Statistical Inference for Transformation Inhomogeneous Point Processes

Abstract.  Statistical inference for exponential inhomogeneous Markov point processes by transformation is discussed. It is argued that the inhomogeneity parameter can be estimated, using a partial likelihood based on an inhomogeneous Poisson point process. The inhomogeneity parameter can thereby be estimated without taking the interaction into account, which simplifies the statistical analysis considerably. Data analysis and simulation experiments support the results.     

A Non‐Parametric Estimator of the Spectral Density of a Continuous‐Time Gaussian Process Observed at Random Times

Abstract. In numerous applications data are observed at random times and an estimated graph of the spectral density may be relevant for characterizing and explaining phenomena. By using a wavelet analysis, one derives a non‐parametric estimator of the spectral density of a Gaussian process with stationary increments (or a stationary Gaussian process) from the observation of one path at random discrete times. For every positive frequency, this estimator is proved to satisfy a central limit theorem with a convergence rate depending on the roughness of the process and the moment of random durations between successive observations. In the case of stationary Gaussian processes, one can compare this estimator with estimators based on the empirical periodogram. Both estimators reach the same optimal rate of convergence, but the estimator based on wavelet analysis converges for a different class of random times. Simulation examples and an application to biological data are also provided.     

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.     

Non-linear models 1

This is the first application of a new method for testing stationary random point processes. Consider the class of all stationary ergodic point processes on the real line with arbitrary dependences among the inter–point distances (spacing).The hypothesis is :The observed process φ is a homogeneous Poisson process or more (resp.less) regular than a Poisson process.The sample is the vector of the first n points t₁, …,t_n.There is a close relation between our method for testing and queueing theory: For finding an appropriate test statistic, we observe the behaviour of a single server queue with the input φ.A table of critical values is given.     

Contrast Estimation for Parametric Stationary Determinantal Point Processes

We study minimum contrast estimation for parametric stationary determinantal point processes. These processes form a useful class of models for repulsive (or regular, or inhibitive) point patterns and are already applied in numerous statistical applications. Our main focus is on minimum contrast methods based on the Ripley's K‐function or on the pair correlation function. Strong consistency and asymptotic normality of theses procedures are proved under general conditions that only concern the existence of the process and its regularity with respect to the parameters. A key ingredient of the proofs is the recently established Brillinger mixing property of stationary determinantal point processes. This work may be viewed as a complement to the study of Y. Guan and M. Sherman who establish the same kind of asymptotic properties for a large class of Cox processes, which in turn are models for clustering (or aggregation).     

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.     

Limit distribution for point processes of high local maxima

We define a local dependence condition which enables us to obtain a sufficient condition for the convergence in distribution of the sequence of point processes of high local maxima generated by a strictly stationary sequence of random variables. The limit point process is an homogeneous Poisson process. The result is applied to a stationary autoregressive sequence of maxima for which, after each upcrossing of a high level, we observe a downward tendency.     

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.     

Detection capability of residual control chart for stationary process data

SUMMARY In recent years, methods for dealing with autocorrelated data in the statistical process control environment have been proposed. A primary method is based on modeling the process data and applying control charts to the residuals. However, the residual charts do not have the same properties as the traditional charts. In the literature, there has been no systematic study on the detection capability of the residual chart for the stationary processes. The article develops a measure of the detection capability of the residual chart for the general stationary processes. Conditions under which the residual chart reduces or increases the detection capability are given. The relationships between the detection capability and the average run length of the residual chart are also established.     

Four general multivariate stationary extremal Markovian processes

Two general multivariate stationary Markovian process with maximization structure (denoted by Max-AR(1) and MaxI-AR(1)) are developed respectively. Max-AR(1) is a subclass of MaxI-AR(1). The characterization of the Max-AR(1) and MaxI-AR(1) to be stationary is studied. Some properties of the two maximization processes are derived. Two more related general multivariate stochastic Markovian process with minification structure are analogously constructed (denoted by Min-AR(1) and MinI-AR(1)). Some well known maximization and minification processes are special cases of these four extermal Markovian processes. Two of them are simulated and some point estimations are provided as an illustration of the wide application of these four processes.     

An efficiency result for the empirical characteristic function in stationary time-series models

It is shown under general conditions that arbitrarily high asymptotic efficiencies can be obtained when the parameters of a stationary time series are estimated by fitting the characteristic functions of the process to their empirical versions. A consistency and a central limit result are also given.     

Time series,point processes,and hybrids

Techniques developed for the study of time series, point processes, and marked point processes can suggest corresponding techniques for each other, and common techniques can be recognized. In this paper connections are drawn based on conceptual foundations, basic parameters, analyses, displays, algorithms, problems, models. The definitions and techniques are brought out by specific scientific problems. The emphasis is on the single-realization stationary case and on the use of second- and third-order moments to help understand the realization. The tool of stacking, at a particular period, is employed in several of the examples.     

The 1970 US draft lottery revisited: a spatial analysis

Summary.  We revise the result of the 1970 selective service draft lottery in the USA following an open question that was suggested by Fienberg in a paper published in Science in 1971. The result of the drawings can be viewed as a particular spatial pattern which can be analysed by using general spatial tools adapted to our context. Approaches for assessing the complete spatial randomness for this spatial process on a finite support are proposed. More specifically, these approaches involve the number of events in a square window and a k ( r )-based function used to analyse stationary spatial point processes.     

Consistent parametric estimation of the intensity of a spatial–temporal point process

We consider conditions under which parametric estimates of the intensity of a spatial–temporal point process are consistent. Although the actual point process being estimated may not be Poisson, an estimate involving maximizing a function that corresponds exactly to the log-likelihood if the process is Poisson is consistent under certain simple conditions. A second estimate based on weighted least squares is also shown to be consistent under quite similar assumptions. The conditions for consistency are simple and easily verified, and examples are provided to illustrate the extent to which consistent estimation may be achieved. An important special case is when the point processes being estimated are in fact Poisson, though other important examples are explored as well.     

Weak approximations for quantile processes of stationary sequences

ABSTRACT By studying the deviations between uniform empirical and quantile processes (the so-called Bahadur-Kiefer representations) of a stationary sequence in properly weighted sup-norm metrics, we find a general approach to obtaining weighted results for uniform quantile processes of stationary sequences. Consequently we are able to obtain weak convergence for weighted uniform quantile processes of stationary mixing and associated sequences. Further, by studying the sup-norm distance of a general quantile process from its corresponding uniform quantile process, we find that information at the two end points of the uniform quantile process can be so utilized that this weighted sup-norm distance converges in probability to zero under the so-called Csörgõ-Révész conditions. This enables us to obtain weak convergence for weighted general quantile processes of stationary mixing and associated sequences.     

Adaptive estimating function inference for nonstationary determinantal point processes

Estimating function inference is indispensable for many common point process models where the joint intensities are tractable while the likelihood function is not. In this article, we establish asymptotic normality of estimating function estimators in a very general setting of nonstationary point processes. We then adapt this result to the case of nonstationary determinantal point processes, which are an important class of models for repulsive point patterns. In practice, often first‐ and second‐order estimating functions are used. For the latter, it is a common practice to omit contributions for pairs of points separated by a distance larger than some truncation distance, which is usually specified in an ad hoc manner. We suggest instead a data‐driven approach where the truncation distance is adapted automatically to the point process being fitted and where the approach integrates seamlessly with our asymptotic framework. The good performance of the adaptive approach is illustrated via simulation studies for non‐stationary determinantal point processes and by an application to a real dataset.     

Efficiency of linear predictors for periodic processes using an incorrect covariance function

Previous work by the author showed that for interpolating a weakly stationary random field, using an incorrect spectral density that has similar high-frequency behavior as the correct spectral density can yield asymptotically optimal linear predictions as the number of observations in a fixed domain increases. However, explicit results on how fast this convergence to optimality occurs could only be obtained for a limited class of processes in one dimension. By considering periodic processes, this work obtains explicit rates of convergence for a broad class of processes in any number of dimensions. These results suggest analogous ones for stationary processes.     

Variance Estimation in Spatial Regression Using a Non-parametric Semivariogram Based on Residuals

Abstract.  The empirical semivariogram of residuals from a regression model with stationary errors may be used to estimate the covariance structure of the underlying process. For prediction (kriging) the bias of the semivariogram estimate induced by using residuals instead of errors has only a minor effect because the bias is small for small lags. However, for estimating the variance of estimated regression coefficients and of predictions, the bias due to using residuals can be quite substantial. Thus we propose a method for reducing this bias. The adjusted empirical semivariogram is then isotonized and made conditionally negative-definite and used to estimate the variance of estimated regression coefficients in a general estimating equations setup. Simulation results for least squares and robust regression show that the proposed method works well in linear models with stationary correlated errors.