Hypotheses Testing: Poisson Versus Self-exciting

Abstract.  We consider the problem of hypotheses testing with the basic simple hypothesis: observed sequence of points corresponds to the stationary Poisson process with known intensity. The alternatives are stationary self-exciting point processes. We consider one-sided parametric and one-sided non-parametric composite alternatives and construct locally asymptotically uniformly most powerful tests. The results of numerical simulations of the tests are presented.     

Estimation of the Intensity Parameter of the Germ‐Grain Quermass‐Interaction Model when the Number of Germs is not Observed

The Quermass‐interaction model allows to generalize the classical germ‐grain Boolean model in adding a morphological interaction between the grains. It enables to model random structures with specific morphologies, which are unlikely to be generated from a Boolean model. The Quermass‐interaction model depends in particular on an intensity parameter, which is impossible to estimate from classical likelihood or pseudo‐likelihood approaches because the number of points is not observable from a germ‐grain set. In this paper, we present a procedure based on the Takacs–Fiksel method, which is able to estimate all parameters of the Quermass‐interaction model, including the intensity. An intensive simulation study is conducted to assess the efficiency of the procedure and to provide practical recommendations. It also illustrates that the estimation of the intensity parameter is crucial in order to identify the model. The Quermass‐interaction model is finally fitted by our method to P. Diggle's heather data set.     

On hypothesis tests in misspecified change-point problems for a Poisson process

Consider an inhomogeneous Poisson process X on [0, T] whose unk-nown intensity function “switches” from a lower function g_* to an upper function h_* at some unknown point ?_* that has to be identified. We consider two known continuous functions g and h such that g_*(t) ? g(t) < h(t) ? h_*(t) for 0 ? t ? T. We describe the behavior of the generalized likelihood ratio and Wald’s tests constructed on the basis of a misspecified model in the asymptotics of large samples. The power functions are studied under local alternatives and compared numerically with help of simulations. We also show the following robustness result: the Type I error rate is preserved even though a misspecified model is used to construct tests.     

Goodness of Fit Tests for Multivariate Counting Process Models with Applications

In this paper, we develop some distribution-free tests for checking the adequacy of the parametric forms of the intensity processes of a multivariate counting process model. The proposed tests, based in Khmaladze's transformations, are derived from the transforms of weighted aggregated martingale residual processes. The transformed processes converge weakly to independent Gaussian martingales under the assumed model. The distribution-free tests, such as Kolmogorov–Smirnov and Cramer–von Mises type tests, are appropriately defined to account for deviations in each of the transformed aggregated martingale residual processes. Consistency of the tests are discussed. The tests are applicable to multiplicative intensity models such as a competing risks model as well as to non-multiplicative intensity models such as a constant relative or excess mortality model. A small simulation study is conducted and an illustration to a real data example is provided.     

STUR过程下危机“传染性”计量检验的有效性研究

文章首先介绍了危机“传染性”计量检验的两种分析框架:基于潜因子的分析模型和结构回归分析模型,同时给出了一些具体的检验方法,主要包括:Forbes和Rigobon的可调节的相关检验和多元“邹”检验,Favero和Giavazzi的“异常点”检验,Pesaran和Pick的阀值检验等。为检验这些方法的有效性,采用随机单位根过程作为经济数据的变迁过程,发现“异常点”检验和阀值检验可以更好地捕捉到危机传染的时间特征。     

On Geometry and Scale of a Stochastic Chemostat

A chemostat is a fixed volume bioreactor in which micro–organisms are grown in a continuously renewed liquid medium. We propose a stochastic model for the evolution of the concentrations in the single species and single substrate case. It is obtained as a diffusion approximation of a pure jump Markov process, whose increments are comparable in mean with the deterministic model. A specific time scale, related to the noise intensity, is considered for each source of variation. The geometric structure of the problem, usable by identification procedures, is preserved both in the drift and diffusion term. We study the properties of this model by numerical experiments.     

Basket CDS pricing with default intensities using a regime-switching shot-noise model

In this paper, we employ an intensity-based credit risk model with regime-switching to study the valuation of basket CDS in a homogeneous portfolio. We assume that the default intensities are described by some dependent regime-switching shot-noise processes and the individual jumps of the intensity are driven by a common factor. By using the conditional Laplace transform of the regime-switching shot-noise process, we obtain the closed form results for pricing the fair spreads of the basket CDS. We present some numerical examples to illustrate the effect of the model parameters on the fair spreads.     

Tests for the Proportional Intensity Assumption Based on the Score Process

Proportional intensity models are widely used for describing the relationship between the intensity of a counting process and associated covariates. A basic assumption in this model is the proportionality, that each covariate has a multiplicative effect on the intensity. We present and study tests for this assumption based on a score process which is equivalent to cumulative sums of the Schoenfeld residuals. Tests within principle power against any type of departure from proportionality can be constructed based on this score process. Among the tests studied, in particular an Anderson-Darling type test turns out to be very useful by having good power properties against general alternatives. A simulation study comparing various tests for proportionality indicates that this test seems to be a good choice for an omnibus test for proportionality.     

Estimation of the intensity function of an inhomogeneous Poisson process with a change‐point

Recent work on point processes includes studying posterior convergence rates of estimating a continuous intensity function. In this article, convergence rates for estimating the intensity function and change‐point are derived for the more general case of a piecewise continuous intensity function. We study the problem of estimating the intensity function of an inhomogeneous Poisson process with a change‐point using non‐parametric Bayesian methods. An Markov Chain Monte Carlo (MCMC) algorithm is proposed to obtain estimates of the intensity function and the change‐point which is illustrated using simulation studies and applications. The Canadian Journal of Statistics 47: 604–618; 2019 © 2019 Statistical Society of Canada     

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.     

Structured Spatio-Temporal Shot-Noise Cox Point Process Models, with a View to Modelling Forest Fires

Abstract.  Spatio-temporal Cox point process models with a multiplicative structure for the driving random intensity, incorporating covariate information into temporal and spatial components, and with a residual term modelled by a shot-noise process, are considered. Such models are flexible and tractable for statistical analysis, using spatio-temporal versions of intensity and inhomogeneous K -functions, quick estimation procedures based on composite likelihoods and minimum contrast estimation, and easy simulation techniques. These advantages are demonstrated in connection with the analysis of a relatively large data set consisting of 2796 days and 5834 spatial locations of fires. The model is compared with a spatio-temporal log-Gaussian Cox point process model, and likelihood-based methods are discussed to some extent.     

Tests for Monotonic Intensity in a Poisson Process

Likelihood ratio tests about the intensity function are obtained by confining the estimated intensity function of a Poisson process to a sample-dependent, left-continuous step function class. These tests have relatively simple test statistics and their distributions are stochastically maximized when the process is homogeneous.     

THE SIGN TEST FOR STOCHASTIC PROCESSES

It is shown that the sign test may be applied to the residuals from the use of model fitting procedures, such as conditional least squares, to make inferences on the predictable part of a stochastic process. Minimal assumptions on the distribution of the process, apart from those already required for the model fitting procedure, are needed. The results are illustrated with an application to first order autoregressive processes.     

Resampling Methods for Testing a Semiparametric Random Censorship Model

This paper presents a goodness-of-fit test for a semiparametric random censorship model proposed by Dikta (1998 ). The test statistic is derived from a model-based process which is asymptotically Gaussian. In addition to test consistency, the proposed test can detect local alternatives distinct n ^-1/2 from the null hypothesis. Due to the intractability of the asymptotic null distribution of the test statistic, we turn to two resampling approximations. We first use the well-known bootstrap method to approximate critical values of the test. We then introduce a so-called random symmetrization method for carrying out the test. Both methods perform very well with a sample of moderate size. A simulation study shows that the latter possesses better empirical powers and sizes for small samples.     

ASYMPTOTIC TESTS FOR THE CONTINUOUS TIME MARKOV BRANCHING PROCESS WITH IMMIGRATION

This paper presents limit distributions for the modified score and the likelihood-ratio (LR) statistic for testing a composite hypothesis involving the split intensity and mean of the offspring distribution of the supercritical continuous time Markov branching process allowing immigration (CBPI). The immigration intensity and mean are treated as nuisance parameters.     

Non‐parametric Bayesian Intensity Model: Exploring Time‐to‐Event Data on Two Time Scales

Time‐to‐event data have been extensively studied in many areas. Although multiple time scales are often observed, commonly used methods are based on a single time scale. Analysing time‐to‐event data on two time scales can offer a more extensive insight into the phenomenon. We introduce a non‐parametric Bayesian intensity model to analyse two‐dimensional point process on Lexis diagrams. After a simple discretization of the two‐dimensional process, we model the intensity by a one‐dimensional piecewise constant hazard functions parametrized by the change points and corresponding hazard levels. Our prior distribution incorporates a built‐in smoothing feature in two dimensions. We implement posterior simulation using the reversible jump Metropolis–Hastings algorithm and demonstrate the applicability of the method using both simulated and empirical survival data. Our approach outperforms commonly applied models by borrowing strength in two dimensions.     

Semiparametric regression for the mean and rate functions of recurrent events

The counting process with the Cox-type intensity function has been commonly used to analyse recurrent event data. This model essentially assumes that the underlying counting process is a time-transformed Poisson process and that the covariates have multiplicative effects on the mean and rate function of the counting process. Recently, Pepe and Cai, and Lawless and co-workers have proposed semiparametric procedures for making inferences about the mean and rate function of the counting process without the Poisson-type assumption. In this paper, we provide a rigorous justification of such robust procedures through modern empirical process theory. Furthermore, we present an approach to constructing simultaneous confidence bands for the mean function and describe a class of graphical and numerical techniques for checking the adequacy of the fitted mean–rate model. The advantages of the robust procedures are demonstrated through simulation studies. An illustration with multiple-infection data taken from a clinical study on chronic granulomatous disease is also provided.     

Statistical inference in nonstationary markov models with embedded explanatory variables

In this paper, we develop an operational nonstationary Markov process model for use with macro aggregate frequency data. Independent, time-variant factors assumed to affect the process of interest are embedded in the model. Transition probabilities are estimated indirectly from the coefficients on the embedded variables. We previously concluded that either the Marquardt or the simplex, derivative-free nonlinear programming algorithm could be used to estimate such a model. Here, we propose a test for parameter stationarity. By means of designed simulation experiments for the two-state model, we find that our test has acceptable Type I error probabilities, and that power rises with the degree of departure from the null hypothesis. Both validity and power performance can be improved by longer time records of data and a greater number of entities observed.     

Hierarchical Bayesian modeling of marked non-homogeneous Poisson processes with finite mixtures and inclusion of covariate information

We investigate marked non-homogeneous Poisson processes using finite mixtures of bivariate normal components to model the spatial intensity function. We employ a Bayesian hierarchical framework for estimation of the parameters in the model, and propose an approach for including covariate information in this context. The methodology is exemplified through an application involving modeling of and inference for tornado occurrences.