A Conditional Approach for Regression Analysis of Longitudinal Data with Informative Observation Time and Non-negligible Observation Duration

Recently, there has been a great interest in the analysis of longitudinal data in which the observation process is related to the longitudinal process. In literature, the observation process was commonly regarded as a recurrent event process. Sometimes some observation duration may occur and this process is referred to as a recurrent episode process. The medical cost related to hospitalization is an example. We propose a conditional modeling approach that takes into account both informative observation process and observation duration. We conducted simulation studies to assess the performance of the method and applied it to a dataset of medical costs.     

Geometric Anisotropic Spatial Point Pattern Analysis and Cox Processes

We consider spatial point processes with a pair correlation function, which depends only on the lag vector between a pair of points. Our interest is in statistical models with a special kind of ‘structured’ anisotropy: the pair correlation function is geometric anisotropic if it is elliptical but not spherical. In particular, we study Cox process models with an elliptical pair correlation function, including shot noise Cox processes and log Gaussian Cox processes, and we develop estimation procedures using summary statistics and Bayesian methods. Our methodology is illustrated on real and synthetic datasets of spatial point patterns.     

Asymptotics of the Finite-time Ruin Probability for the Sparre Andersen Risk Model Perturbed by an Inflated Stationary Chi-process

In this article, we consider the Sparre Andersen risk model that is perturbed by an inflated chi-process with non-negative random inflator R. Under some conditions on the perturbation and the random inflator, which allow for both small and large fluctuations, exact asymptotic behaviour of the finite-time ruin probability is obtained when initial reserve tends to infinity.     

Uniform Convergence of Compactly Supported Wavelet Expansions of Gaussian Random Processes

New results on uniform convergence in probability for expansions of Gaussian random processes using compactly supported wavelets are given. The main result is valid for general classes of non stationary processes. An application of the obtained results to stationary processes is also presented. It is shown that the convergence rate of the expansions is exponential.     

Nonparametric Bayesian inference for multivariate density functions using Feller priors

Multivariate density estimation plays an important role in investigating the mechanism of high-dimensional data. This article describes a nonparametric Bayesian approach to the estimation of multivariate densities. A general procedure is proposed for constructing Feller priors for multivariate densities and their theoretical properties as nonparametric priors are established. A blocked Gibbs sampling algorithm is devised to sample from the posterior of the multivariate density. A simulation study is conducted to evaluate the performance of the procedure.     

A Full Bayesian Non-parametric Analysis Involving a Neutral to the Right Process

Implementation of a full Bayesian non-parametric analysis involving neutral to the right processes (apart from the special case of the Dirichlet process) has been difficult for two reasons: first, the posterior distributions are complex and therefore only Bayes estimates (posterior expectations) have previously been presented; secondly, it is difficult to obtain an interpretation for the parameters of a neutral to the right process. In this paper we extend Ferguson & Phadia (1979) by presenting a general method for specifying the prior mean and variance of a neutral to the right process, providing the interpretation of the parameters. Additionally, we provide the basis for a full Bayesian analysis, via simulation, from the posterior process using a hybrid of new algorithms that is applicable to a large class of neutral to the right processes (Ferguson & Phadia only provide posterior means). The ideas are exemplified through illustrative analyses.     

A test of normality using nonparametrlic residuals

In this paper, we develop a test of the normality assumption of the errors using the residuals from a nonparametric kernel regression. Contrary to the existing tests based on the residuals from a parametric regression, our test is thus robust to misspecification of the regression function. The test statistic proposed here is a Bera-Jarque type test of skewness and kurtosis. We show that the test statistic has the usual x²(2) limit distribution under the null hypothesis. In contrast to the results of Rilstone (1992), we provide a set of primitive assumptions that allow weakly dependent observations and data dependent bandwidth parameters. We also establish consistency property of the test. Monte Carlo experiments show that our test has reasonably good size and power performance in small samples and perfornu better than some of the alternative tests in various situations.     

Bayesian quadrature with non-normal approximating functions

We consider an efficient Bayesian approach to estimating integration-based posterior summaries from a separate Bayesian application. In Bayesian quadrature we model an intractable posterior density function f(·) as a Gaussian process, using an approximating function g(·), and find a posterior distribution for the integral of f(·), conditional on a few evaluations of f (·) at selected design points. Bayesian quadrature using normal g (·) is called Bayes-Hermite quadrature. We extend this theory by allowing g(·) to be chosen from two wider classes of functions. One is a family of skew densities and the other is the family of finite mixtures of normal densities. For the family of skew densities we describe an iterative updating procedure to select the most suitable approximation and apply the method to two simulated posterior density functions.     

Simplified Estimating Functions for Diffusion Models with a High-dimensional Parameter

We consider estimating functions for discretely observed diffusion processes of the following type: for one part of the parameter of interest we propose to use a simple and explicit estimating function of the type studied by Kessler (2000); for the remaining part of the parameter we use a martingale estimating function. Such an approach is particularly useful in practical applications when the parameter is high-dimensional. It is also often necessary to supplement a simple estimating function by another type of estimating function because only the part of the parameter on which the invariant measure depends can be estimated by a simple estimating function. Under regularity conditions the resulting estimators are consistent and asymptotically normal. Several examples are considered in order to demonstrate the idea of the estimating procedure. The method is applied to two data sets comprising wind velocities and stock prices. In one example we also propose a general method for constructing diffusion models with a prescribed marginal distribution which have a flexible dependence structure.     

On Non-parametric Estimation of the Survival Function with Competing Risks

The non-parametric maximum likelihood estimators (MLEs) are derived for survival functions associated with individual risks or system components in a reliability framework. Lifetimes are observed for systems that contain one or more of those components. Analogous to a competing risks model, the system is assumed to fail upon the first instance of any component failure; i.e. the system is configured in series. For any given risk or component type, the asymptotic distribution is shown to depend explicitly on the unknown survival function of the other risks, as well as the censoring distribution. Survival functions with increasing failure rate are investigated as a special case. The order restricted MLE is shown to be consistent under mild assumptions of the underlying component lifetime distributions.