On functional central limit theorems of Bayesian nonparametric priors

A general approach to derive the weak convergence, when centered and rescaled, of certain Bayesian nonparametric priors is proposed. This method may be applied to a wide range of processes including, for instance, nondecreasing nonnegative pure jump Lévy processes and normalized nondecreasing nonnegative pure jump Lévy processes with known finite dimensional distributions. Examples clarifying this approach involve the beta process in latent feature models and the Dirichlet process.     

Parametric Estimation for Subordinators and Induced OU Processes

Abstract.  Consider a stationary sequence of random variables with infinitely divisible marginal law, characterized by its Lévy density. We analyse the behaviour of a so-called cumulant M-estimator, in case this Lévy density is characterized by a Euclidean (finite dimensional) parameter. Under mild conditions, we prove consistency and asymptotic normality of the estimator. The estimator is considered in the situation where the data are increments of a subordinator as well as the situation where the data consist of a discretely sampled Ornstein–Uhlenbeck (OU) process induced by the subordinator. We illustrate our results for the Gamma-process and the Inverse-Gaussian OU process. For these processes we also explain how the estimator can be computed numerically.     

Bayesian Survival Analysis in Proportional Hazard Models with Logistic Relative Risk

Abstract.  The traditional Cox proportional hazards regression model uses an exponential relative risk function. We argue that under various plausible scenarios, the relative risk part of the model should be bounded, suggesting also that the traditional model often might overdramatize the hazard rate assessment for individuals with unusual covariates. This motivates our working with proportional hazards models where the relative risk function takes a logistic form. We provide frequentist methods, based on the partial likelihood, and then go on to semiparametric Bayesian constructions. These involve a Beta process for the cumulative baseline hazard function and any prior with a density, for example that dictated by a Jeffreys-type argument, for the regression coefficients. The posterior is derived using machinery for Lévy processes, and a simulation recipe is devised for sampling from the posterior distribution of any quantity. Our methods are illustrated on real data. A Bernshtĕn–von Mises theorem is reached for our class of semiparametric priors, guaranteeing asymptotic normality of the posterior processes.     

Goodness-of-fit tests based on the distance between the Dirichlet process and its base measure

The Dirichlet process is a fundamental tool in studying Bayesian nonparametric inference. The Dirichlet process has several sum representations, where each one of these representations highlights some aspects of this important process. In this paper, we use the sum representations of the Dirichlet process to derive explicit expressions that are used to calculate Kolmogorov, Lévy, and Cramér–von Mises distances between the Dirichlet process and its base measure. The derived expressions of the distance are used to select a proper value for the concentration parameter of the Dirichlet process. These tools are also used in a goodness-of-fit test. Illustrative examples and simulation results are included.     

Random spectral measure for non Gaussian moving averages

We study the distribution of phases and amplitudes for the spectral representation of weighted moving averages of a general noise measure. The simple independent structure, known for the Gaussian case, and involving Rayleigh amplitude and uniform phase distributions, is lost for the non Gaussian noise case. We show that the amplitude/phase distributions exhibit a rich and more complex structure depending not just on the covariance of the process but specifically on the form of the kernel and the noise distribution. We present a theoretical tool for studying these distributions that follows from a proof of the spectral theorem that yields an explicit expression for the spectral measure. The main interest is in noise measures based on second-order Lévy motions since such measures are easily available through independent sampling. We approximate the spectral stochastic measure by independent noise increments which allows us to obtain amplitude/phase distributions that is of fundamental interest for analyzing processes in the frequency domain. For the purpose of approximating the moving average process through sums of trigonometric functions, we assess the mean square error of discretization of the spectral representation. For a specified accuracy, the approximation is explicitly given. We illustrate the method for the moving averages driven by the Laplace motion.     

Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling

The normal inverse Gaussian distribution is defined as a variance-mean mixture of a normal distribution with the inverse Gaussian as the mixing distribution. The distribution determines an homogeneous Lévy process, and this process is representable through subordination of Brownian motion by the inverse Gaussian process. The canonical, Lévy type, decomposition of the process is determined. As a preparation for developments in the latter part of the paper the connection of the normal inverse Gaussian distribution to the classes of generalized hyperbolic and inverse Gaussian distributions is briefly reviewed. Then a discussion is begun of the potential of the normal inverse Gaussian distribution and Lévy process for modelling and analysing statistical data, with particular reference to extensive sets of observations from turbulence and from finance. These areas of application imply a need for extending the inverse Gaussian Lévy process so as to accommodate certain, frequently observed, temporal dependence structures. Some extensions, of the stochastic volatility type, are constructed via an observation-driven approach to state space modelling. At the end of the paper generalizations to multivariate settings are indicated.     

Absolute Moments of Generalized Hyperbolic Distributions and Approximate Scaling of Normal Inverse Gaussian Lévy Processes

Abstract.  Expressions for (absolute) moments of generalized hyperbolic and normal inverse Gaussian (NIG) laws are given in terms of moments of the corresponding symmetric laws. For the (absolute) moments centred at the location parameter μ explicit expressions as series containing Bessel functions are provided. Furthermore, the derivatives of the logarithms of absolute μ -centred moments with respect to the logarithm of time are calculated explicitly for NIG Lévy processes. Computer implementation of the formulae obtained is briefly discussed. Finally, some further insight into the apparent scaling behaviour of NIG Lévy processes is gained.     

On the likelihood function of small time variance Gamma Lévy processes

We investigate the likelihood function of small generalized Laplace laws and variance gamma Lévy processes in the short time framework. We prove the local asymptotic normality property in statistical inference for the variance gamma Lévy process under high-frequency sampling with its associated optimal convergence rate and Fisher information matrix. The location parameter is required to be given in advance for this purpose, while the remaining three parameters are jointly well behaved with an invertible Fisher information matrix. The results are discussed with relation to equivalent formulations of the variance gamma Lévy process, that is, as a time-changed Brownian motion and as a difference of two independent gamma processes.     

A mixture of beta–Dirichlet processes prior for Bayesian analysis of event history data

In this paper, we propose a mixture of beta–Dirichlet processes as a nonparametric prior for the cumulative intensity functions of a Markov process. This family of priors is a natural extension of a mixture of Dirichlet processes or a mixture of beta processes which are devised to compromise advantages of parametric and nonparametric approaches. They give most of their prior mass to the small neighborhood of a specific parametric model. We show that a mixture of beta–Dirichlet processes prior is conjugate with Markov processes. Formulas for computing the posterior distribution are derived. Finally, results of analyzing credit history data are given.     

Mixture reduction via predictive scores

This paper explores the problem of reducing a mixture of conjugate priors to a smaller mixture, from the perspective of the application of a distance measure between priors. The analysis focuses on mixtures of Dirichlet priors, but it has wider applicability. In respect to the proposed scheme, it emerges that for mixtures of -distributions a simple moment-matching reduction procedure is optimal and very good for the more general case of Dirichlet mixtures.     

Slice sampling mixture models

We propose a more efficient version of the slice sampler for Dirichlet process mixture models described by Walker (Commun. Stat., Simul. Comput. 36:45–54, 2007). This new sampler allows for the fitting of infinite mixture models with a wide-range of prior specifications. To illustrate this flexibility we consider priors defined through infinite sequences of independent positive random variables. Two applications are considered: density estimation using mixture models and hazard function estimation. In each case we show how the slice efficient sampler can be applied to make inference in the models. In the mixture case, two submodels are studied in detail. The first one assumes that the positive random variables are Gamma distributed and the second assumes that they are inverse-Gaussian distributed. Both priors have two hyperparameters and we consider their effect on the prior distribution of the number of occupied clusters in a sample. Extensive computational comparisons with alternative “conditional” simulation techniques for mixture models using the standard Dirichlet process prior and our new priors are made. The properties of the new priors are illustrated on a density estimation problem.     

A note on non-negative continuous time processes

Summary.  Recently there has been much work on developing models that are suitable for analysing the volatility of a continuous time process. One general approach is to define a volatility process as the convolution of a kernel with a non-decreasing Lévy process, which is non-negative if the kernel is non-negative. Within the framework of time continuous autoregressive moving average (CARMA) processes, we derive a necessary and sufficient condition for the kernel to be non-negative. This condition is in terms of the Laplace transform of the CARMA kernel, which has a simple form. We discuss some useful consequences of this result and delineate the parametric region of stationarity and non-negative kernel for some lower order CARMA models.     

Exact and approximate sum representations for the Dirichlet process

The Dirichlet process can be regarded as a random probability measure for which the authors examine various sum representations. They consider in particular the gamma process construction of Ferguson (1973) and the “stick‐breaking” construction of Sethuraman (1994). They propose a Dirichlet finite sum representation that strongly approximates the Dirichlet process. They assess the accuracy of this approximation and characterize the posterior that this new prior leads to in the context of Bayesian nonpara‐metric hierarchical models.     

Lévy Processes,Phase-Type Distributions,and Martingales

Lévy processes are defined as processes with stationary independent increments and have become increasingly popular as models in queueing, finance, etc.; apart from Brownian motion and compound Poisson processes, some popular examples are stable processes, variance gamma processes, CGMY Lévy processes (tempered stable processes), NIG (normal inverse Gaussian) Lévy processes, and hyperbolic Lévy processes. We consider here a dense class of Lévy processes, compound Poisson processes with phase-type jumps in both directions and an added Brownian component. Within this class, we survey how to explicitly compute a number of quantities that are traditionally studied in the area of Lévy processes, in particular two-sided exit probabilities and associated Laplace transforms, the closely related scale function, one-sided exit probabilities and associated Laplace transforms coming up in queueing problems, and similar quantities for a Lévy process with reflection in 0. The solutions are in terms of roots to polynomials, and the basic equations are derived by purely probabilistic arguments using martingale optional stopping; a particularly useful martingale is the so-called Kella-Whitt martingale. Also, the relation to fluid models with a Brownian component is discussed.     

Sampling schemes for generalized linear Dirichlet process random effects models

We evaluate MCMC sampling schemes for a variety of link functions in generalized linear models with Dirichlet process random effects. First, we find that there is a large amount of variability in the performance of MCMC algorithms, with the slice sampler typically being less desirable than either a Kolmogorov–Smirnov mixture representation or a Metropolis–Hastings algorithm. Second, in fitting the Dirichlet process, dealing with the precision parameter has troubled model specifications in the past. Here we find that incorporating this parameter into the MCMC sampling scheme is not only computationally feasible, but also results in a more robust set of estimates, in that they are marginalized-over rather than conditioned-upon. Applications are provided with social science problems in areas where the data can be difficult to model, and we find that the nonparametric nature of the Dirichlet process priors for the random effects leads to improved analyses with more reasonable inferences.     

On the Posterior Consistency of Mixtures of Dirichlet Process Priors with Censored Data

Mixtures of Dirichlet process priors offer a reasonable compromise between purely parametric and purely non‐parametric models, and are popularly used in survival analysis and for testing problems with non‐parametric alternatives. In this paper, we study large sample properties of the posterior distribution with a mixture of Dirichlet process priors. We show that the posterior distribution of the survival function is consistent with right censored data.     

Tempered Mittag-Leffler Lévy processes

In this article, we introduce tempered Mittag-Leffler Lévy processes (TMLLP). TMLLP is represented as tempered stable subordinator delayed by a gamma process. Its probability density function and Lévy density are obtained in terms of infinite series and Mittag-Leffler function, respectively. Asymptotic forms of the tails and moments are given. A step-by-step procedure of the parameters estimation and simulation of sample paths is given. We also provide main results available for Mittag-Leffler Lévy processes (MLLP) and some extensions which are not available in a collective way in a single article. Our results generalize and complement the results available on Mittag-Leffler distribution and MLLP in several directions. Further, the asymptotic forms of the moments of the first-exit times of the TMLLP are also discussed.     

On a Class of Random Probability Measures with General Predictive Structure

Abstract. In this study, we investigate a recently introduced class of non‐parametric priors, termed generalized Dirichlet process priors. Such priors induce (exchangeable random) partitions that are characterized by a more elaborate clustering structure than those arising from other widely used priors. A natural area of application of these random probability measures is represented by species sampling problems and, in particular, prediction problems in genomics. To this end, we study both the distribution of the number of distinct species present in a sample and the distribution of the number of new species conditionally on an observed sample. We also provide the Bayesian Non‐parametric estimator for the number of new species in an additional sample of given size and for the discovery probability as function of the size of the additional sample. Finally, the study of its conditional structure is completed by the determination of the posterior distribution.     

A blocked Gibbs sampler for NGG-mixture models via a priori truncation

We define a new class of random probability measures, approximating the well-known normalized generalized gamma (NGG) process. Our new process is defined from the representation of NGG processes as discrete measures where the weights are obtained by normalization of the jumps of Poisson processes and the support consists of independent identically distributed location points, however considering only jumps larger than a threshold \(\varepsilon \). Therefore, the number of jumps of the new process, called \(\varepsilon \)-NGG process, is a.s. finite. A prior distribution for \(\varepsilon \) can be elicited. We assume such a process as the mixing measure in a mixture model for density and cluster estimation, and build an efficient Gibbs sampler scheme to simulate from the posterior. Finally, we discuss applications and performance of the model to two popular datasets, as well as comparison with competitor algorithms, the slice sampler and a posteriori truncation.