Convergence analysis of parametric identification methods for jump processes

The asymptotic properties of recursive estimate for the parameter of the signal process of a jump process are investigated. When increasing the observation period, conditions are proposed that guarantee the consistency and asymptotic normality of the estimate. An estimate of the unknown parameter for the random intensity of an extended gamma process model is discussed by way of example.     

An Additive–Multiplicative Cox–Aalen Regression Model

We propose an additive–multiplicative intensity model that extends the Cox regression model as well as the additive Aalen model. Instead of having a simple baseline intensity the extended model uses an additive Aalen model as its covariate dependent baseline. Approximate maximum likelihood estimators of the baseline intensity functions and the relative risk parameters of the Cox model are suggested by solving the score equations. The derived estimator is efficient. We establish the large sample properties of the estimator. The model provides a simple pragmatic way of including time-varying covariate effects.     

On the estimation of the stochastic intensity and the parameters of neyman-scott trigger processes

In this paper we are concerned with a class of simple point processes, whose unobservable stochastic intensity is a shot-noise process. We derive a stochastic equation for the conditional moment generating function of the intensity, which can be solved in a recursive way. This yields explicit expression for the minimum variance estimate of the intensity as well as the likelihood ration with respect to the reference measure, on the basis of point process observations.     

Intensity Estimation for Spatial Point Processes Observed with Noise

Abstract.  This article introduces a kernel estimator of the intensity function of spatial point processes taking into account location errors. The asymptotic properties of the estimator are derived and a bandwidth selection procedure is described. A simulation study compares our results with that of the classical kernel estimator and shows that the edge-corrected deconvoluting kernel estimator is more appropriate.     

On Statistical Models for d-Dimensional Stable Processes, and Some Generalizations

In statistical models where jumps of a d -dimensional stable process ( S _t ) _{t ≥0} are observed in windows with certain asymptotic properties, and where parameters appearing in the Levy measure of S are to be estimated, we have asymptotically efficient estimators. If Poisson random measure μ on (0, ∞) × ( R ^d \{0}) with intensity dt Λ( dx ) replaces the jump measure of S , where Λ is a ε-finite measure on R ^d \{0} admitting tail parameters in a suitable sense, we specify a notion of neighbourhood which allows to treat efficiency in statistical experiments of the second type by switching to accompanying sequences of the stable process type considered first.     

Algorithms for Point Processes Analysis

A time point process can be defined either by the statistical properties of the time intervals between successive points or by those of the number of points in arbitrary time intervals. There are mathematical expressions to link up these two points of view, but they are in many cases too complicated to be used in practice. In this article, we present an algorithmic procedure to obtain the number of points of a stationary point process recorded in some time intervals by processing the values of the distances between successive points. We present some results concerning the statistical analysis of these numbers of points and when analytical calculations are possible the experimental results obtained with our algorithms are in excellent agreement with those predicted by the theory. Some properties of point processes in which theoretical calculations are almost impossible are also presented.     

A Note on Empirical Process Methods in the Theory of Poisson Point Processes

For a sample taken from an i.i.d. sequence of Poisson point processes with not necessarily finite unknown intensity measure the arithmetic mean is shown to be an estimator which is consistent uniformly on certain classes of functions. The method is a reduction to the case of finite intensity measure, which in turn can be dealt with using empirical process methods. A functional central limit theorem is also established in this context.     

Some comments on sample quantiles for dependent observations

Sample quantiles obtained from a linear process are examined and their asymptotic properties are derived by using simple elementary arguments.     

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.     

Simulations of Some Doubly Stochastic Poisson Point Processes

Computer simulations of point processes are important either to verify the results of certain theoretical calculations that can be very awkward at times or to obtain practical results when these calculations become almost impossible. One of the most common methods for the simulation of nonstationary Poisson processes is random thinning. Its extension when the intensity becomes random (doubly stochastic Poisson processes) depends on the structure of this intensity. If the random density takes only discrete values, which is a common situation in many physical problems where quantum mechanics introduces discrete states, it is shown that the thinning method can be applied without error. We study in particular the case of binary density and present the kind of theoretical calculations that then become possible. The results of various experiments realized with data obtained by simulation show a fairly good agreement with the theoretical calculations.     

Continuous-Time Stochastic Processes with Cyclical Long-Range Dependence

This paper introduces continuous‐time random processes whose spectral density is unbounded at some non‐zero frequencies. The discretized versions of these processes have asymptotic properties similar to those of discrete‐time Gegenbauer processes. The paper presents some properties of the covariance function and spectral density as well as a theory of statistical estimation of the mean and covariance function of such processes. Some directions for further generalizations of the results are indicated.     

Inversion of the Renewal Density with Dead-time

Stationary renewal point processes are defined by the probability distribution of the distances between successive points (lifetimes) that are independent and identically distributed random variables. For some applications it is also interesting to define the properties of a renewal process by using the renewal density. There are well-known expressions of this density in terms of the probability density of the lifetimes. It is more difficult to solve the inverse problem consisting in the determination of the density of the lifetimes in terms of the renewal density. Theoretical expressions between their Laplace transforms are available but the inversion of these transforms is often very difficult to obtain in closed form. We show that this is possible for renewal processes presenting a dead-time property characterized by the fact that the renewal density is zero in an interval including the origin. We present the principle of a recursive method allowing the solution of this problem and we apply this method to the case of some processes with input dead-time. Computer simulations on Poisson and Erlang (2) processes show quite good agreement between theoretical calculations and experimental measurements on simulated data.     

Bayesian hierarchical models for linear networks

The purpose of this study is to highlight dangerous motorways via estimating the intensity of accidents and study its pattern across the UK motorway network. Two methods have been developed to achieve this aim. First, the motorway-specific intensity is estimated by using a homogeneous Poisson process. The heterogeneity across motorways is incorporated using two-level hierarchical models. The data structure is multilevel since each motorway consists of junctions that are joined by grouped segments. In the second method, the segment-specific intensity is estimated. The homogeneous Poisson process is used to model accident data within grouped segments but heterogeneity across grouped segments is incorporated using three-level hierarchical models. A Bayesian method via Markov Chain Monte Carlo is used to estimate the unknown parameters in the models and the sensitivity to the choice of priors is assessed. The performance of the proposed models is evaluated by a simulation study and an application to traffic accidents in 2016 on the UK motorway network. The deviance information criterion (DIC) and the widely applicable information criterion (WAIC) are employed to choose between models.     

FUNCTIONAL ESTIMATION BY ASYMPTOTIC REGRESSION

Through an appeal to asymptotic Gaussian representations of certain empirical stochastic processes, the techniques of continuous regression are applied to derive estimates for underlying parametric probability laws. This asymptotic regression approach yields estimates for a wide range of statistical problems, including estimation based on the empirical quantile function, Poisson process intensity estimation, and parametric density estimation.     

MOMENT ESTIMATION IN THE CLASS OF BISEXUAL BRANCHING PROCESSES WITH POPULATION–SIZE DEPENDENT MATING

This paper concerns the estimation of the offspring mean vector, the covariance matrix and the growth rate in the class of bisexual branching processes with population‐size dependent mating. For the proposed estimators, some unconditional moments and some conditioned to non‐extinction are determined and asymptotic properties are established. Confidence intervals are obtained and, as illustration, a simulation example is given.     

A Note on Continuous Spatial-Temporal Dynamics of Stochastic Processes

We propose a general form to analyze the space-time interdependency of continuous space-time stochastic processes. We present a new space-time approach based on the intensity function of the underlying point process. These formulations can be, to some extent, analytically solved to obtain explicit formulae of interest. We define a general function that controls the space-time interaction and allows for closed forms depending on the particular choice of several mathematical tools playing a role in this interaction function. In particular, we make use of copulas and Laplace transforms to provide interesting examples of the dynamics of the random intensity function and, in turn, of the number of points contained in a given region.     

Mixed-Poisson Point Process with Partially-Observed Covariates: Ecological Momentary Assessment of Smoking

Ecological Momentary Assessment is an emerging method of data collection in behavioral research that may be used to capture the times of repeated behavioral events on electronic devices, and information on subjects' psychological states through the electronic administration of questionnaires at times selected from a probability-based design as well as the event times. A method for fitting a mixed Poisson point process model is proposed for the impact of partially-observed, time-varying covariates on the timing of repeated behavioral events. A random frailty is included in the point-process intensity to describe variation among subjects in baseline rates of event occurrence. Covariate coefficients are estimated using estimating equations constructed by replacing the integrated intensity in the Poisson score equations with a design-unbiased estimator. An estimator is also proposed for the variance of the random frailties. Our estimators are robust in the sense that no model assumptions are made regarding the distribution of the time-varying covariates or the distribution of the random effects. However, subject effects are estimated under gamma frailties using an approximate hierarchical likelihood. The proposed approach is illustrated using smoking data.     

Third order point process intensity estimation for reaction time experiment data

The authors consider a simple parametric model for data from a point process version of a reaction time experiment to test the nature of thinning mechanisms in the presence of nonlinear inhibition in the eye‐brain‐hand system. In particular, they study the effects of two such mechanisms on the third order intensity estimate. An alternative nonparametric estimate is also considered in this context. These various approaches are applied on twelve runs of real data, but the analysis turns out to be inconclusive: neither mechanism can be ruled out, nor the possibility that their effects could be combined.     

Semiparametric multiparameter regression survival modeling

We consider a log-linear model for survival data, where both the location and scale parameters depend on covariates, and the baseline hazard function is completely unspecified. This model provides the flexibility needed to capture many interesting features of survival data at a relatively low cost in model complexity. Estimation procedures are developed, and asymptotic properties of the resulting estimators are derived using empirical process theory. Finally, a resampling procedure is developed to estimate the limiting variances of the estimators. The finite sample properties of the estimators are investigated by way of a simulation study, and a practical application to lung cancer data is illustrated.