Semiparametric Mixtures of Generalized Exponential Families

Abstract.  A semiparametric mixture model is characterized by a non-parametric mixing distribution Q (with respect to a parameter θ ) and a structural parameter β common to all components. Much of the literature on mixture models has focused on fixing β and estimating Q . However, this can lead to inconsistent estimation of both Q and the order of the model m . Creating a framework for consistent estimation remains an open problem and is the focus of this article. We formulate a class of generalized exponential family (GEF) models and establish sufficient conditions for the identifiability of finite mixtures formed from a GEF along with sufficient conditions for a nesting structure. Finite identifiability and nesting structure lead to the central result that semiparametric maximum likelihood estimation of Q and β fails. However, consistent estimation is possible if we restrict the class of mixing distributions and employ an information-theoretic approach. This article provides a foundation for inference in semiparametric mixture models, in which GEFs and their structural properties play an instrumental role.     

Asymptotic results for model robust regression

Since the mid 1980's many statisticians have studied methods for combining parametric and nonparametric models to improve the quality of fits in a regression problem. Notably Einsporn (1987) proposed the Model Robust Regression 1 estimate (MRRl) in which the parametric function, f, and the nonparametric functiong were combined in a straightforward fashion via the use of a mixing parameter, λ This technique was studied extensively atsmall samples and was shown to be quite effective at modeling various unusual functions. In this paper we have asymptotic results for the MRRl estimate in the case where λ is theoretically optimal, is asymptotically optimal and data driven, and is chosen with the PRESS statistic (Allen, 1971) We demonstrate that the MRRl estimate with λchosen by the PRESS statistic is slightly inferior asymptotically to the other two estimates, but, nevertheless possesses positive asymptotic qualities.     

Histogram estimation of radon-nikodym derivatives for strong mixing data

Nonparametric inference for point processes is discussed by way of histograms, which provide a nice tool for the analysis of on-line data. The construction of histograms depends on a sequence of partitions, which we take tc be nonenibedded to allow partitions with sets of equal measure. This presents some theoretical problems, which are addressed with an assumption on the decomposition of second order moments. In another direction, we drop the usual independence assumption on the sample, replacing it by a strong mixing assumption. Under this setting, we study the convergence of the histogram in probability, which depends on approximation conditions between the distributions of random pairs and the product of their marginal distributions, and^almost completely, which is based on the decomposition of the second order moments. This last convergence is stated on two versions according to the assumption of Laplace transforms or the Cramer moment conditions. These are somewhat stronger, but enable us to recover the usual condition on the decrease rate of sets on each partition. In the final section we prove that the finite dimensional distributions converge in distribution to a Gaussian centered vector with a specified covariance.     

An overview of model-robust regression

The proper combination of parametric and nonparametric regression procedures can improve upon the shortcomings of each when used individually. Considered is the situation where the researcher has an idea of which parametric model should explain the behavior of the data, but this model is not adequate throughout the entire range of the data. An extension of partial linear regression and two other methods of model-robust regression are developed and compared in this context. The model-robust procedures each involve the proportional mixing of a parametric fit to the data and a nonparametric fit to either the data or residuals. The emphasis of this work is on fitting in the small-sample situation, where nonparametric regression alone has well-known inadequacies. Performance is based on bias and variance considerations, and theoretical mean squared error formulas are developed for each procedure. An example is given that uses generated data from an underlying model with defined misspecification to provide graphical comparisons of the fits and to show the theoretical benefits of the model-robust procedures. Simulation results are presented which establish the accuracy of the theoretical formulas and illustrate the potential benefits of the model-robust procedures. Simulations are also used to illustrate the advantageous properties of a data-driven selector developed in this work for choosing the smoothing and mixing parameters. It is seen that the model-robust procedures (the final proposed method, in particular) give much improved fits over the individual parametric and nonparametric fits.     

早期在华传教士"独厚儒学"原因初探

受文艺复兴和宗教改革的影响和怀着对中华悠久文明的敬仰,以及出于发展自身、与新教抗衡之目的,以意大利耶稣会会士利玛窦为代表的基督教士在明末清初来华传教。在传教期间,他们基于传教现实和教义融合的考虑,“借儒宣教”、“独厚儒学”,成功地促成了中西哲学思想的一次大融合。他们的传教策略与在传统儒学接触中所采取的承认、包容和尊重态度对于我们今天的中西方文化交流有巨大的启示作用。     

后殖民理论及其意义

后殖民理论的先驱是一批旅居西欧的非洲知识分子 ,他们在 2 0世纪 5 0、6 0年代对殖民主义给殖民地遗留的问题进行思考。 2 0世纪 80、90年代 ,后殖民主义成为流行世界的文化思潮。作为后殖民主义的原创性理论 ,主要包括赛义德的“东方主义“、“文化霸权”理论 ;斯皮瓦克的“边缘中心化”理论 ;霍米·巴巴的“含混矛盾的杂糅”、“文化差异性发布”、“第三度空间”理论。后殖民理论呈现出复杂性、理论对抗性和两重性的特点 ,但其文化贡献和社会意义非常明显 :对“欧洲中心论”文化立场的批判、东方民族的文化建设和民主进程的推进都具有重要价值。     

On multivariate variable-kernel density estimates for time series

Let X₁ be a strictly stationary multiple time series with values in R^d and with a common density f. Let X₁,.,.,X_n, be n consecutive observations of X₁. Let k = k_n, be a sequence of positive integers, and let H_ni be the distance from X_i to its kth nearest neighbour among X_j, j i. The multivariate variable-kernel estimate f_n, of f is defined by where K is a given density. The complete convergence of f_n, to f on compact sets is established for time series satisfying a dependence condition (referred to as the strong mixing condition in the locally transitive sense) weaker than the strong mixing condition. Appropriate choices of k are explicitly given. The results apply to autoregressive processes and bilinear time-series models.     

Nonparametric mixture analysis of rock crab of the genus Leptograpsus

A nonparametric mixture analysis has been applied to study morphological characteristics of Leptograpsus crab. Two Gaussian models were also considered, one with the assumption of independent components and one with arbitrary relationship between the components. The three models then fitted to several combination of variables based on selecting different morphological characteristics. It has been observed that the nonparametric method gave the best result overall.     

The Bahadur representation for sample quantiles under strongly mixing sequence

Sun [2006, The Bahadur representation for sample quantiles under weak dependence. Statist. Probab. Lett. 76, 1238-1244] established the Bahadur representation for sample quantiles under the strongly mixing sequence. But there are some problems in the proofs of the main results. In the paper, we further investigate the Bahadur representation for sample quantiles under strongly mixing sequence and get better bound than that in Sun (2006).