Limit theorems for Markovian Hawkes processes with a large initial intensity

Hawkes process is a simple point process that is self-exciting and has clustering effect. The intensity of this point process depends on its entire past history. It has wide applications in finance, neuroscience, social networks, criminology, seismology, and many other fields. In this paper, we study the linear Hawkes process with an exponential exciting function in the asymptotic regime where the initial intensity of the Hawkes process is large. We derive limit theorems under this asymptotic regime as well as the regime when both the initial intensity and the time are large.     

Bayesian Inference for Hawkes Processes

The Hawkes process is a practically and theoretically important class of point processes, but parameter-estimation for such a process can pose various problems. In this paper we explore and compare two approaches to Bayesian inference. The first approach is based on the so-called conditional intensity function, while the second approach is based on an underlying clustering and branching structure in the Hawkes process. For practical use, MCMC (Markov chain Monte Carlo) methods are employed. The two approaches are compared numerically using three examples of the Hawkes process.     

Computationally Efficient Estimation for the Generalized Odds Rate Mixture Cure Model With Interval-Censored Data

For semiparametric survival models with interval-censored data and a cure fraction, it is often difficult to derive nonparametric maximum likelihood estimation due to the challenge in maximizing the complex likelihood function. In this article, we propose a computationally efficient EM algorithm, facilitated by a gamma-Poisson data augmentation, for maximum likelihood estimation in a class of generalized odds rate mixture cure (GORMC) models with interval-censored data. The gamma-Poisson data augmentation greatly simplifies the EM estimation and enhances the convergence speed of the EM algorithm. The empirical properties of the proposed method are examined through extensive simulation studies and compared with numerical maximum likelihood estimates. An R package “GORCure” is developed to implement the proposed method and its use is illustrated by an application to the Aerobic Center Longitudinal Study dataset. Supplementary material for this article is available online.     

Some asymptotic results for nonlinear Hawkes processes

Hawkes process is a class of simple point processes with self-exciting and clustering properties. Hawkes process has been widely applied in finance, neuroscience, social networks, criminology, seismology, and many other fields. In this paper, we study fluctuations, large deviations and moderate deviations nonlinear Hawkes processes in a new asymptotic regime, the large intensity function and the small exciting function regime. It corresponds to the large baseline intensity asymptotics for the linear case, and can also be interpreted as the asymptotics for the mean process of Hawkes processes on a large network.     

Model-based clustering and segmentation of time series with changes in regime

Mixture model-based clustering, usually applied to multidimensional data, has become a popular approach in many data analysis problems, both for its good statistical properties and for the simplicity of implementation of the Expectation?CMaximization (EM) algorithm. Within the context of a railway application, this paper introduces a novel mixture model for dealing with time series that are subject to changes in regime. The proposed approach, called ClustSeg, consists in modeling each cluster by a regression model in which the polynomial coefficients vary according to a discrete hidden process. In particular, this approach makes use of logistic functions to model the (smooth or abrupt) transitions between regimes. The model parameters are estimated by the maximum likelihood method solved by an EM algorithm. This approach can also be regarded as a clustering approach which operates by finding groups of time series having common changes in regime. In addition to providing a time series partition, it therefore provides a time series segmentation. The problem of selecting the optimal numbers of clusters and segments is solved by means of the Bayesian Information Criterion. The ClustSeg approach is shown to be efficient using a variety of simulated time series and real-world time series of electrical power consumption from rail switching operations.     

Functional central limit theorems for stationary Hawkes processes and application to infinite-server queues

A univariate Hawkes process is a simple point process that is self-exciting and has a clustering effect. The intensity of this point process is given by the sum of a baseline intensity and another term that depends on the entire past history of the point process. Hawkes processes have wide applications in finance, neuroscience, social networks, criminology, seismology, and many other fields. In this paper, we prove a functional central limit theorem for stationary Hawkes processes in the asymptotic regime where the baseline intensity is large. The limit is a non-Markovian Gaussian process with dependent increments. We use the resulting approximation to study an infinite-server queue with high-volume Hawkes traffic. We show that the queue length process can be approximated by a Gaussian process, for which we compute explicitly the covariance function and the steady-state distribution. We also extend our results to multivariate stationary Hawkes processes and establish limit theorems for infinite-server queues with multivariate Hawkes traffic.     

An efficient ECM algorithm for maximum likelihood estimation in mixtures of t-factor analyzers

Mixture of t factor analyzers (MtFA) have been shown to be a sound model-based tool for robust clustering of high-dimensional data. This approach, which is deemed to be one of natural parametric extensions with respect to normal-theory models, allows for accommodation of potential noise components, atypical observations or data with longer-than-normal tails. In this paper, we propose an efficient expectation conditional maximization (ECM) algorithm for fast maximum likelihood estimation of MtFA. The proposed algorithm inherits all appealing properties of the ordinary EM algorithm such as its stability and monotonicity, but has a faster convergence rate since its CM steps are governed by a much smaller fraction of missing information. Numerical experiments based on simulated and real data show that the new procedure outperforms the commonly used EM and AECM algorithms substantially in most of the situations, regardless of how the convergence speed is assessed by the computing time or number of iterations.     

A reduction method for semi-infinite programming by means of a global stochastic approach†

We describe a reduction algorithm for solving semi-infinite programming problems. The proposed algorithm uses the simulated annealing method equipped with a function stretching as a multi-local procedure, and a penalty technique for the finite optimization process. An exponential penalty merit function is reduced along each search direction to ensure convergence from any starting point. Our preliminary numerical results seem to show that the algorithm is very promising in practice.     

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As a new reliability test plan, generalized progressive hybrid censoring can improve test efficiency by allowing experimenters to observe a pre-specified number of failure samples before the final termination point. Based on a class of widely used life distribution in life data analysis --- generalized exponential distribution, this paper discusses its parameters inference issue under generalized progressive hybrid censoring scheme. EM Algorithm is used to estimate parameters of the considered model. Simulation studies and a real-data analysis are carried out to illustrate the performance of finite sample for the proposed procedure.     

排序集抽样下指数分布的产品可靠度研究

为了提高指数分布产品可靠度的估计效率,研究了基于排序集抽样方法的极大似然估计量(Maximum likelihood estimator,MLE),证明了新MLE具有存在性、唯一性和渐近正态性,并通过排序集样本的Fisher信息得到MLE的渐近方差。针对似然方程没有显式解的问题,利用部分期望法对MLE进行修正,并给出其具体表达式。渐近相对效率和模拟相对效率的研究结果表明:排序集抽样下MLE和修正MLE的估计效率都一致高于简单随机抽样下MLE。最后,将推荐方法应用到转移性肾癌的临床研究中。     

An EM Algorithm for Batch Markovian Arrival Processes and its Comparison to a Simpler Estimation Procedure

Although the concept of Batch Markovian Arrival Processes (BMAPs) has gained widespread use in stochastic modelling of communication systems and other application areas, there are few statistical methods of parameter estimation proposed yet. However, in order to practically use BMAPs for modelling, statistical model fitting from empirical time series is an essential task. The present paper contains a specification of the classical EM algorithm for MAPs and BMAPs as well as a performance comparison to the computationally simpler estimation procedure recently proposed by Breuer and Gilbert. Furthermore, it is shown how to adapt the latter to become an estimator for hidden Markov models.     

An effective selection of regression variables when the error distribution is incorrectly specified

Summary An asymptotically efficient selection of regression variables is considered in the situation where the statistician estimates regression parameters by the maximum likelihood method but fails to choose a likelihood function matching the true error distribution. The proposed procedure is useful when a robust regression technique is applied but the data in fact do not require that treatment. Examples and a Monte Carlo study are presented and relationships to other selectors such as Mallows'C _p are investigated. Research supported by Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 123 “Stochastische Mathematische Modelle” and AFOSR Contract No. F49620 82 C 0009.     

非线性再生散度模型的Bayes估计

非线性再生散度模型是指数族非线性模型、广义线性模型和正态非线性回归模型的推广和发展,唐年胜等人研究了该模型参数的极大似然估计及其统计诊断。本文基于Gibbs抽样和MH抽样算法讨论非线性再生散度模型参数的Bayes估计。模拟研究和实例分析被用来说明该方法的有效性。     

Regression analysis of right-censored failure time data with missing censoring indicators

This paper discusses regression analysis of right-censored failure time data when censoring indicators are missing for some subjects. Several methods have been developed for the analysis under different situations and especially, Goetghebeur and Ryan considered the situation where both the failure time and the censoring time follow the proportional hazards models marginally and developed an estimating equation approach. One limitation of their approach is that the two baseline hazard functions were assumed to be proportional to each other. We consider the same problem and present an efficient estimation procedure for regression parameters that does not require the proportionality assumption. An EM algorithm is developed and the method is evaluated by a simulation study, which indicates that the proposed methodology performs well for practical situations. An illustrative example is provided.     

双边定时截尾样本下广义逆指数分布形状参数的估计

先给出了广义逆指数分布在双边定时截尾样本下形状参数的最大似然估计,并不能得到估计的显式表达式,但证明了参数在(0,+∞)上最大似然估计是唯一存在的.其次提出用EM算法求出形状参数的估计且该估计具有良好的收敛性,还给出了形状参数的EM估计的渐近方差和近似置信区间;最后通过数值模拟,对形状参数的最大似然估计和EM估计的效果进行了比较,说明了用EM算法求形状参数的估计是可行的,并且模拟效果相对比较好.     

Simple Accurate Approximation of Likelihood Profiles

Likelihood profiles for arbitrary functions of the model parameters are useful for constructing likelihood ratio confidence intervals, diagnosing linear approximation intervals, suggesting linearizing transforms, and many other purposes. This article investigates a simple integration method for producing accurate approximations to likelihood profiles that avoid problems associated with producing exact profiles. A basic theorem guarantees that the approximation can achieve any desired precision. In many cases the method requires no more than adding a few lines of code to that required to produce the maximum likelihood estimate of the parameter vector. Standard methods for computing likelihood profiles are based on solving a sequence of constrained maximum likelihood problems. Each problem generates a point on the profile. An earlier article proposed an integration method that generates the entire profile directly by solving a differential equation. That method, however, requires the Hessian of the log-likelihood which can be difficult to produce. The method considered here requires only the gradient. It may be particularly useful in conjunction with the EM algorithm which typically does not produce any type of confidence interval.     

Heteroscedastic replicated measurement error models under asymmetric heavy-tailed distributions

We propose a heteroscedastic replicated measurement error model based on the class of scale mixtures of skew-normal distributions, which allows the variances of measurement errors to vary across subjects. We develop EM algorithms to calculate maximum likelihood estimates for the model with or without equation error. An empirical Bayes approach is applied to estimate the true covariate and predict the response. Simulation studies show that the proposed models can provide reliable results and the inference is not unduly affected by outliers and distribution misspecification. The method has also been used to analyze a real data of plant root decomposition.     

A Cautionary Note on Likelihood Ratio Tests in Mixture Models

We show that iterative methods for maximizing the likelihood in a mixture of exponentials model depend strongly on their particular implementation. Different starting strategies and stopping rules yield completely different estimators of the parameters. This is demonstrated for the likelihood ratio test of homogeneity against two-component exponential mixtures, when the test statistic is calculated by the EM algorithm.     

Weighted Empirical Likelihood Ratio Confidence Intervals for the Mean with Censored Data

We propose a procedure to construct the empirical likelihood ratio confidence interval for the mean using a resampling method. This approach leads to the definition of a likelihood function for censored data, called weighted empirical likelihood function. With the second order expansion of the log likelihood ratio, a weighted empirical likelihood ratio confidence interval for the mean is proposed and shown by simulation studies to have comparable coverage accuracy to alternative methods, including the nonparametric bootstrap-t. The procedures proposed here apply in a unified way to different types of censored data, such as right censored data, doubly censored data and interval censored data, and computationally more efficient than the bootstrap-t method. An example of a set of doubly censored breast cancer data is presented with the application of our methods.     

Identification and estimation of superposed Neyman–Scott spatial cluster processes

This paper proposes an estimation method for superposed spatial point patterns of Neyman–Scott cluster processes of different distance scales and cluster sizes. Unlike the ordinary single Neyman–Scott model, the superposed process of Neyman–Scott models is not identified solely by the second-order moment property of the process. To solve the identification problem, we use the nearest neighbor distance property in addition to the second-order moment property. In the present procedure, we combine an inhomogeneous Poisson likelihood based on the Palm intensity with another likelihood function based on the nearest neighbor property. The derivative of the nearest neighbor distance function is regarded as the intensity function of the rotation invariant inhomogeneous Poisson point process. The present estimation procedure is applied to two sets of ecological location data.