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Variance related premium principle is one of the most important principles not only in practice applications but also in research field of actuarial science. In this paper, the Bayesian models are established under variance related premium principle. The Bayesian estimate and credibility estimate of risk premium are derived. Furthermore, some statistical properties of estimators are discussed. In the models with multitude contract data, the unbiased consistent estimates of the structure parameters are proposed. Finally, the empirical Bayes estimator are proved to be asymptotically optimal.     

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This paper considers the asymptotics of randomly weighted sums and their maxima, where the increments {X_i,i\geq1\} is a sequence of independent, identically distributed and real-valued random variables and the weights {\theta_i,i\geq1\} form another sequence of non-negative and independent random variables, and the two sequences of random variables follow some dependence structures. When the common distribution F of the increments belongs to dominant variation class, we obtain some weakly asymptotic estimations for the tail probability of randomly weighted sums and their maxima. In particular, when the F belongs to consistent variation class, some asymptotic formulas is presented. Finally, these results are applied to the asymptotic estimation for the ruin probability.     

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本文给出了上期望空间中独立随机变量部分和的最大不等式、指数 不等式、Marcinkiewicz-Zygmund不等式. 并且应用指数不等式和Marcinkiewicz-Zygmund不等式 研究了随机变量部分和序列完备收敛的性质.     

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??In this paper we describe the excursions from a set explicitly for
recurrent Markov chain with discrete time. A new exit system is presented through using a
law conditioned by specifying the starting point and ending point of excursions. In a simple
case, we verify that our conditioned excursion law is a discrete approximation for that of
a diffusion.     

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??In classical credibility theory, the claim amounts of different
insurance policies in a portfolio are assumed to be independent and the premiums are derived
under squared-error loss function. Wen et al. (2012) studied the credibility models with a
dependence structure among the claim amounts of one insurance policy that is called time
changeable effects and obtained the credibility formula. In this paper, we generalized this
dependence structure called time changeable effects to the claim amounts of different
insurance policies in a portfolio. Credibility premiums are obtained for Buhlmann and
Buhlmann-Straub credibility models with dependence structure under balanced loss function.     

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??In this paper, we studied the inverse probability weighted least squares estimation of single-index model with response variable missing at random. Firstly, the B-spline technique is used to approximate the unknown single-index function, and then the objective function is established based on the inverse probability weighted least squares method. By the two-stage Newton iterative algorithm, the estimation of index parameters and the B-spline coefficients can be obtained. Finally, through many simulation examples and a real data application, it can be concluded that the method proposed in this paper performs very well for moderate sample     

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Clustered interval-censored failure time data often arises in medical studies when study subjects come from the same cluster. Furthermore, the failure time may be related to the cluster size. A simple and common approach is to simplify interval-censored data due to the lack of proper inference procedures for direct analysis. For this reason, we proposed the within-cluster resampling-based method to consider the case II interval-censored data under the additive hazards model. With-cluster resampling is simple but computationally intensive. A major advantage of the proposed approach is that the estimator can be easily implemented when the cluster size is informative. Asymptotic properties and some simulation results are provided and indicate that the proposed approach works well.     

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本文研究了一种可用于分析非负数据的新的分布族. 它用一个随机表示来定义, 是一个半正态随机变量与一个指数随机变量的幂的混合. 其密度函数和性质, 包括风险函数, 矩和矩母函数被导出. 该分布族的实用性和灵活性通过一个实例用最大似然程序被阐明.     

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本文在传统Lee-Carter人口死亡率模型的框架下, 引入同出生年人群死亡率之间的相关性效应, 从而对未来死亡率的动态变化进行更加具体的刻画. 同时借鉴Lin和Cox(2005)所提出的长寿债券构造机制, 基于中国的实际人口死亡率数据, 运用多维概率扭转变换对不完全市场下长寿债券的定价结果进行比较分析.     

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In this paper, we introduce the definitions of geometric strongly ergodic, strongly ergodic and weakly ergodic for continuous-state Markov chains, then we give a primary proof of equivalence of the ergodicities for continuous-state Markov chains.     

A Localized Implementation of the Iterative Proportional Scaling Procedure for Gaussian Graphical Models

In this article, we propose localized implementations of the iterative proportional scaling (IPS) procedure by the strategy of partitioning cliques for computing maximum likelihood estimations in large Gaussian graphical models. We first divide the set of cliques into several nonoverlapping and nonempty blocks, and then adjust clique marginals in each block locally. Thus, high-order matrix operations can be avoided and the IPS procedure is accelerated. We modify the Swendsen–Wang Algorithm and apply the simulated annealing algorithm to find an approximation to the optimal partition which leads to the least complexity. This strategy of partitioning cliques can also speed up the existing IIPS and IHT procedures. Numerical experiments are presented to demonstrate the competitive performance of our new implementations and strategies.     

Conditional Iterative Proportional Fitting for Gaussian Distributions

A Gaussian version of the iterative proportional fitting procedure (IFP-P) was applied by Speed and Kiiveri to solve the likelihood equations in graphical Gaussian models. The calculation of the maximum likelihood estimates can be seen as the problem to find a Gaussian distribution with prescribed Gaussian marginals. We extend the Gaussian version of the IPF-P so that additionally given conditionals of Gaussian type are taken into account. The convergence of both proposed procedures, called conditional iterative proportional fitting procedures (CIPF-P), is proved.     

High-Dimensional Mixed Graphical Models

While graphical models for continuous data (Gaussian graphical models) and discrete data (Ising models) have been extensively studied, there is little work on graphical models for datasets with both continuous and discrete variables (mixed data), which are common in many scientific applications. We propose a novel graphical model for mixed data, which is simple enough to be suitable for high-dimensional data, yet flexible enough to represent all possible graph structures. We develop a computationally efficient regression-based algorithm for fitting the model by focusing on the conditional log-likelihood of each variable given the rest. The parameters have a natural group structure, and sparsity in the fitted graph is attained by incorporating a group lasso penalty, approximated by a weighted lasso penalty for computational efficiency. We demonstrate the effectiveness of our method through an extensive simulation study and apply it to a music annotation dataset (CAL500), obtaining a sparse and interpretable graphical model relating the continuous features of the audio signal to binary variables such as genre, emotions, and usage associated with particular songs. While we focus on binary discrete variables for the main presentation, we also show that the proposed methodology can be easily extended to general discrete variables.     

Clique trees of chordal graphs: leafage and 3-asteroidals

Chordal graphs were characterized as those graphs having a tree, called clique tree, whose vertices are the cliques of the graph and for every vertex in the graph, the set of cliques that contain it form a subtree of clique tree. In this work, we study the relationship between the clique trees of a chordal graph and its subgraphs. We will prove that clique trees can be described locally and all clique trees of a graph can be obtained from clique trees of subgraphs. In particular, we study the leafage of chordal graphs, that is the minimum number of leaves among the clique trees of the graph. It is known that interval graphs are chordal graphs without 3-asteroidals. We will prove a generalization of this result using the framework developed in the present article. We prove that in a clique tree that realizes the leafage, for every vertex of degree at least 3, and every choice of 3 branches incident to it, there is a 3−asteroidal in these branches.     

Qualitative inequalities for squared partial correlations of a Gaussian random vector

We describe various sets of conditional independence relationships, sufficient for qualitatively comparing non-vanishing squared partial correlations of a Gaussian random vector. These sufficient conditions are satisfied by several graphical Markov models. Rules for comparing degree of association among the vertices of such Gaussian graphical models are also developed. We apply these rules to compare conditional dependencies on Gaussian trees. In particular for trees, we show that such dependence can be completely characterised by the length of the paths joining the dependent vertices to each other and to the vertices conditioned on. We also apply our results to postulate rules for model selection for polytree models. Our rules apply to mutual information of Gaussian random vectors as well.     

基于最大惩罚似然的高斯混合模型无监督分类研究

本文提出了一个基于高斯混合模型的无监督分类算法. 考虑到利用EM算法求解高斯混合模型的参数参数估计问题容易陷入局部最优解, 我们引入逆Wishart分布来代替传统的Jeffery先验. 几个实验数据的结果表明, 采用该方法估计无监督分类的成分数, 无论是估计的正确率, 还是运算速度, 都有较大提高.     

Precision Matrix Estimation by Inverse Principal Orthogonal Decomposition

We investigate the structure of a large precision matrix in Gaussian graphical models by decomposing it into a low rank component and a remainder part with sparse precision matrix.Based on the decomposition,we propose to estimate the large precision matrix by inverting a principal orthogonal decomposition(IPOD).The IPOD approach has appealing practical interpretations in conditional graphical models given the low rank component,and it connects to Gaussian graphical models with latent variables.Specifically,we show that the low rank component in the decomposition of the large precision matrix can be viewed as the contribution from the latent variables in a Gaussian graphical model.Compared with existing approaches for latent variable graphical models,the IPOD is conveniently feasible in practice where only inverting a low-dimensional matrix is required.To identify the number of latent variables,which is an objective of its own interest,we investigate and justify an approach by examining the ratios of adjacent eigenvalues of the sample covariance matrix?Theoretical properties,numerical examples,and a real data application demonstrate the merits of the IPOD approach in its convenience,performance,and interpretability.     

Epistemic irrelevance in credal nets: The case of imprecise Markov trees

We focus on credal nets, which are graphical models that generalise Bayesian nets to imprecise probability. We replace the notion of strong independence commonly used in credal nets with the weaker notion of epistemic irrelevance, which is arguably more suited for a behavioural theory of probability. Focusing on directed trees, we show how to combine the given local uncertainty models in the nodes of the graph into a global model, and we use this to construct and justify an exact message-passing algorithm that computes updated beliefs for a variable in the tree. The algorithm, which is linear in the number of nodes, is formulated entirely in terms of coherent lower previsions, and is shown to satisfy a number of rationality requirements. We supply examples of the algorithm’s operation, and report an application to on-line character recognition that illustrates the advantages of our approach for prediction. We comment on the perspectives, opened by the availability, for the first time, of a truly efficient algorithm based on epistemic irrelevance.     

Positivity for Gaussian graphical models

Gaussian graphical models are parametric statistical models for jointly normal random variables whose dependence structure is determined by a graph. In previous work, we introduced trek separation, which gives a necessary and sufficient condition in terms of the graph for when a subdeterminant is zero for all covariance matrices that belong to the Gaussian graphical model. Here we extend this result to give explicit cancellation-free formulas for the expansions of non-zero subdeterminants.