Bayesian autoregressive spectral analysis

The article describes an operational Bayesian approach to making inferences for the spectral density function for univariate autoregressive processes and for the AR operator of multivariate autoregressive processes. The derivation of the approach is described. Numerical examples, including the Wolfer Sunspot numbers, are used to demonstrate the practical usefulness of the approach.     

Sums of weighted spacings and a test for the logistic distribution

A simple proof for a theorem of Csörgö and Révész (1981b and 1984) concerning sums of weighted spacings is given, The conditions of the theorem are relaxed. As an application, a goodness-of-fit test for the logistic distribution is proposed. The percentage points of the proposed test statistic are obtained by a simulation experiment.     

Nonparametric bayes estimation of the survival function from a record of failures and follow-ups

In some observational studies, we have random censoring model. However, the data available may be partially observable censored data consisting of the observed failure times and only those nonfailure times which are subject to follow-up. Suzuki (1985) discussed the problem of nonparametric estimation of the survival function from such partially observable censored data. In this article, we derive a nonparametric Bayes estimator of the survival function for such data of failures and follow-ups under a Dirichlet process prior and squared error loss. The limiting properties such as the mean square consistency, weak convergence and strong consistency of the Bayes estimator are studied. Finally, the procedures developed are illustrated by means of an example.     

First order autoregressive time series with negative binomial and geometric marginals

In this paper we present first order autoregressive (AR(1)) time series with negative binomial and geometric marginals. These processes are the discrete analogues of the gamma and exponential processes introduced by Sim (1990). Many properties of the processes discussed here, such as autocorrelation, regression and joint distributions, are studied.     

A plug-in technique in nonparametric regression with dependence

The problem addressed is that of smoothing parameter selection in kernel nonparametric regression in the fixed design regression model with dependent noise. An asymptotic expression of the optimum bandwidth parameter has been obtained in recent studies, where this takes the form h = C ₀ n ^?1/5. This paper proposes to use a plug-in methodology, in order to obtain an optimum estimation of the bandwidth parameter, through preliminary estimation of the unknown value of C ₀.     

On a bayesian approach for the shiftpoint problem

Using a direct resampling process, a Bayesian approach is developed for the analysis of the shiftpoint problem. In many problems it is straight forward to isolate the marginal posterior distribution of the shift-point parameter and the conditional distribution of some of the parameters given the shift point and the other remaining parameters. When this is possible, a direct sampling approach is easily implemented whereby standard random number generators can be used to generate samples from the joint posterior distribution of aii the parameters in the model. This technique is illustrated with examples involving one shift for Poisson processes and regression models.     

Hypotheses testing for a multidimensional parameter of inhomogeneous Poisson processes

We observe a realization of an inhomogeneous Poisson process whose intensity function depends on an unknown multidimensional parameter. We consider the asymptotic behaviour of the Rao score test for a simple null hypothesis against the multilateral alternative. By using the Edgeworth type expansion (under the null hypothesis) for a vector of stochastic integrals with respect to the Poisson process, we refine the (classic) threshold of the test (obtained by the central limit theorem), which improves the first type probability of error. The expansion allows us to describe the power of the test under the local alternative, i.e. a sequence of alternatives, which converge to the null hypothesis with a certain rate. The rates can be different for components of the parameter.     

Level-Phase Independence for Fluid Queues

ABSTRACT

A Markov-modulated fluid queue is a two-dimensional Markov process; the first dimension is continuous and is usually called the level, and the second is the state of a Markov process that determines the evolution of the level, it is usually called the phase. We show that it is always possible to modify the transition rules at the boundary level of the fluid queue in order to obtain independence between the level and the phase under the stationary distribution. We obtain this result by exploiting the similarity between fluid queues and Quasi-Birth-and-Death (QBD) processes.     

A semiparametric maximum likelihood ratio test for the change point in copula models

In the present paper, a semiparametric maximum-likelihood-type test statistic is proposed and proved to have the same limit null distribution as the classical parametric likelihood one. Under some mild conditions, the limiting law of the proposed test statistic, suitably normalized and centralized, is shown to be double exponential, under the null hypothesis of no change in the parameter of copula models. We also discuss the Gaussian-type approximations for the semiparametric likelihood ratio. The asymptotic distribution of the proposed statistic under specified alternatives is shown to be normal, and an approximation to the power function is given. Simulation results are provided to illustrate the finite sample performance of the proposed statistical tests based on the double exponential and Gaussian-type approximations.     

Alternative posterior consistency results in nonparametric binary regression using Gaussian process priors

We establish consistency of posterior distribution when a Gaussian process prior is used as a prior distribution for the unknown binary regression function. Specifically, we take the work of Ghosal and Roy [2006. Posterior consistency of Gaussian process prior for nonparametric binary regression. Ann. Statist. 34, 2413–2429] as our starting point, and then weaken their assumptions on the smoothness of the Gaussian process kernel while retaining a stronger yet applicable condition about design points. Furthermore, we extend their results to multi-dimensional covariates under a weaker smoothness condition on the Gaussian process. Finally, we study the extent to which posterior consistency can be achieved under a general model where, when additional hyperparameters in the covariance function of a Gaussian process are involved.