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.     

Monitoring Variance Change in Infinite Order Moving Average Processes and Nonstationary Autoregressive Processes

This article considers the sequential monitoring problem of variance change in stationary and non stationary time series. We suggest a CUSUM of squares procedure to detect variance change in infinite order moving average processes, and a residual CUSUM of squares procedure to detect variance change in non stationary autoregressive processes. Moreover, we introduce a bandwidth parameter to improve the monitoring power when change point does not occur at the early stage of monitoring. It is shown that both procedures have the same null distribution. The procedures are illustrated via a simulation study and an investigation of daily Mexico/US exchange rates.     

The efficient estimation of tail probabilities for extremes of moving average processes using conditional simulation

This paper describes a conditional simulation technique which can be used to estimate probabilities associated with the distribution of the maximum of a real-valued process which can be written in the form of a moving average. The class of processes to which the technique applies includes non-stationary and spatial processes, and autoregressive processes. The technique is shown to achieve a considerable variance reduction compared with the obvious simulation-based estimator, particularly for estimating small upper-tail probabilities.     

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.     

Approximate wavelet-based simulation of long memory processes

In this article, we investigate an algorithm for the fast O(N) and approximate simulation of long memory (LM) processes of length N using the discrete wavelet transform. The algorithm generates stationary processes and is based on the notion that we can improve standard wavelet-based simulation schemes by noting that the decorrelation property of wavelet transforms is not perfect for certain LM process. The method involves the simulation of circular autoregressive process of order one. We demonstrate some of the statistical properties of the processes generated, with some focus on four commonly used LM processes. We compare this simulation method with the white noise wavelet simulation scheme of Percival and Walden [Percival, D. and Walden, A., 2000, Wavelet Methods for Time Series Analysis (Cambridge: Cambridge University Press).].     

Compound-Estimator Based Cause-Selecting Control Chart for Monitoring Multistage Processes

Although there exists an increasing interest in monitoring and diagnosing multistage processes through the recent years, this issue has been overlooked to a large extent in cascade processes where the quality characteristics are liable to outliers. The presence of outliers has a debilitating effect on the detect-ability of the traditional cause selecting control charts and thus makes them unreliable. Therefore, the purpose of this article is to provide a robust approach to quality control in multistage processes. It is assumed that the process consists of two stages and the historical data with regard to both dependent quality characteristics contain outliers. A robust fitting procedure based on compound-estimator is employed to build the relationship between the quality variables and a robust monitoring approach is presented. Subsequently, simulation studies are undertaken to assess the performance of the robust scheme by means of the average run length (ARL) criterion. It is shown that the proposed robust procedure can much faster detect diverse types of shift.     

Brownian Bridge and Other Path-dependent Gaussian Processes Vectorial Simulation

The iterative simulation of the Brownian bridge is well known. In this article, we present a vectorial simulation alternative based on Gaussian processes for machine learning regression that is suitable for interpreted programming languages implementations. We extend the vectorial simulation of path-dependent trajectories to other Gaussian processes, namely, sequences of Brownian bridges, geometric Brownian motion, fractional Brownian motion, and Ornstein–Ulenbeck mean reversion process.     

Autoregressive moving-average convolution processes on the unit circle

ARMA convolution models for processes in continuous space (in this case the unit circle) and discrete time are derived as a natural extension of the usual Box-Jenkins models. Both weakly time-stationary and nonstationary processes are considered. Sufficient conditions for the existence of weakly time-stationary AR_cMA_c processes are derived, and the covariance functions for some processes are computed. It is demonstrated that the usual scalar and multivariate ARMA processes can be embedded within the larger class of AR_CMA_c models. A possible application of these models to sea-surface temperature prediction is discussed.     

AN ALGORITHM FOR CHOOSING THE SAMPLING TIMES FOR AN EXPERIMENT ON A CONTINUOUS PROCESS

Saunders & Eccleston (1992) and Saunders, Eccleston & Spessa (1992) developed an approach to the design of factorial experiments on continuous processes that allows for the correlation present in such processes. Their methods concentrated on identifying the order of application of treatments in such experiments, assuming that the spacing between experiments is constant. On a continuous process, there is no necessity to maintain equally spaced sampling times. This paper gives an algorithm for choosing the optimal sampling times for a factorial experiment aimed at estimating a particular parameter or set of parameters. It is shown that in practical situations the optimal sampling times give considerable improvements in the accuracy of the parameter estimates.     

Supervised Classification for a Family of Gaussian Functional Models

Abstract. In the framework of supervised classification (discrimination) for functional data, it is shown that the optimal classification rule can be explicitly obtained for a class of Gaussian processes with ‘triangular’ covariance functions. This explicit knowledge has two practical consequences. First, the consistency of the well‐known nearest neighbours classifier (which is not guaranteed in the problems with functional data) is established for the indicated class of processes. Second, and more important, parametric and non‐parametric plug‐in classifiers can be obtained by estimating the unknown elements in the optimal rule. The performance of these new plug‐in classifiers is checked, with positive results, through a simulation study and a real data example.     

Estimation in long memory time series models

A regression type estimator of the parameter d in fractionally differenced ARMA (p,q) processes is presented. The proposed estimator is shown to be mean square consistent. Its performance is compared with some of the existing estimators via a simulation study.     

A law of large numbers for the scaled age distribution of linear birth-and-death processes

It is shown that certain measure-valued stochastic processes describing the age distribution of particles whose development is controlled by linear critical birth-and-death processes converge in distribution to a deterministic positive bounded measure.     

The role of the drift in I(2) systems

Summary This paper discussed the role of the drift in vector autoregressive processes allowing for integrated components up to order 2. It is shown how the drift can generate linear and quadratic deterministic trends. A two-stage statistical analysis of the system in the presence of quadratic trends is proposed. The analysis of the system allows to define a consistent sequence of tests on the numbers of common components integrated of a given order, called the integration indices of the system. The relevant asymptotic distributions are non-standard, belong to the Limiting Gaussian Functional family and are tabulated by simulation. The proposed procedure can also be consistently combined with other procedures proposed by the author for the cases of a linear trend and of no deterministic trends in the system. Invited paper at the Conference held in Bologna, Italy, 27–28 May 1993, on “Statistical Tests: Methodology and Econometric Applications”. Partial financial support is acknowledged from Italian MURST grants 40% and 60%.     

Interacting neighbour point processes: Some models for clustering

We introduce a class of spatial point processesinteracting neighbour point (INP) processes, where the density of the process can be written by means of local interactions between a point and subsets of its neighbourhood but where the processes may not be Ripley-Kelly Markov processes with respect to this neighbourhood. We show that the processes are iterated Markov processes defined by Hayat and Gubner (1996). Furthermore, we pay special attention to a subclass of interacting neighbour processes, where the density belongs to the exponential family and all neighbours of a point affect it simultaneously. A simulation study is presented to show that some simple processes of this subclass can produce clustered patterns of great variety. Finally, an empirical example is given.     

Graphical Models for Composable Finite Markov Processes

Abstract.  Composable Markov processes were introduced by Schweder (1970) in order to capture the idea that a process can be composed of different components where some of these only depend on a subset of the other components. Here we propose a graphical representation of this kind of dependence which has been called 'local dependence'. It is shown that the graph allows to read off further independencies characterizing the underlying Markov process. Also, some standard methods for inference are adapted to exploit the graphical representation, e.g. for testing local independence.     

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.     

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.     

A method for checking regression models in survival analysis based on the risk score

We propose to perform model check for the Cox and Aalen regression models using martingale residual processes grouped after the risk score. Asymptotic distributions of the grouped martingale residual processes are deduced, so both formal and graphical model check can be performed. The method is validated by stochastic simulation. A data example with patients with primary biliary cirrhosis of the liver is discussed.     

Convergence of compensators of simple point processes

Processes are viewed as random variables with values in the space of càdlàg functions endowed with the J₁ topology. For a sequence of point processes with their minimal filtration, the convergence of their compensators is studied. It is shown that convergence in probability of the processes and of the corresponding conditional distributions of their jump times implies, with some additional hypothesis, the convergence in probability of the compensators. The result is then applied to convergence to a quasi-left-continuous point process with independent increments.     

Test for Parameter Change in Linear Processes Based on Whittle's Estimator

In this article, we develop a cusum test for testing for parameter changes in linear processes based on Whittle's estimator. It is shown that under regularity conditions, the test statistic converges to the sup of a Brownian bridge. The result is particularly useful in handling the change point test in stationary ARMA processes. A simulation result is provided for illustration.