A class of accelerated means regression models for recurrent event data

In this article, we propose a general class of accelerated means regression models for recurrent event data. The class includes the proportional means model, the accelerated failure time model and the accelerated rates model as special cases. The new model offers great flexibility in formulating the effects of covariates on the mean functions of counting processes while leaving the stochastic structure completely unspecified. For the inference on the model parameters, estimating equation approaches are developed and both large and final sample properties of the proposed estimators are established. In addition, some graphical and numerical procedures are presented for model checking. An illustration with multiple-infection data from a clinic study on chronic granulomatous disease is also provided.     

The Pearson Diffusions: A Class of Statistically Tractable Diffusion Processes

Abstract.  The Pearson diffusions form a flexible class of diffusions defined by having linear drift and quadratic squared diffusion coefficient. It is demonstrated that for this class explicit statistical inference is feasible. A complete model classification is presented for the ergodic Pearson diffusions. The class of stationary distributions equals the full Pearson system of distributions. Well-known instances are the Ornstein–Uhlenbeck processes and the square root (CIR) processes. Also diffusions with heavy-tailed and skew marginals are included. Explicit formulae for the conditional moments and the polynomial eigenfunctions are derived. Explicit optimal martingale estimating functions are found. The discussion covers GMM, quasi-likelihood, non-linear weighted least squares estimation and likelihood inference too. The analytical tractability is inherited by transformed Pearson diffusions, integrated Pearson diffusions, sums of Pearson diffusions and Pearson stochastic volatility models. For the non-Markov models, explicit optimal prediction-based estimating functions are found. The estimators are shown to be consistent and asymptotically normal.     

Inférence statistique dans les processus stochastiques: Aperçu historique

The study of statistical inference for stochastic processes has evolved along two paths. Some problems related to particular processes have been studied, and also some trials to extend general results obtained for independent identically distributed random variables have been made. We retrace the first main contributions, evaluate their influence, and give an idea of the evolution of the research in the field of statistical inference made with observations coming from a stochastic process.     

A pathwise inference method for the parameters of diffusion terms

We consider inference of the parameters of the diffusion term for continuous time stochastic processes with a power-type dependence of the diffusion coefficient from the underlying process such as Cox–Ingersoll–Ross, CKLS, and similar processes. We suggest some original pathwise estimates for this coefficient and for the power index based on an analysis of an auxiliary continuous time complex-valued process generated by the underlying real-valued process. These estimates do not rely on the distribution of the underlying process and on a particular choice of the drift. Some numerical experiments are used to illustrate the feasibility of the suggested method.     

Inference for Observations of Integrated Diffusion Processes

Abstract.  Estimation of parameters in diffusion models is investigated when the observations are integrals over intervals of the process with respect to some weight function. This type of observations can, for example, be obtained when the process is observed after passage through an electronic filter. Another example is provided by the ice-core data on oxygen isotopes used to investigate paleo-temperatures. Finally, such data play a role in connection with the stochastic volatility models of finance. The integrated process is not a Markov process. Therefore, prediction-based estimating functions are applied to estimate parameters in the underlying diffusion model. The estimators are shown to be consistent and asymptotically normal. The theory developed in the paper also applies to integrals of processes other than diffusions. The method is applied to inference based on integrated data from Ornstein–Uhlenbeck processes and from the Cox–Ingersoll–Ross model, for both of which an explicit optimal estimating function is found.     

Random variate generation from D-distributions

Within the context of non-parametric Bayesian inference, Dykstra and Laud (1981) define an extended gamma (EG) process and use it as a prior on increasing hazard rates. The attractive features of the extended gamma (EG) process, among them its capability to index distribution functions that are absolutely continuous, are offset by the intractable nature of the computation that needs to be performed. Sampling based approaches such as the Gibbs Sampler can alleviate these difficulties but the EG processes then give rise to the problem of efficient random variate generation from a class of distributions called D-distributions. In this paper, we describe a novel technique for sampling from such distributions, thereby providing an efficient computation procedure for non-parametric Bayesian inference with a rich class of priors for hazard rates.     

RECENT TRENDS IN ASYMPTOTIC OPTIMAL INFERENCE FOR DEPENDENT OBSERVATIONS1

An overview of some recent developments in the area of asymptotic inference for non-ergodic type stochastic processes is presented. Both local and global formulations of the asymptotic model are given, and non-local optimality results are reviewed. Recent results on conditional inference are briefly discussed. Some open problems and possibilities for new developments are also mentioned.     

Geostochastic calculus

A stochastic calculus for a family of continuous measure-valued Markov processes is developed. Such processes arise naturally in the construction of stochastic models of spatially distributed populations. The stochastic calculus is a tool whereby a class of density-dependent models can be studied in terms of the multiplicative measure diffusion process. In this paper the stochastic integral is introduced in the space-time setting and a Cameron-Martin-Girsanov theorem is established.     

Gaussian models for degradation processes-part I: Methods for the analysis of biomarker data

We present two stochastic models that describe the relationship between biomarker process values at random time points, event times, and a vector of covariates. In both models the biomarker processes are degradation processes that represent the decay of systems over time. In the first model the biomarker process is a Wiener process whose drift is a function of the covariate vector. In the second model the biomarker process is taken to be the difference between a stationary Gaussian process and a time drift whose drift parameter is a function of the covariates. For both models we present statistical methods for estimation of the regression coefficients. The first model is useful for predicting the residual time from study entry to the time a critical boundary is reached while the second model is useful for predicting the latency time from the infection until the time the presence of the infection is detected. We present our methods principally in the context of conducting inference in a population of HIV infected individuals.     

Spatiotemporal prediction for log-Gaussian Cox processes

Space–time point pattern data have become more widely available as a result of technological developments in areas such as geographic information systems. We describe a flexible class of space–time point processes. Our models are Cox processes whose stochastic intensity is a space–time Ornstein–Uhlenbeck process. We develop moment-based methods of parameter estimation, show how to predict the underlying intensity by using a Markov chain Monte Carlo approach and illustrate the performance of our methods on a synthetic data set.     

A class of Box-Cox transformation models for recurrent event data

In this article, we propose a class of Box-Cox transformation models for recurrent event data, which includes the proportional means models as special cases. The new model offers great flexibility in formulating the effects of covariates on the mean functions of counting processes while leaving the stochastic structure completely unspecified. For the inference on the proposed models, we apply a profile pseudo-partial likelihood method to estimate the model parameters via estimating equation approaches and establish large sample properties of the estimators and examine its performance in moderate-sized samples through simulation studies. In addition, some graphical and numerical procedures are presented for model checking. An example of application on a set of multiple-infection data taken from a clinic study on chronic granulomatous disease (CGD) is also illustrated.     

Threshold selection in jump-discriminant filter for discretely observed jump processes

Threshold estimation is one of the useful techniques in the inference for jump-type stochastic processes from discrete observations. In this method, a jump-discriminant filter is used to infer the continuous part and the jump part separately. Although there are several choices for the filter, statistics constructed via filters are often sensitive to the choice. This paper presents some numerical procedures for selecting a suitable filter based on observations.     

Modeling of semi-competing risks by means of first passage times of a stochastic process

In semi-competing risks one considers a terminal event, such as death of a person, and a non-terminal event, such as disease recurrence. We present a model where the time to the terminal event is the first passage time to a fixed level c in a stochastic process, while the time to the non-terminal event is represented by the first passage time of the same process to a stochastic threshold S, assumed to be independent of the stochastic process. In order to be explicit, we let the stochastic process be a gamma process, but other processes with independent increments may alternatively be used. For semi-competing risks this appears to be a new modeling approach, being an alternative to traditional approaches based on illness-death models and copula models. In this paper we consider a fully parametric approach. The likelihood function is derived and statistical inference in the model is illustrated on both simulated and real data.     

Asymptotic distribution of martingale estimators for a class of epidemic models

This article is a contribution to the asymptotic inference on the parameters of a quite general class of stochastic models for the spread of epidemics developing in closed populations. Various epidemic models are contained within our framework, for instance, a stochastic version of the Kermack and McKendrick model and the SIS epidemic model. Each model belonging to this class, which consists in a family of discrete-time stochastic process, contains certain parameters to be estimated by means of martingale estimators. Some particular cases defined by means of Markov chains are included in our setting. The main aim of this work is to prove consistency and asymptotic normality of these estimators. Some hypothesis tests based on the main results are also shown.     

Semiparametric Stochastic Modeling of the Rate Function in Longitudinal Studies

In longitudinal biomedical studies, there is often interest in the rate functions, which describe the functional rates of change of biomarker profiles. This paper proposes a semiparametric approach to model these functions as the realizations of stochastic processes defined by stochastic differential equations. These processes are dependent on the covariates of interest and vary around a specified parametric function. An efficient Markov chain Monte Carlo algorithm is developed for inference. The proposed method is compared with several existing methods in terms of goodness-of-fit and more importantly the ability to forecast future functional data in a simulation study. The proposed methodology is applied to prostate-specific antigen profiles for illustration. Supplementary materials for this paper are available online.     

Statistical Inference for Curved Fibrous Objects in 3D – Based on Multiple Short Observations of Multivariate Autoregressive Processes

This paper deals with statistical inference on the parameters of a stochastic model, describing curved fibrous objects in three dimensions, that is based on multivariate autoregressive processes. The model is fitted to experimental data consisting of a large number of short independently sampled trajectories of multivariate autoregressive processes. We discuss relevant statistical properties (e.g. asymptotic behaviour as the number of trajectories tends to infinity) of the maximum likelihood (ML) estimators for such processes. Numerical studies are also performed to analyse some of the more intractable properties of the ML estimators. Finally the whole methodology, i.e., the fibre model and its statistical inference, is applied to appropriately describe the tracking of fibres in real materials.     

DYNAMIC FACTOR MODELS

This paper introduces nonlinear dynamic factor models for various applications related to risk analysis. Traditional factor models represent the dynamics of processes driven by movements of latent variables, called the factors. Our approach extends this setup by introducing factors defined as random dynamic parameters and stochastic autocorrelated simulators. This class of factor models can represent processes with time varying conditional mean, variance, skewness and excess kurtosis. Applications discussed in the paper include dynamic risk analysis, such as risk in price variations (models with stochastic mean and volatility), extreme risks (models with stochastic tails), risk on asset liquidity (stochastic volatility duration models), and moral hazard in insurance analysis.

We propose estimation procedures for models with the marginal density of the series and factor dynamics parameterized by distinct subsets of parameters. Such a partitioning of the parameter vector found in many applications allows to simplify considerably statistical inference. We develop a two- stage Maximum Likelihood method, called the Finite Memory Maximum Likelihood, which is easy to implement in the presence of multiple factors. We also discuss simulation based estimation, testing, prediction and filtering.     

Estimation for Non-Negative Lévy-Driven CARMA Processes

Continuous-time autoregressive moving average (CARMA) processes with a nonnegative kernel and driven by a nondecreasing Lévy process constitute a useful and very general class of stationary, nonnegative continuous-time processes that have been used, in particular, for the modeling of stochastic volatility. Brockwell, Davis, and Yang (2007) derived efficient estimates of the parameters of a nonnegative Lévy-driven CAR(1) process and showed how the realization of the underlying Lévy process can be estimated from closely-spaced observations of the process itself. In this article we show how the ideas of that article can be generalized to higher order CARMA processes with nonnegative kernel, the key idea being the decomposition of the CARMA process into a sum of dependent Ornstein–Uhlenbeck processes.     

Hidden Second‐order Stationary Spatial Point Processes

In the existing statistical literature, the almost default choice for inference on inhomogeneous point processes is the most well‐known model class for inhomogeneous point processes: reweighted second‐order stationary processes. In particular, the K‐function related to this type of inhomogeneity is presented as the inhomogeneous K‐function. In the present paper, we put a number of inhomogeneous model classes (including the class of reweighted second‐order stationary processes) into the common general framework of hidden second‐order stationary processes, allowing for a transfer of statistical inference procedures for second‐order stationary processes based on summary statistics to each of these model classes for inhomogeneous point processes. In particular, a general method to test the hypothesis that a given point pattern can be ascribed to a specific inhomogeneous model class is developed. Using the new theoretical framework, we reanalyse three inhomogeneous point patterns that have earlier been analysed in the statistical literature and show that the conclusions concerning an appropriate model class must be revised for some of the point patterns.