedie models in brain image segmentation. Expectation- Method of moments and Maximisation algorithm results from the combination of both EM (Expectation-Maximisation) algorithm and the method of moments. The algorithm was validated on synthetic data and tested on real images. The basic merit of our proposed algorithm of segmentation lies in its ability to identify the brain tumour. Experiments using Tweedie hidden Markov random field showed that the proposed method produces more intuitive results with a recognition rate of 93.72. In fact, the experimental results highlighted that our approach outperforms other approaches that are reported in literature.

Analytical approach for designing accelerated degradation tests under an exponential dispersion model

To provide timely lifetime information of manufactured products to potential customers, designing an efficient accelerated degradation test (ADT) is an important task for reliability analysts. In the literature, several papers have addressed a general k-level ADT design problem (including the determinations of the number of stress levels (k), testing stresses, allocation of test units and termination times) when the underlying degradation path follows an exponential dispersion model. The results are practical and interesting. However, most of those studies only addressed the case of k≤4. Generally, if a direct proof (such as using the Karush–Kuhn–Tucker conditions) for the case of k>4 is intractable, we can consider an indirect proof via the general equivalence theorem.In this paper, we first propose a conjecture optimal design for a k-level ADT design problem and then apply the general equivalence theorem to show that this conjecture design turns out to be the global V-optimal design. In addition, an example is used to illustrate the proposed procedure. The main contribution of this work is that this analytical approach can provide reliability analysts with better insight for designing an efficient ADT plan.

Exponential dispersion models for overdispersed zero-inflated count data

We consider two classes of exponential dispersion models of discrete probability distributions which are defined by specifying their variance functions in their mean value parameterization. These classes were considered in our earlier paper as models of overdispersed zero-inflated distributions. In this paper we analyze the application of these classes to fit count data having overdispersed and zero-inflated statistics. For this reason, we first elaborate on the computational aspects of the probability distributions, before we consider the data fitting with our models. We execute an extensive comparison with other statistical models that are recently proposed, on both real data sets, and simulated data sets. Our findings are that our framework is a flexible tool that gives excellent results in a wide range of cases. Moreover, specifically when the data characteristics show also large skewness and kurtosis our models perform best.

Spatial Tweedie exponential dispersion models: an application to insurance rate-making

In this paper we propose a statistical modeling framework that contributes to advancing methods for modeling insurance policy premium in the actuarial literature. Specification of separate frequency and severity models, accounting for territorial risk and performing accurate inference, are some of the challenges an actuary faces while modeling policy premium. Policy premiums are characterized to follow a semi-continuous probability distribution, featuring a non-zero probability at zero along with a positive continuous support. Interpretability is a concern when quantifying unobserved risks premiums face from spatial variation. Commonly used strategies in the literature are known to successfully quantify this risk, but do not necessarily produce interpretable estimates. Resorting to frequency-severity models leaves the actuary indecisive about the specification of covariates and spatial effects. The novelty of our proposed approach lies in the development of a parsimonious and interpretable zero-adjusted modeling framework that allows for joint estimation of the effect of policy and individual characteristics on the mean premium and dispersion, while quantifying spatial variability in the mean model. The developed methods are applied to a database featuring premiums arising from the collision coverage in insurance policies for motor vehicles within the state of Connecticut, USA, for the year 2008.

Integrating the EM algorithm with particle filter for image restoration with exponential dispersion noise

Images are often degraded during the data acquisition process. The degradation may involve blurring, information loss due to sampling, and various sources of noise. The purpose of image restoration is to estimate the original image from the degraded data. The present work sets forward a restoration technique for exponential dispersion noise based on Particle filtering (PF) using Hidden Markov Model. In order not to take observation information into account in general, the PF algorithm produced an incorrect sample from a discrete approximation distribution. To resolve this problem, we propose in the resampling stage of PF, samples which are generated from a continuous distribution rather than a discrete one based on Exponential Dispersion Models (EDM). An iterative approach, called the Expectation-Maximization (EM) algorithm, is used to find the maximum likelihood estimates of the relevant unknown parameters of the EDM. Moreover, under some conditions, the concavity of the conditional expected log-likelihood function is established in the maximization step of the EM algorithm. The proposed approach is rooted in ideas from statistics, control theory and signal processing. Experimental results are eventually displayed with simulation and satellite images, which demonstrate the good performance of the proposed approach.

The snap, crackle and pop of solar flares explained

The irregular fluctuations of solar flare emissions, as determined from terrestrial neutron monitors, remains poorly understood. These records empirically revealed a temporally-related variance to mean power law, $1/f$ noise and a non-Gaussian distribution, all features indicative of self-organized criticality, a theory of how derministic dynamical systems can spontaneously evolve to unstable states that express erratic changes. The non-Gaussian distribution found here approximated a Tweedie compound Poisson exponential dispersion model, a statistical distribution characterized by a variance to mean power law that itself can imply $1/f$ noise. Tweedie exponential dispersion models serve a primary role in statistical theory as foci for weak convergence for a wide range of random distributions, a role which supports an alternative conjecture to explain the solar flare fluctuations as being based on random processes rather than a deterministic system.

Self-organized criticality of aggregated animals attributed to Tweedie convergence

Ecologists have had an ongoing interest in a variance to mean power law that governs the clustering of individuals of animal and plant species. This same power law has been reported from disparate biological, physical and mathematical systems, and also characterizes a family of statistical distributions known as the Tweedie exponential dispersion models. Its widespread appearance can be explained by fundamental statistical convergence effects on random data that cause this, and related, power laws to emerge and provide mechanistic insight into its origin, as well as the origin of 1/$f$ noise, multifractality and other phenomena attributable to self-organized criticality. A meta-analysis of ecological field data was conducted here to examine how such statistical convergence might affect the power law. These findings provided conjectural insight into a form of self-organized criticality, driven and modulated by the statistical convergence of random data, which could underlie the power law’s emergence.

New exponential dispersion models for count data: the ABM and LM classes

In their fundamental paper on cubic variance functions (VFs), Letac and Mora (The Annals of Statistics, 1990) presented a systematic, rigorous and comprehensive study of natural exponential families (NEFs) on the real line, their characterization through their VFs and mean value parameterization. They presented a section that for some reason has been left unnoticed. This section deals with the construction of VFs associated with NEFs of counting distributions on the set of nonnegative integers and allows to find the corresponding generating measures. As EDMs are based on NEFs, we introduce in this paper two new classes of EDMs based on their results. For these classes, which are associated with simple VFs, we derive their mean value parameterization and their associated generating measures. We also prove that they have some desirable properties. Both classes are shown to be overdispersed and zero inflated in ascending order, making them as competitive statistical models for those in use in both, statistical and actuarial modeling. To our best knowledge, the classes of counting distributions we present in this paper, have not been introduced or discussed before in the literature. To show that our classes can serve as competitive statistical models for those in use (e.g., Poisson, Negative binomial), we include a numerical example of real data. In this example, we compare the performance of our classes with relevant competitive models.

An introduction to Bent Jørgensen’s ideas

We briefly expose some key aspects of the theory and use of dispersion models, for which Bent Jørgensen played a crucial role as a driving force and an inspiration source. Starting with the general notion of dispersion models, built using minimalistic mathematical assumptions, we specialize in two classes of families of distributions with different statistical flavors: exponential dispersion and proper dispersion models. The construction of dispersion models involves the solution of integral equations that are, in general, untractable. These difficulties disappear when more mathematical structure is assumed: it reduces to the calculation of a moment generating function or of a Riemann–Stieltjes integral for the exponential dispersion and the proper dispersion models, respectively. A new technique for constructing dispersion models based on characteristic functions is introduced turning the integral equations above into a tractable convolution equation and yielding examples of dispersion models that are neither proper dispersion nor exponential dispersion models. A corollary is that the cardinality of regular and non-regular dispersion models are both large. Some selected applications are discussed including exponential families non-linear models (for which generalized linear models are particular cases) and several models for clustered and dependent data based on a latent Lévy process.

Multivariate risk measures based on conditional expectation and systemic risk for Exponential Dispersion Models

Exponential dispersion models are well used and studied in quantitative risk management and actuarial science. One of the main interests is the risk measurement analysis of such models when facing extreme loss events. In this paper, we propose two multivariate risk measures based on conditional expectation and derive the explicit formulae for exponential dispersion models. In particular, our multivariate risk measures could facilitate a systemic risk measure with explicit expressions for exponential dispersion models subject to any pre-specified “systemic event.” We provide two numerical examples based on practical data to show the advantages of our approach in the context of exponential dispersion models.

Independent, Tough Identical Results: The Class of Tweedie on Power Variance Functions and the Class of Bar-Lev and Enis on Reproducible Natural Exponential Families

The Rao-Blackwell theorem has had a fundamental role in statistical theory. However, as opposed to what seems natural, Rao and Blackwell did not investigate and write the theorem jointly. In fact, they both published the same result independently, two years apart. Indeed, as C.R. Rao writes in Wikipedia: &amp;rdquo;the result on one parameter case was published by Rao (1945) in the Bulletin of the Calcutta Mathematical Society and by Blackwell (1947) in The Annals of Mathematical Statistics. Only Lehmann and Sche &amp;acute;e (1950) called the result as Rao-Blackwell theorem&amp;rdquo;. Forty years later, a situation very similar to the previous one seems to have happened. Tweedie (1984) in a paper published in a proceedings to a conference held in Calcutta and Bar-Lev and Enis (1986) in a paper published in The Annals of Statistics both presented for the first time, albeit two years apart, independently and in di erent contexts, the class of natural exponential families having power variance functions (NEF-PVFs). Tweedie&amp;rsquo;s results were then mentioned by Jorgensen (1987) in his fundamental paper on exponential dispersion models published in the Journal of the Royal Statistical Society, Series B. Jorgensen, however, mentioned also other researchers, including Bar-Lev and Enis, as dealt with the same problem. Nonetheless, Jorgensen (1987) stated in his paper that &amp;rdquo;The most complete study&amp;rdquo; of NEF-PVFs was given by Tweedie (1984), a statement which has led to naming the class of NEF-PVFs as the Tweedie class. This statement of Jorgensen is entirely and utterly incorrect. Accordingly, one of the goals of this note is to &amp;#39;prove&amp;#39; such incorrectness. Based on this &amp;#39;proof&amp;#39; it will be evident, so I trust, that both Bar-Lev and Enis should have received the appropriate credit by re-naming the class of NEF-PVFs via the exploitation of the names of Tweedie, Bar-Lev and Enis. This would resemble the dignified and elegant manner Lehmann and Sche &amp;acute;e acted on the Rao-Blackwell Theorem. Notwithstanding, the main aim of the note is to encourage young researchers to present their results with self-confidence and to get the credit they deserve.

Exponential Dispersion Process for Degradation Analysis

The methods for analyzing degradation data are usually based on a specific degradation model. However, this may result in a large bias to estimate the reliability of the product when the assumed model is wrong. In this paper, a new stochastic process, called exponential dispersion process, is developed to describe the degradation path of the product's physical or chemical characteristics. Exponential dispersion process includes the most used degradation models (Wiener process, gamma process, inverse Gaussian process, and compound Poisson process) as special cases. Thus, it is useful for suggesting an appropriate degradation model for a specific dataset. We investigate the exponential dispersion model with nonlinear degradation path, and use approximated maximum likelihood method and bootstrap approach to obtain the point and confidence interval estimates of the unknown model parameters. Furthermore, exponential dispersion process models with acceleration factors and random effects are also explored. The developed methodologies are then applied to real data analysis.

Spatial and Seasonal Variations and Inter-Relationship in Fitted Model Parameters for Rainfall Totals across Australia at Various Timescales

Probabilistic models are useful tools in understanding rainfall characteristics, generating synthetic data and predicting future events. This study describes the results from an analysis on comparing the probabilistic nature of daily, monthly and seasonal rainfall totals using data from 1327 rainfall stations across Australia. The main objective of this research is to develop a relationship between parameters obtained from models fitted to daily, monthly and seasonal rainfall totals. The study also examined the possibility of estimating the parameters for daily data using fitted parameters to monthly rainfall. Three distributions within the Exponential Dispersion Model (EDM) family (Normal, Gamma and Poisson-Gamma) were found to be optimal for modelling the daily, monthly and seasonal rainfall total. Within the EDM family, Poisson-Gamma distributions were found optimal in most cases, whereas the normal distribution was rarely optimal except for the stations from the wet region. Results showed large differences between regional and seasonal ϕ-index values (dispersion parameter), indicating the necessity of fitting separate models for each season. However, strong correlations were found between the parameters of combined data and those derived from individual seasons (0.70–0.81). This indicates the possibility of estimating parameters of individual season from the parameters of combined data. Such relationship has also been noticed for the parameters obtained through monthly and daily models. Findings of this research could be useful in understanding the probabilistic features of daily, monthly and seasonal rainfall and generating daily rainfall from monthly data for rainfall stations elsewhere.