Asymptotic distributions of weighted divergence between discrete distributions

A divergence measure between discrete probability distributions introduced by Csiszar (1967) generalizes the Kullback-Leibler information and several other information measures considered in the literature. We introduce a weighted divergence which generalizes the weighted Kullback-Leibler information considered by Taneja (1985). The weighted divergence between an empirical distribution and a fixed distribution and the weighted divergence between two independent empirical distributions are here investigated for large simple random samples, and the asymptotic distributions are shown to be either normal or equal to the distribution of a linear combination of independent X²-variables     

Quasi- and pseudo-maximum likelihood estimators for discretely observed continuous-time Markov branching processes

This article deals with quasi- and pseudo-likelihood estimation for a class of continuous-time multi-type Markov branching processes observed at discrete points in time. “Conventional” and conditional estimation are discussed for both approaches. We compare their properties and identify situations where they lead to asymptotically equivalent estimators. Both approaches possess robustness properties, and coincide with maximum likelihood estimation in some cases. Quasi-likelihood functions involving only linear combinations of the data may be unable to estimate all model parameters. Remedial measures exist, including the resort either to non-linear functions of the data or to conditioning the moments on appropriate sigma-algebras. The method of pseudo-likelihood may also resolve this issue. We investigate the properties of these approaches in three examples: the pure birth process, the linear birth-and-death process, and a two-type process that generalizes the previous two examples. Simulations studies are conducted to evaluate performance in finite samples.     

A Monte Carlo Markov chain algorithm for a class of mixture time series models

This article generalizes the Monte Carlo Markov Chain (MCMC) algorithm, based on the Gibbs weighted Chinese restaurant (gWCR) process algorithm, for a class of kernel mixture of time series models over the Dirichlet process. This class of models is an extension of Lo’s (Ann. Stat. 12:351–357, 1984) kernel mixture model for independent observations. The kernel represents a known distribution of time series conditional on past time series and both present and past latent variables. The latent variables are independent samples from a Dirichlet process, which is a random discrete (almost surely) distribution. This class of models includes an infinite mixture of autoregressive processes and an infinite mixture of generalized autoregressive conditional heteroskedasticity (GARCH) processes.     

The Ornstein-Uhlenbeck Dirichlet process and other time-varying processes for Bayesian nonparametric inference

This paper introduces a new class of time-varying, measure-valued stochastic processes for Bayesian nonparametric inference. The class of priors is constructed by normalising a stochastic process derived from non-Gaussian Ornstein-Uhlenbeck processes and generalises the class of normalised random measures with independent increments from static problems. Some properties of the normalised measure are investigated. A particle filter and MCMC schemes are described for inference. The methods are applied to an example in the modelling of financial data.     

Nonparametric Bayesian modelling using skewed Dirichlet processes

We introduce a new class of discrete random probability measures that extend the definition of Dirichlet process (DP) by explicitly incorporating skewness. The asymmetry is controlled by a single parameter in such a way that symmetric DPs are obtained as a special case of the general construction. We review the main properties of skewed DPs and develop appropriate Polya urn schemes. We illustrate the modelling in the context of linear regression models of the capital asset pricing model (CAPM) type, where assessing symmetry for the error distribution is important to check validity of the model.     

A functional central limit theorem for Markov additive processes with an application to the closed Lu–Kumar network

A functional central limit theorem for a class of time-homogeneous continuous-time Markov processes (X,Y) is proved. The process X is a positive recurrent Markov process on a countable-state space and the process Y has conditionally independent increments given X. The pair (X,Y) is called a Markov additive process. This paper unifies and generalizes several functional central limit theorems for Markov additive processes. An explicit expression for the variance parameter of the limit process is calculated using the local characteristics of the X process. The functional central limit theorem is then used to prove a heavy traffic limit theorem for the closed Lu–Kumar network.     

Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum

This paper defines and studies a new class of non-stationary random processes constructed from discrete non-decimated wavelets which generalizes the Cramér (Fourier) representation of stationary time series. We define an evolutionary wavelet spectrum (EWS) which quantifies how process power varies locally over time and scale. We show how the EWS may be rigorously estimated by a smoothed wavelet periodogram and how both these quantities may be inverted to provide an estimable time-localized autocovariance. We illustrate our theory with a pedagogical example based on discrete non-decimated Haar wavelets and also a real medical time series example.     

BERNOULLI SAMPLING

The concepts of the Bernoulli count process of a point process and Bernoulli sampling of a discrete parameter stochastic process are introduced. The Bernoulli count process determines the stochastic structure of the point process, and a process obtained by thinning a discrete parameter stochastic process by Bernoulli sampling satisfies the same property. Stationarity and the Markov property remain invariant under Bernoulli sampling.     

A class of neutral to the right priors induced by superposition of beta processes

A random distribution function on the positive real line which belongs to the class of neutral to the right priors is defined. It corresponds to the superposition of independent beta processes at the cumulative hazard level. The definition is constructive and starts with a discrete time process with random probability masses obtained from suitably defined products of independent beta random variables. The continuous time version is derived as the corresponding infinitesimal weak limit and is described in terms of completely random measures. It takes the interpretation of the survival distribution resulting from independent competing failure times. We discuss prior specification and illustrate posterior inference on a real data example.     

Markov Sampling

A discrete parameter stochastic process is observed at epochs of visits to a specified state in an independent two-state Markov chain. It is established that the family of finite dimensional distributions of the process derived in this way, referred to as Markov sampling, uniquely determines the stochastic structure of the original process. Using this identifiability, it is shown that if the derived process is Markov, then the original process is also Markov and if the derived process is strictly stationary then so is the original.     

Multivariate discrete exponential family of distributions and their properties

The paper generalizes the univariate discrete exponential family of distributions to the multivariate situation, and this generalization includes the multivariate power series distributions, the multivariate Lagrangian distributions, and the modified multivariate power-series distributions. This provides a unified approach for the study of these three classes of distributions. We obtain recurrence relations for moments and cumulants, and the maximum likelihood estimation for the discrete exponential family. These results are applied to some multivariate discrete distributions like the Lagrangian Poisson, Lagrangian (negative) multinomial, logarithmic series distributions and multivariate Lagrangian negative binomial distribution.     

APPROXIMATION OF EXCEEDANCE PROCESSES IN LARGE POPULATIONS

We consider a population of n individuals. Each of these individuals generates a discrete time branching stochastic process. We study the number of ancestors S(n,t) whose offspring at time t exceeds level θ(t), where θ(t) is some positive valued function. It is proved that S(n,t) may be approximated as t → ∞ and n → ∞ by some stochastic processes with independent increments.

     

Modelling Dependence within Joint Tail Regions

Standard approaches for modelling dependence within joint tail regions are based on extreme value methods which assume max-stability, a particular form of joint tail dependence. We develop joint tail models based on a broader class of dependence structure which provides a natural link between max-stable models and weaker forms of dependence including independence and negative association. This approach overcomes many of the problems that are encountered with standard methods and is the basis for a Poisson process representation that generalizes existing bivariate results. We apply the new techniques to simulated and environmental data, and demonstrate the marked advantage that the new approach offers for joint tail extrapolation.     

Nonparametric maximum-likelihood estimation of probability measures: existence and consistency

This paper formulates the nonparametric maximum-likelihood estimation of probability measures and generalizes the consistency result on the maximum-likelihood estimator (MLE). We drop the independent assumption on the underlying stochastic process and replace it with the assumption that the stochastic process is stationary and ergodic. The present proof employs Birkhoff's ergodic theorem and the martingale convergence theorem. The main result is applied to the parametric and nonparametric maximum-likelihood estimation of density functions.     

On assessing independence of Competing Risks when failure times are discrete

The traditional approach to modelling for Competing Risks, via a multivariate distribution of latent failure times, is very natural for many applications but suffers from a well-documented problem of identifiability. However, the demonstrations of this problem in the literature apply to essentially continuous latent failure times where any atoms of probability in their distributions are not too intrusive. It is shown in this paper that for discrete failure times the classic results on the identifiability problem concerning the existence of equivalent independent risks are incomplete.     

Fractional Lévy stable motion time-changed by gamma subordinator

Abstract

In this paper a new stochastic process is introduced by subordinating fractional Lévy stable motion (FLSM) with gamma process. This new process incorporates stochastic volatility in the parent process FLSM. Fractional order moments, tail asymptotic, codifference and persistence of signs long-range dependence of the new process are discussed. A step-by-step procedure for simulations of sample trajectories and estimation of the parameters of the introduced process are given. Our study complements and generalizes the results available on variance-gamma process and fractional Laplace motion in various directions, which are well studied processes in literature.     

Modified likelihood ratio statistics for inflated beta regressions

The class of inflated beta regression models generalizes that of beta regressions [S.L.P. Ferrari and F. Cribari-Neto, Beta regression for modelling rates and proportions, J. Appl. Stat. 31 (2004), pp. 799–815] by incorporating a discrete component that allows practitioners to model data on rates and proportions with observations that equal an interval limit. For instance, one can model responses that assume values in (0, 1]. The likelihood ratio test tends to be quite oversized (liberal, anticonservative) in inflated beta regressions estimated with a small number of observations. Indeed, our numerical results show that its null rejection rate can be almost twice the nominal level. It is thus important to develop alternative testing strategies. This paper develops small-sample adjustments to the likelihood ratio and signed likelihood ratio test statistics in inflated beta regression models. The adjustments do not require orthogonality between the parameters of interest and the nuisance parameters and are fairly simple since they only require first- and second-order log-likelihood cumulants. Simulation results show that the modified likelihood ratio tests deliver much accurate inference in small samples. An empirical application is presented and discussed.     

A GENERALIZED EMERSON RECURRENCE RELATION

The Emerson (1968, Biometrics 24 , 695–701) recurrence relation has many important applications in statistics. However, the original derivation applied only to discrete distributions. In the following, a simple derivation is given that generalizes the Emerson recurrence relation to any distribution for which the necessary expectations exist. A modern application is outlined.     

The discretely observed immigration-death process: Likelihood inference and spatiotemporal applications

ABSTRACT

We consider a stochastic process, the homogeneous spatial immigration-death (HSID) process, which is a spatial birth-death process with as building blocks (i) an immigration-death (ID) process (a continuous-time Markov chain) and (ii) a probability distribution assigning iid spatial locations to all events. For the ID process, we derive the likelihood function, reduce the likelihood estimation problem to one dimension, and prove consistency and asymptotic normality for the maximum likelihood estimators (MLEs) under a discrete sampling scheme. We additionally prove consistency for the MLEs of HSID processes. In connection to the growth-interaction process, which has a HSID process as basis, we also fit HSID processes to Scots pine data.     

Comparison of performance measures for multivariate discrete models

For multivariate probit models, Spiess and Tutz suggest three alternative performance measures, which are all based on the decomposition of the variation. The multivariate probit model can be seen as a special case of the discrete copula model. This paper proposes some new measures based on the value of the likelihood function and the prediction-realization table. In addition, it generalizes the measures from Spiess and Tutz for the discrete copula model. Results of a simulation study designed to compare the different measures in various situations are presented.