The distribution of the quasispecies for the Wright–Fisher model on the sharp peak landscape

We consider the classical Wright–Fisher model with mutation and selection. Mutations occur independently in each locus, and selection is performed according to the sharp peak landscape. In the asymptotic regime studied in Cerf (2014), a quasispecies is formed. We find explicitly the distribution of this quasispecies, which turns out to be the same distribution as for the Moran model.     

On the number of countable models of complete theories with finite Rudin-Keisler preorders

The aim of this article is to generalize the classification of complete theories with finitely many countable models with respect to two principal characteristics, Rudin-Keisler preorders and the distribution functions of the number of limit models, to an arbitrary case with a finite Rudin-Keisler preorder. We establish that the same characteristics play a crucial role in the case we consider. We prove the compatibility of arbitrary finite Rudin-Keisler preorders with arbitrary distribution functions f satisfying the condition rang f?ω∪{ω, 2^ω}.     

Inference for the Number of Topics in the Latent Dirichlet Allocation Model via Bayesian Mixture Modeling

In latent Dirichlet allocation, the number of topics, T, is a hyperparameter of the model that must be specified before one can fit the model. The need to specify T in advance is restrictive. One way of dealing with this problem is to put a prior on T, but unfortunately the distribution on the latent variables of the model is then a mixture of distributions on spaces of different dimensions, and estimating this mixture distribution by Markov chain Monte Carlo is very difficult. We present a variant of the Metropolis–Hastings algorithm that can be used to estimate this mixture distribution, and in particular the posterior distribution of the number of topics. We evaluate our methodology on synthetic data and compare it with procedures that are currently used in the machine learning literature. We also give an illustration on two collections of articles from Wikipedia. Supplemental materials for this article are available online.     

On the posterior median estimators of possibly sparse sequences

We adopt the Bayesian paradigm and discuss certain properties of posterior median estimators of possibly sparse sequences. The prior distribution considered is a mixture of an atom of probability at zero and a symmetric unimodal distribution, and the noise distribution is taken as another symmetric unimodal distribution. We derive an explicit form of the corresponding posterior median and show that it is an antisymmetric function and, under some conditions, a shrinkage and a thresholding rule. Furthermore we show that, as long as the tails of the nonzero part of the prior distribution are heavier than the tails of the noise distribution, the posterior median, under some constraints on the involved parameters, has the bounded shrinkage property, extending thus recent results to larger families of prior and noise distributions. Expressions of posterior distributions and posterior medians in particular cases of interest are obtained. The asymptotes of the derived posterior medians, which provide valuable information of how the corresponding estimators treat large coefficients, are also given. These results could be particularly useful for studying frequentist optimality properties and developing statistical techniques of the resulting posterior median estimators of possibly sparse sequences for a wider set of prior and noise distributions.     

A maximum entropy approach to the newsvendor problem with partial information

In this paper, we consider the newsvendor model under partial information, i.e., where the demand distribution D is partly unknown. We focus on the classical case where the retailer only knows the expectation and variance of D. The standard approach is then to determine the order quantity using conservative rules such as minimax regret or Scarf’s rule. We compute instead the most likely demand distribution in the sense of maximum entropy. We then compare the performance of the maximum entropy approach with minimax regret and Scarf’s rule on large samples of randomly drawn demand distributions. We show that the average performance of the maximum entropy approach is considerably better than either alternative, and more surprisingly, that it is in most cases a better hedge against bad results.     

Joint modelling of the total amount and the number of claims by conditionals

In the risk theory context, let us consider the classical collective model. The aim of this paper is to obtain a flexible bivariate joint distribution for modelling the couple (S,N), where N is a count variable and S=X₁+?+X_N is the total claim amount. A generalization of the classical hierarchical model, where now we assume that the conditional distributions of S|N and N|S belong to some prescribed parametric families, is presented. A basic theorem of compatibility in conditional distributions of the type S given N and N given S is stated. Using a known theorem for exponential families and results from functional equations new models are obtained. We describe in detail the extension of two classical collective models, which now we call Poisson-Gamma and the Poisson-Binomial conditionals models. Other conditionals models are proposed, including the Poisson-Lognormal conditionals distribution, the Geometric-Gamma conditionals model and a model with inverse Gaussian conditionals. Further developments of collective risk modelling are given.     

一个机器服务模型的可靠性分析

本文讨论了由Ｎ个同型部件和一个服务设备组成的机器服务模型，在服务设备绝对可靠，或服务设备具有指数寿命，修理工对其修理的时间为ＰＨ分布，这两种情况下求得了系统的平稳概率分布，从而求得机器系统和服务设备的稳态可靠性指标，并证明了它们的首次失效时间均服从ＰＨ分布，服务设备和修理工的忙期亦服从ＰＨ分布．     

New prior near-ignorance models on the simplex

The aim of this paper is to derive new near-ignorance models on the probability simplex, which do not directly involve the Dirichlet distribution and, thus, are alternative to the Imprecise Dirichlet Model (IDM). We focus our investigation on a particular class of distributions on the simplex which is known as the class of Normalized Infinitely Divisible (NID) distributions; it includes the Dirichlet distribution as a particular case. For this class it is possible to derive general formulae for prior and posterior predictive inferences, by exploiting the Lévy–Khintchine representation theorem. This allows us to generally characterize the near-ignorance properties of the NID class. After deriving these general properties, we focus our attention on three members of this class. We will show that one of these near-ignorance models satisfies the representation invariance principle and, for a given value of the prior strength, always provides inferences that encompass those of the IDM. The other two models do not satisfy this principle, but their imprecision depends linearly or almost linearly on the number of observed categories; we argue that this is sometimes a desirable property for a predictive model.     

Buckley–James-type of estimators under the classical case cohort design

We consider the estimation problem with classical case-cohort data. The case-cohort design was first proposed by Prentice (Biometrics 73:1–11, 1986). Most studies focus on the Cox regression model. In this paper, we consider the linear regression model. We propose an estimator which extends the Buckley–James estimator to the classical case-cohort design. In order to derive the BJE, there is an additional problem of finding the generalized maximum likelihood estimator (GMLE) of the underlying distribution functions. We propose a self-consistent algorithm for the GMLE. We also justify that the GMLE is consistent and asymptotically normally distributed under certain regularity conditions. We further present some simulation results on the asymptotic properties of the BJE and apply our procedure to a data set used in the literature.     

Estimating the Innovation Distribution in Nonlinear Autoregressive Models

The usual estimator for the expectation of a function under the innovation distribution of a nonlinear autoregressive model is the empirical estimator based on estimated innovations. It can be improved by exploiting that the innovation distribution has mean zero. We show that the resulting estimator is efficient if the innovations are estimated with an efficient estimator for the autoregression parameter. Efficiency of this estimator is necessary except when the expectation of the function can be estimated adaptively. Analogous results hold for heteroscedastic models.     

Sojourn time distributions in the queue defined by a general QBD process

We consider a general QBD process as defining a FIFO queue and obtain the stationary distribution of the sojourn time of a customer in that queue as a matrix exponential distribution, which is identical to a phase-type distribution under a certain condition. Since QBD processes include many queueing models where the arrival and service process are dependent, these results form a substantial generalization of analogous results reported in the literature for queues such as the PH/PH/c queue. We also discuss asymptotic properties of the sojourn time distribution through its matrix exponential form.     

Phase-type distribution of customer relationship with Markovian response and marketing expenditure decision on the customer lifetime value

In recent years, more and more companies have adopted relationship marketing (RM). At the core of RM is the development and maintenance of long-term relationships with valuable customers. In RM, the customer lifetime value (CLV) is the discounted profit streams of a customer across the entire customer life cycle. The CLV plays a key role in customer acquisition and retention decisions. In this paper, we present a general mathematical framework for RM, and introduce a Markov chain model which is appropriate in modeling RM because of its flexibility and probabilistic nature. We also develop the life distribution of the customer relationship, which is a phase-type distribution since it is the distribution of the first arrival stopping state. And we obtain the expectation of the CLV, which is an important statistic for good decision-making. Finally, we illustrate how to find the optimal remarketing policy numerically. The framework developed for RM systems in this paper should be seen as a practical approach to RM where one can directly apply the results of phase-type distribution and expectation of CLV to marketing decisions.     

A local limit theorem for the probability of ruin

In this paper, we give a result on the local asymptotic behaviour of the probability of ruin in a continuous-time risk model in which the inter-claim times have an Erlang distribution and the individual claim sizes have a distribution that belongs to S(v) with v≥ 0, but where the Lundberg exponent of the underlying risk process does not exist.     

On the lifetime behavior of a discrete time shock model

In this article, we study a shock model in which the shocks occur according to a binomial process, i.e. the interarrival times between successive shocks follow a geometric distribution with mean 1/p$1/p$. According to the model, the system fails when the time between two consecutive shocks is less than a prespecified level. This is the discrete time version of the so-called δ$\delta$-shock model which has been previously studied for the continuous case. We obtain the probability mass function and probability generating function of the system’s lifetime. We also present an extension of the results to the case where the shock occurrences are dependent in a Markovian fashion.     

On the credibility of insurance claim frequency: Generalized count models and parametric estimators

We analyze the concept of credibility in claim frequency in two generalized count models–Mittag-Leffler and Weibull count models–which can handle both underdispersion and overdispersion in count data and nest the commonly used Poisson model as a special case. We find evidence, using data from a Danish insurance company, that the simple Poisson model can set the credibility weight to one even when there are only three years of individual experience data resulting from large heterogeneity among policyholders, and in doing so, it can thus break down the credibility model. The generalized count models, on the other hand, allow the weight to adjust according to the number of years of experience available. We propose parametric estimators for the structural parameters in the credibility formula using the mean and variance of the assumed distributions and a maximum likelihood estimation over a collective data. As an example, we show that the proposed parameters from Mittag-Leffler provide weights that are consistent with the idea of credibility. A simulation study is carried out investigating the stability of the maximum likelihood estimates from the Weibull count model. Finally, we extend the analyses to multidimensional lines and explain how our approach can be used in selecting profitable customers in cross-selling; customers can now be selected by estimating a function of their unknown risk profiles, which is the mean of the assumed distribution on their number of claims.     

Random graphs with given vertex degrees and switchings

Random graphs with a given degree sequence are often constructed using the configuration model, which yields a random multigraph. We may adjust this multigraph by a sequence of switchings, eventually yielding a simple graph. We show that, assuming essentially a bounded second moment of the degree distribution, this construction with the simplest types of switchings yields a simple random graph with an almost uniform distribution, in the sense that the total variation distance is o(1). This construction can be used to transfer results on distributional convergence from the configuration model multigraph to the uniform random simple graph with the given vertex degrees. As examples, we give a few applications to asymptotic normality. We show also a weaker result yielding contiguity when the maximum degree is too large for the main theorem to hold.     

Mathematical modelling and computational analysis of enzymatic degradation of xenobiotic polymers

We study the enzymatic degradation of xenobiotic polymers mathematically. As a mathematical model, we derive a linear second-order hyperbolic partial differential equation which governs the evolution of the weight distribution with respect to the molecular weight. Given an initial weight distribution and a final weight distribution, we formulate a problem to determine a degradation rate. We establish a necessary and sufficient condition for which the problem has a local solution. We also introduce a numerical technique based on our analysis, and present a numerical result that we obtained applying weight distributions before and after enzymatic degradation of polyvinyl alcohol.     

Multiple Meixner–Pollaczek polynomials and the six-vertex model

We study multiple orthogonal polynomials of Meixner–Pollaczek type with respect to a symmetric system of two orthogonality measures. Our main result is that the limiting distribution of the zeros of these polynomials is one component of the solution to a constrained vector equilibrium problem. We also provide a Rodrigues formula and closed expressions for the recurrence coefficients. The proof of the main result follows from a connection with the eigenvalues of (locally) block Toeplitz matrices, for which we provide some general results of independent interest.The motivation for this paper is the study of a model in statistical mechanics, the so-called six-vertex model with domain wall boundary conditions, in a particular regime known as the free fermion line. We show how the multiple Meixner–Pollaczek polynomials arise in an inhomogeneous version of this model.     

半参数ACD模型及其在中国股票市场中的应用研究

本文提出了半参数ACD模型并基于模拟样本与调整后的中国股票市场的价格时间间隔样本对模型进行实证分析.半参数ACD模型对条件期望的函数形式与随机误差项的分布形式要求都没有参数ACD模型强,因此不会像参数ACD模型那样因模型形式设定错误而得出错误结论.这一点在我们的实证分析中可以得到证实.与非参数ACD模型相比,半参数ACD模型能够估计出参数,这增加了模型的解释能力.半参数ACD模型估计出来的各个可加部分图形的形状对于正确设定参数ACD模型具有一定的指导作用.     

On the width of hybrid zones

Hybrid zones occur when two species are found in close proximity and interbreeding occurs, but the species’ characteristics remain distinct. These systems have been treated in the biology literature using partial differential equations models. Here we investigate a stochastic spatial model and prove the existence of a stationary distribution that represents the hybrid zone in equilibrium. We calculate the width of the hybrid zone, which agrees with the PDE formula only in dimensions d≥3$d\ge 3$. Our results also give insight into properties of hybrid zones in patchy environments.