过度离散型数据的统计模拟与分析

针对车险索赔次数数据经常出现的过度离散问题,采用数值模拟的方法,分别使用泊松模型(Poisson)、负二项回归模型(NB)以及广义泊松模型(GP)对不同程度的过度离散车险索赔次数数据进行拟合,并用均方误差、偏差以及AIC和BIC准则对Poisson、NB、GP三种模型的优良性进行比较分析,得到了不同条件下三种模型的优良性,并针对不同的条件给出了模型选择的建议.     

Smoothness Properties and Gradient Analysis Under Spatial Dirichlet Process Models

When analyzing point-referenced spatial data, interest will be in the first order or global behavior of associated surfaces. However, in order to better understand these surfaces, we may also be interested in second order or local behavior, e.g., in the rate of change of a spatial surface at a given location in a given direction. In a Bayesian parametric setting, such smoothness analysis has been pursued by Banerjee and Gelfand (2003) and Banerjee et al. (2003). We study continuity and differentiability of random surfaces in the Bayesian nonparametric setting proposed by Gelfand et al. (2005), which is based on the formulation of a spatial Dirichlet process (SDP). We provide conditions under which the random surfaces sampled from a SDP are smooth. We also obtain complete distributional theory for the directional finite difference and derivative processes associated with those random surfaces. We present inference under a Bayesian framework and illustrate our methodology with a simulated dataset.     

Logic of agreement: Foundations,semantic system and proof theory

In this paper a multi-valued propositional logic — logic of agreement — in terms of its model theory and inference system is presented. This formal system is the natural consequence of a new way to approach concepts as commonsense knowledge, uncertainty and approximate reasoning — the point of view of agreement. Particularly, it is discussed a possible extension of the Classical Theory of Sets based on the idea that, instead of trying to conceptualize sets as “fuzzy” or “vague” entities, it is more adequate to define membership as the result of a partial agreement among a group of individual agents. Furthermore, it is shown that the concept of agreement provides a framework for the development of a formal and sound explanation for concepts (e.g. fuzzy sets) which lack formal semantics. According to the definition of agreement, an individual agent agrees or not with the fact that an object possesses a certain property. A clear distinction is then established, between an individual agent — to whom deciding whether an element belongs to a set is just a yes or no matter — and a commonsensical agent — the one who interprets the knowledge shared by a certain group of people. Finally, the logic of agreement is presented and discussed. As it is assumed the existence of several individual agents, the semantic system is based on the perspective that each individual agent defines her/his own conceptualization of reality. So the semantics of the logic of agreement can be seen as being similar to a semantics of possible worlds, one for each individual agent. The proof theory is an extension of a natural deduction system, using supported formulas and incorporating only inference rules. Moreover, the soundness and completeness of the logic of agreement are also presented.     

Auxiliary Likelihood-Based Approximate Bayesian Computation in State Space Models

A computationally simple approach to inference in state space models is proposed, using approximate Bayesian computation (ABC). ABC avoids evaluation of an intractable likelihood by matching summary statistics for the observed data with statistics computed from data simulated from the true process, based on parameter draws from the prior. Draws that produce a “match” between observed and simulated summaries are retained, and used to estimate the inaccessible posterior. With no reduction to a low-dimensional set ofsufficient statistics being possible in the state space setting, we define the summaries as the maximum of an auxiliary likelihood function, and thereby exploit the asymptotic sufficiency of this estimator for the auxiliary parameter vector. We derive conditions under which this approach—including a computationally efficient version based on the auxiliary score—achieves Bayesian consistency. To reduce the well-documented inaccuracy of ABC in multiparameter settings, we propose the separate treatment of each parameter dimension using an integrated likelihood technique. Three stochastic volatility models for which exact Bayesian inference is either computationally challenging, or infeasible, are used for illustration. We demonstrate that our approach compares favorably against an extensive set of approximate and exact comparators. An empirical illustration completes the article. Supplementary materials for this article are available online.     

Cornerstone Mathematics: designing digital technology for teacher adaptation and scaling

We report the results of a design-based research project in England that embeds digital technology. The research followed from two phases in the USA: (1) a design phase that used dynamic representations to foster conceptual understanding of hard-to-teach mathematical ideas, and (2) a research phase that measured the efficacy of the resulting technology-based curriculum units as implemented in Texas schools. The goal of the third phase in England was initially to “scale up” the US approach. We determined, however, that the materials had to be re-designed for adaptability by English teachers. We report how the features of the innovation—particularly its technological infrastructure—could be leveraged, not only to achieve positive learning outcomes, but also to lay the foundations for change in pedagogy and learning at scale. We identify an emergent framework of design affordances for teacher adaptability that are particularly salient when technology is a critical element.     

Young students’ subjective interpretations of mathematical diagrams: elements of the theoretical construct “frame-based interpreting competence”

During the last few decades several studies have showed that mathematical visual aids are not at all self-explanatory. Nevertheless, students do make sense of those representations spontaneously and—as a matter of course—cannot avoid their own sense-making. Further, the function of visual aids as “re-presentation” of a given structure is complemented through an epistemological function to explore mathematical structures and generate new meaning. But in which way do socially learned interpreting schemes (frames) influence children’s subjective interpretations of mathematical diagrams? The CORA project investigates which frames can be reconstructed in young pupils’ interpretations of visual diagrams. This paper presents central ideas, theoretical background and—by means of short sequences from pre- and post-interviews—first aspects of “frame-based interpreting competence”. We describe children’s subjective frames in a range between “object-oriented” (focus on the diagram’s visible elements) and “system-oriented” (focus on relation between those elements).     

Palindromic Resolutions of Simple Modules

For a finite dimensional hereditary k-algebra H of Dynkin type, it is known that the trivial extension T(H) has the property that minimal projective resolutions of T(H)-modules are periodic. We examine conditions in which T(H)-modules have additional symmetry in their resolutions—namely, their resolutions are “palindromic.” A more general framework in which “palindromic” resolutions occur is also presented.     

Evolutionary games with facilitators: When does selection favor cooperation?

We study the combined influence of selection and random fluctuations on the evolutionary dynamics of two-strategy (“cooperation” and “defection”) games in populations comprising cooperation facilitators. The latter are individuals that support cooperation by enhancing the reproductive potential of cooperators relative to the fitness of defectors. By computing the fixation probability of a single cooperator in finite and well-mixed populations that include a fixed number of facilitators, and by using mean field analysis, we determine when selection promotes cooperation in the important classes of prisoner’s dilemma, snowdrift and stag-hunt games. In particular, we identify the circumstances under which selection favors the replacement and invasion of defection by cooperation. Our findings, corroborated by stochastic simulations, show that the spread of cooperation can be promoted through various scenarios when the density of facilitators exceeds a critical value whose dependence on the population size and selection strength is analyzed. We also determine under which conditions cooperation is more likely to replace defection than vice versa.     

Towards a More General Concept of Inference

The main objective of this paper is to sketch unifying conceptual and formal framework for inference that is able to explain various proof techniques without implicitly changing the underlying notion of inference rules. We base this framework upon the so-called two-dimensional, i.e., deduction to deduction, account of inference introduced by Tichý in his seminal work The Foundation’s of Frege’s Logic (1988). Consequently, it will be argued that sequent calculus provides suitable basis for such general concept of inference and therefore should not be seen just as technical tool, but philosophically well-founded system that can rival natural deduction in terms of its “naturalness”.     

Generalized Solutions to Semilinear Elliptic PDE with Applications to the Lichnerowicz Equation

In this article we investigate the existence of a solution to a semi-linear, elliptic, partial differential equation with distributional coefficients and data. The problem we consider is a generalization of the Lichnerowicz equation that one encounters in studying the constraint equations in general relativity. Our method for solving this problem consists of solving a net of regularized, semi-linear problems with data obtained by smoothing the original, distributional coefficients. In order to solve these regularized problems, we develop a priori L ^∞-bounds and sub- and super-solutions to apply a fixed point argument. We then show that the net of solutions obtained through this process satisfies certain decay estimates by determining estimates for the sub- and super-solutions and utilizing classical, a priori elliptic estimates. The estimates for this net of solutions allow us to regard this collection of functions as a solution in a Colombeau-type algebra. We motivate this Colombeau algebra framework by first solving an ill-posed critical exponent problem. To solve this ill-posed problem, we use a collection of smooth, “approximating” problems and then use the resulting sequence of solutions and a compactness argument to obtain a solution to the original problem. This approach is modeled after the more general Colombeau framework that we develop, and it conveys the potential that solutions in these abstract spaces have for obtaining classical solutions to ill-posed non-linear problems with irregular data.     

Generalized self-adjustment method for statistical mechanics of composite materials

A new method is developed for the statistical mechanics of composite materials — the generalized selfadjustment method — which makes it possible to reduce the problem of predicting effective elastic properties of composites with random structures to the solution of two simpler “averaged” problems of an inclusion with transitional layers in a medium with the desired effective elastic properties. The inhomogeneous elastic properties and dimensions of the transitional layers take into account both the “approximate” order of mutual positioning, and also the variation in the dimensions and elastics properties of inclusions through appropriate special averaged indicator functions of the random structure of the composite. A numerical calculation of averaged indicator functions and effective elastic characteristics is performed by the generalized self-adjustment method for a unidirectional fiberglass on the basis of various models of actual random structures in the plane of isotropy.     

Chaos in M(atrix) theory

We consider the classical and quantum dynamics in M(atrix) theory. Using a simple ansatz we show that a classical trajectory exhibits a chaotic motion. We argue that the holographic feature of M(atrix) theory is related with the repulsive feature of energy eigenvalues in quantum chaotic system. Chaotic dynamics in N = 2 supersymmetric Yang—Mills theory is also discussed. We demonstrate that after the separation of “slow” and “fast” modes there is a singular contribution from the “slow” modes to the Hamiltonian of the “fast” modes.     

Ancestors and descendants in evolving k‐tree models

We consider several random graph models based on k‐trees, which can be generated by applying the probabilistic growth rules “uniform attachment”, “preferential attachment”, or a “saturation”‐rule, respectively, but which also can be described in a combinatorial way. For all of these models we study the number of ancestors and the number of descendants of nodes in the graph by carrying out a precise analysis which leads to exact and limiting distributional results. © 2014 Wiley Periodicals, Inc. Random Struct. Alg. 44, 465–489, 2014     

Orthocomplemented difference lattices in association with generalized rings

Orthocomplemented difference lattices (ODLs) are orthocomplemented lattices endowed with an additional operation of “abstract symmetric difference”. In studying ODLs as universal algebras or instances of quantum logics, several results have been obtained (see the references at the end of this paper where the explicite link with orthomodularity is discussed, too). Since the ODLs are “nearly Boolean”, a natural question arises whether there are “nearly Boolean rings” associated with ODLs. In this paper we find such an association — we introduce some difference ring-like algebras (the DRAs) that allow for a natural one-to-one correspondence with the ODLs. The DRAs are defined by only a few rather plausible axioms. The axioms guarantee, among others, that a DRA is a group and that the association with ODLs agrees, for the subrings of DRAs, with the famous Stone (Boolean ring) correspondence.     

COM-negative binomial distribution: modeling overdispersion and ultrahigh zero-inflated count data

We focus on the COM-type negative binomial distribution with three parameters, which belongs to COM-type (a, b, 0) class distributions and family of equilibrium distributions of arbitrary birth-death process. Besides, we show abundant distributional properties such as overdispersion and underdispersion, log-concavity, log-convexity (infinite divisibility), pseudo compound Poisson, stochastic ordering, and asymptotic approximation. Some characterizations including sum of equicorrelated geometrically distributed random variables, conditional distribution, limit distribution of COM-negative hypergeometric distribution, and Stein’s identity are given for theoretical properties. COM-negative binomial distribution was applied to overdispersion and ultrahigh zero-inflated data sets. With the aid of ratio regression, we employ maximum likelihood method to estimate the parameters and the goodness-of-fit are evaluated by the discrete Kolmogorov-Smirnov test.     

TO LEARN OR TO CHANGE: OPTIMAL R&D INVESTMENTS UNDER UNCERTAINTY IN THE CASE OF CLIMATE CHANGE

When studying R&D investments in technologies that address potential damage from climate change (termed as “research to change” or RTC), current literature overlooks the effects of purchased learning (i.e., learn through scientific research, termed as “research to learn” or RTL) about climate change. We investigate interactions between optimal R&D investments in RTC and RTL under uncertainty in climate change and research outcomes, while accounting for the positive impact that successful RTL may have on RTC outcome. We find that simultaneously investing in both RTL and RTC may be optimal when the probability that climate change imposes a specific level of damage is either moderate or very high and when RTL cost is relatively low. Whenever RTL and RTC are conducted simultaneously, then they substitute. However, when it is not optimal to conduct RTC and RTL simultaneously, then an increase in RTC cost decreases, at least weakly, RTL investment (i.e., RTL and RTC complement). When the probability that climate change imposes damage increases, then the optimal RTL investment may first decrease and then increase. Moreover, we identify conditions under which either the precautionary principle or the learn‐then‐act principle should be followed regarding R&D investments.     

Queue-length balance equations in multiclass multiserver queues and their generalizations

A classical result for the steady-state queue-length distribution of single-class queueing systems is the following: The distribution of the queue length just before an arrival epoch equals the distribution of the queue length just after a departure epoch. The constraint for this result to be valid is that arrivals, and also service completions, with probability one occur individually, i.e., not in batches. We show that it is easy to write down somewhat similar balance equations for multidimensional queue-length processes for a quite general network of multiclass multiserver queues. We formally derive those balance equations under a general framework. They are called distributional relationships and are obtained for any external arrival process and state-dependent routing as long as certain stationarity conditions are satisfied and external arrivals and service completions do not simultaneously occur. We demonstrate the use of these balance equations, in combination with PASTA, by (1) providing very simple derivations of some known results for polling systems and (2) obtaining new results for some queueing systems with priorities. We also extend the distributional relationships for a nonstationary framework.     

Die Entwicklung kausaler Hypothesen in der qualitativen Sozialforschung

The disregard of causal inference in the methodological literature about qualitative research is highly problematic, since the category of causality is closely linked to the concept of social action. However, it is also clear that causal analysis is burdened with certain difficulties and methodological challenges in the realm of social research. Some of these problems are discussed in this article using Mackie—s concept of 3 “INUS”-conditions. Thereby it will be shown that strategies of causal analysis based on comparative methods proposed for qualitative research, namely “Analytic Induction” and “Qualitative Comparative Analysis” have great difficulties in dealing adequately with these problems. They can only be solved, if case-comparative methods are combined with explorative research strategies which support the researcher in gaining access to the local knowledge of the research field.     

U-factorization of ideals

We study the factorization of ideals of a commutative ring, in the context of the U-factorization framework introduced by Fletcher. This leads to several “U-factorability” properties weaker than unique U-factorization. We characterize these notions, determine the implications between them, and give several examples to illustrate the differences. For example, we show that a ring is a general ZPI-ring if and only if its monoid of ideals has unique factorization in the sense of Fletcher. We also examine how these “U-factorability” properties behave with respect to several ring-theoretic constructions.     

Polynomial entropies and integrable Hamiltonian systems

We introduce two numerical conjugacy invariants of dynamical systems — the polynomial entropy and the weak polynomial entropy — which are well-suited for the study of “completely integrable” Hamiltonian systems. These invariants describe the polynomial growth rate of the number of balls (for the usual “dynamical” distances) of covers of the ambient space. We give explicit examples of computation of these polynomial entropies for generic Hamiltonian systems on surfaces.