Auswahl der wissenschaftlichen Literatur zum Thema „Poisson-Distributed observations“

The mean and the variance of the time S(t) spent by a system below a random threshold until t are obtained when the system level is modelled by the current value of a sequence of independent and identically distributed random variables appearing at the epochs of a nonhomogeneous Poisson process. In the case of the homogeneous Poisson process, the asymptotic distribution of S(t)/t as t → ∞ is derived.

The mean and the variance of the time S(t) spent by a system below a random threshold until t are obtained when the system level is modelled by the current value of a sequence of independent and identically distributed random variables appearing at the epochs of a nonhomogeneous Poisson process. In the case of the homogeneous Poisson process, the asymptotic distribution of S(t)/t as t → ∞ is derived.

AbstractThe cross-classified chain ladder has a number of versions, depending on the distribution to which observations are subject. The simplest case is that of Poisson distributed observations, and then maximum likelihood estimates of parameters are explicit. Most other cases, however, including Bayesian chain ladder models, lead to implicit MAP (Bayesian) or MLE (non-Bayesian) solutions for these parameter estimates, raising questions as to their existence and uniqueness. The present paper investigates these questions in the case where observations are distributed according to some member of the exponential dispersion family.

A GLR (generalized likelihood ratio) chart for Poisson distributed process with individual observations is proposed and the design procedure of the GLR chart is discussed. The performance of the GLR charts is compared to the exponentially weighted moving average (EWMA) chart and the GWMA chart. The numerical experiments show that the GLR chart has comparable performance as the other two charts. However, the GLR chart is much easier to design and implement since there are more design parameters in these two charts.

When data is derived under a single or multiple lower limits of quantification (LLOQ), estimation of distribution parameters as well as precision of these estimates appear to be challenging, as the way to account for unquantifiable observations due to LLOQs needs particular attention. The aim of this investigation is to characterize the precision of censored sample maximum likelihood estimates of the mean for normal, exponential and Poisson distribution affected by one or two LLOQs using confidence intervals (CI). In a simulation study, asymptotic and bias-corrected accelerated bootstrap CIs for the location parameter mean are compared with respect to coverage proportion and interval width. To enable this examination, we derived analytical expressions of the maximum likelihood location parameter estimate for the assumption of exponentially and Poisson distributed data, where the censored sample method and simple imputation method are used to account for LLOQs. Additionally, we vary the proportion of observations below the LLOQs. When based on the censored sample estimate, the bootstrap CI led to higher coverage proportions and narrower interval width than the asymptotic CI. The results differed by underlying distribution. Under the assumption of normality, the CI’s coverage proportion and width suffered most from high proportions of unquantifiable observations. For exponentially and Poisson distributed data, both CI approaches delivered similar results. To derive the CIs, the point estimates from the censored sample method are preferable, because the point estimate of the simple imputation method leads to higher bias for all investigated distributions. This biased simple imputation estimate impairs the coverage proportion of the respective CI. The bootstrap CI surpassed the asymptotic CIs with respect to coverage proportion for the investigated choice of distributional assumptions. The variety of distributions for which the methods are suitable gives the applicant a widely usable tool to handle LLOQ affected data with appropriate approaches.

Political scientists often study dollar-denominated outcomes that are zero for some observations. These zeros can arise because the data-generating process is granular: The observed outcome results from aggregation of a small number of discrete projects or grants, each of varying dollar size. This article describes the use of a compound distribution in which each observed outcome is the sum of a Poisson—distributed number of gamma distributed quantities, a special case of the Tweedie distribution. Regression models based on this distribution estimate loglinear marginal effects without either the ad hoc treatment of zeros necessary to use a log-dependent variable regression or the change in quantity of interest necessary to use a tobit or selection model. The compound Poisson—gamma regression is compared with commonly applied approaches in an application to data on high-speed rail grants from the United States federal government to the states, and against simulated data from several data-generating processes.

For τ, a stopping rule adapted to a sequence ofnindependent and identically distributed observations, we define the loss to be E[q(Rτ)], whereRjis the rank of thejth observation andqis a nondecreasing function of the rank. This setting covers both the best-choice problem, withq(r) =1(r> 1), and Robbins' problem, withq(r) =r. Asntends to ∞, the stopping problem acquires a limiting form which is associated with the planar Poisson process. Inspecting the limit we establish bounds on the stopping value and reveal qualitative features of the optimal rule. In particular, we show that the complete history dependence persists in the limit; thus answering a question asked by Bruss (2005) in the context of Robbins' problem.

La théorie du filtrage concerne essentiellement l’estimation optimale de l’état dans les systèmes stochastiques, surtout avec des mesures partielles et bruitées. Ce domaine, fortement lié à la théorie du contrôle, se concentre sur la synthèse d'estimateurs effectuant des calculs en temps réel pour minimiser l'erreur quadratique moyenne. La nécessité de telles estimations devient de plus en plus critique avec la prolifération de systèmes contrôlés par réseau, tels que les véhicules autonomes et les processus industriels complexes, où les processus d'observations sont soumis au caractère aléatoire de la transmission, ce qui donne lieu à des modèles d'information variables pour résoudre le problème d'estimation.Cette thèse aborde la tâche importante de l'estimation d'état dans les systèmes dynamiques stochastiques en temps continu lorsque les mesures sont disponible aux certains instants discrets defini par un processus aléatoire. En adaptant les méthodes d'estimation classiques, nous développons des équations pour un estimateur optimal d'état, explorons leurs propriétés et les aspect pratiques, et proposons et analysons des alternatives sous-optimales, présentant des parallèles avec les techniques existantes dans le domaine d'estimation classique lorsqu'elles sont appliquées aux processus d'observation Poisson-distribués.L'étude couvre trois classes de modèles mathématiques pour le système dynamique en temps continu et le processus d'observation discret. Tout d’abord, nous considérons des équations différentielles Ito-stochastiques avec le champ de vecteur Lipschitz et un coefficient de diffusion constant, alors que le processus d’observation discrète de dimension inférieure comprend la fonction nonlinéaire de l’état et un bruit Gaussien additif. Nous proposons des estimateurs d’état sous-optimaux continus-discrets, qui sont faciles à implémenter pour cette classe de systèmes. En supposant qu'un compteur de Poisson décrit les instants discrets auxquels les observations sont disponibles, nous calculons le processus de covariance d’erreur d’estimation. L'analyse est effectuée pour fournir les conditions de limitation du processus de covariance d'erreur, ainsi que la dépendance au taux d'échantillonnage moyen.Deuxièmement, nous considérons les systèmes dynamiques décrits par des chaînes de Markov en temps continu avec un espace d'état fini, et le processus d'observation est obtenu en discrétisant un processus stochastique conventionnel piloté par un processus de Wiener. Dans ce cas, nous montrons la convergence $L_1$ de l'estimateur optimal vers l'estimateur optimal classique (purement continu) (filtre de Wonham) quand l'intensité des processus de Poisson augmente.Enfin, nous étudions les filtres à particules continus-discrets pour les processus d'Ornstein-Uhlenbeck avec des observations discrètes décrites par des fonctions d'état linéaires et un bruit Gaussien additif. Les filtres à particules ont gagné beaucoup d'intérêt pour l'estimation d'état dans les modèles à grande échelle avec des mesures bruitées où le calcul du gain optimal est soit coûteux en calcul, soit pas entièrement réalisable en raison de la complexité de la dynamique. Dans cette thèse, nous proposons des processus de diffusion de type McKean–Vlasov continus-discrets, qui servent de modèle de champ moyen pour décrire la dynamique des particules. Nous étudions plusieurs types de processus de champ moyen en fonction de la manière dont les termes de bruit sont inclus pour l'imitation du processus d'état et du modèle d'observation. Les particules résultantes sont couplées via des covariances empiriques qui sont mises à jour en temps discrets avec l'arrivée de nouvelles observations. Avec une analyse appropriée des premier et deuxième instants, nous montrons que sous certaines conditions sur les paramètres du système, les performances des filtres à particules se rapprochent du filtre optimal quand le nombre de particules augmente
Filtering theory basically relates to optimal state estimation in stochastic dynamical systems, particularly when faced with partial and noisy data. This field, closely intertwined with control theory, focuses on designing estimators doing real-time computation while maintaining an acceptable level of accuracy as measured by the mean square error. The necessity for such estimates becomes increasingly critical with the proliferation of network-controlled systems, such as autonomous vehicles and complex industrial processes, where the observation processes are subject to randomness in transmission and this gives rise to varying information patterns under which the estimation must be carried out.This thesis addresses the important task of state estimation in continuous-time stochastic dynamical systems when the observation process is available only at some discrete time instants governed by a random process. By adapting classical estimation methods, we derive equations for optimal state estimator, explore their properties and practicality, and propose and evaluates sub-optimal alternatives, showcasing parallels to the existing techniques within the classical estimation domain when applied to Poisson-distributed observation processes.The study covers three classes of mathematical models for the continuous-time dynamical system and the discrete observation process. First, we consider Ito-stochastic differential equations with Lipschitz drift terms and constant diffusion coefficient, whereas the lower-dimensional discrete observation process comprises the nonlinear mapping of the state and additive Gaussian noise. We propose easy-to-implement continuous-discrete suboptimal state estimators for this system class. Assuming that a Poisson counter governs discrete times at which the observations are available, we compute the expectation or error covariance process. Analysis is carried out to provide conditions for boundedness of the error covariance process, as well as, the dependence on the mean sampling rate.Secondly, we consider the dynamical systems described by continuous-time Markov chains with finite state space, and the observation process is obtained by discretizing a conventional stochastic process driven by a Wiener process. For this case, the $L_1$-convergence of the derived optimal estimator to the classical (purely continuous) optimal estimator (Wonham filter) is shown with respect to increasing intensity of Poisson processes.Lastly, we study continuous-discrete particle filters for Ornstein-Uhlenbeck processes with discrete observations described by linear functions of state and additive Gaussian noise. Particle filters have gained a lot of interest for state estimation in large-scale models with noisy measurements where the computation of optimal gain is either computationally expensive or not entirely feasible due to complexity of the dynamics. In this thesis, we propose continuous-discrete McKean–Vlasov type diffusion processes, which serve as the mean-field model for describing the particle dynamics. We study several kinds of mean-field processes depending on how the noise terms are included in mimicking the state process and the observation model. The resulting particles are coupled through empirical covariances which are updated at discrete times with the arrival of new observations. With appropriate analysis of the first and second moments, we show that under certain conditions on system parameters, the performance of the particle filters approaches the optimal filter as the number of particles gets larger