Testing of two sample proportional intensity assumption for non-homogeneous Poisson processes

For two independent non-homogeneous Poisson processes with unknown intensities we propose a test for testing the hypothesis that the ratio of the intensities is constant versus it is increasing on (0,t]. The existing test procedures for testing such relative trends are based on conditioning on the number of failures observed in (0,t] from the two processes. Our test is unconditional and is based on the original time truncated data which enables us to have meaningful asymptotics. We obtain the asymptotic null distribution (as t becomes large) of the proposed test statistic and show that the proposed test is consistent against several large classes of alternatives. It was observed by Park and Kim (IEEE. Trans. Rehab. 40 (1), 1992, 107–111) that it is difficult to distinguish between the power-law and log-linear processes for certain parameter values. We show that our test is consistent for such alternatives also.     

Measurement and analysis of demographic processes: the characteristics of generation

Since it would take too long (100 years) to ascertain all demographic data about a given age group, i.e., all those born in a given year, these data are determined hypothetically by measuring the various characteristics of persons of all ages in a given time period (1-2 years). Also needed is an indicator of the population as a whole; cumulative coefficients are used for this purpose. One of these is the overall coefficient of births, meaning the number of children a women would have over her whole period of fertility if she had the precise number of children at each period in her life as other women of that age. An analogous indicator is used for measuring mortality--the average life expectancy of a person at each stage in his life. The crude coefficient of population reproduction represents the number of girls to which each woman will give birth between the ages of 15 and 50. This has to be corrected by the number of those who will not live to reproductive age. The result is the corrected coefficient of reproduction of the female population. This coefficient is often thought to reflect the population's growth prospects; if less than unity, therefore, the population will not reproduce itself. This is an incorrect interpretation. The impact of immigration and emigration on the population must also be incorportated. In addition to the above hypothetical indicators, we must also develop real population indicators. Techniques must also be employed to evaluate the reliability of these demographic indicators.     

Modelling spatial intensity for replicated inhomogeneous point patterns in brain imaging

Summary.  Pharmacological experiments in brain microscopy study patterns of cellular activ- ation in response to psychotropic drugs for connected neuroanatomic regions. A typical ex- perimental design produces replicated point patterns having highly complex spatial variability. Modelling this variability hierarchically can enhance the inference for comparing treatments. We propose a semiparametric formulation that combines the robustness of a nonparametric kernel method with the efficiency of likelihood-based parameter estimation. In the convenient framework of a generalized linear mixed model, we decompose pattern variation by kriging the intensities of a hierarchically heterogeneous spatial point process. This approximation entails discretizing the inhomogeneous Poisson likelihood by Voronoi tiling of augmented point patterns. The resulting intensity-weighted log-linear model accommodates spatial smoothing through a reduced rank penalized linear spline. To correct for anatomic distortion between subjects, we interpolate point locations via an isomorphic mapping so that smoothing occurs relative to common neuroanatomical atlas co-ordinates. We propose a criterion for choosing the degree and spatial locale of smoothing based on truncating the ordered set of smoothing covariates to minimize residual extra-dispersion. Additional spatial covariates, experimental design factors, hierarchical random effects and intensity functions are readily accommodated in the linear predictor, enabling comprehensive analyses of the salient properties underlying replicated point patterns. We illustrate our method through application to data from a novel study of drug effects on neuronal activation patterns in the brain of rats.     

On the superposition of overlapping Poisson processes and nonparametric estimation of their intensity function

It is shown that, when measuring time in the Total Time on Test scale, the superposition of overlapping realizations of a nonhomogeneous Poisson process is also a Poisson process and is sufficient for inferential purposes. Hence, many nonparametric procedures which are available when only one realization is observed can be easily extended for the overlapping realizations setup. These include, for instance, the constrained maximum likelihood estimator of a monotonic intensity and bootstrap confidence bands based on Kernel estimates of the intensity. The kernel estimate proposed here performs the smoothing in the Total Time on Test scale and it is shown to behave approximately as a usual kernel estimate but with a variable bandwidth which is inversely proportional to the number of realizations at-risk. Likewise, bootstrap samples can be obtained from the single realization of the superimposed process. The methods are illustrated using a real data set consisting of the failure histories of 40 electrical power transformers.     

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.     

Forecasting using locally stationary wavelet processes

Locally stationary wavelet (LSW) processes, built on non-decimated wavelets, can be used to analyse and forecast non-stationary time series. They have been proved useful in the analysis of financial data. In this paper, we first carry out a sensitivity analysis, then propose some practical guidelines for choosing the wavelet bases for these processes. The existing forecasting algorithm is found to be vulnerable to outliers, and a new algorithm is proposed to overcome the weakness. The new algorithm is shown to be stable and outperforms the existing algorithm when applied to real financial data. The volatility forecasting ability of LSW modelling based on our new algorithm is then discussed and shown to be competitive with traditional GARCH models.     

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.     

Estimation of spatial processes using local scoring rules

We display pseudo-likelihood as a special case of a general estimation technique based on proper scoring rules. Such a rule supplies an unbiased estimating equation for any statistical model, and this can be extended to allow for missing data. When the scoring rule has a simple local structure, as in many spatial models, the need to compute problematic normalising constants is avoided. We illustrate the approach through an analysis of data on disease in bell pepper plants.     

Nonparametric Bayesian modelling using skewed Dirichlet processes

We introduce a new class of discrete random probability measures that extend the definition of Dirichlet process (DP) by explicitly incorporating skewness. The asymmetry is controlled by a single parameter in such a way that symmetric DPs are obtained as a special case of the general construction. We review the main properties of skewed DPs and develop appropriate Polya urn schemes. We illustrate the modelling in the context of linear regression models of the capital asset pricing model (CAPM) type, where assessing symmetry for the error distribution is important to check validity of the model.     

A two-phase method for controlling Erlang-failure processes with high reliability

Monitoring a failure process and measuring its performance are important issues for complex nonrepairable and repairable systems. For a highly reliable process, traditional methods for reliability monitoring and performance measuring become inapplicable. This paper proposes a new two-phase controlling method for monitoring and measuring an Erlang-failure process (EFP). In the first-phase controlling method, a control chart is used to monitor the EFP condition. When special causes of variation have been removed from the EFP and all of the failure times plotted on the control chart lie within the control limits, the EFP is considered to be in control. However, the in-control EFP still likely carries out bad or out-of-lifetime-specification conditions. Thus, its lifetime-specification limit is taken into consideration as the second-phase controlling method for measuring the in-control EFP performance. We propose a lifetime-capability index. Its value has a one-to-one corresponding relationship with the lifetime-conforming rate, which indicates the lifetime performance of this EFP. Without collecting additional data efforts, in-control data gathered from the control chart in the first phase is employed to estimate the lifetime-capability index. To realize main lifetime-capability of the EFP impacting on downstream customers, the lower confidence bound of the estimate of the lifetime-capability index, capturing its minimum lifetime capability, is considered. The advantage of this two-phase method for controlling the failure processes can motivate the manufacturer to develop a reliability-monitoring technique, establish an adequate reliability improvement program and implement an appropriate analysis to ensure its lifetime performance meeting the customers requirement.     

Space-time estimation of a particle system model

Let X be a discrete time contact process (CP) on ?², as defined by Durrett and Levin (1994, Stochastic spatial models: a user's guide to ecological applications. Philosophical Transactions of the Royal Society of London Series B, 343, 329–350). We study the estimation of the model based on space-time evolution of X, that is, T + 1 successive observations of X on a finite subset S of sites. We consider the maximum marginal pseudo-likelihood (MPL) estimator and show that, when T→∞, this estimator is consistent and asymptotically normal for a non-vanishing supercritical CP. Numerical studies confirm these theoretical ones.     

Bayesian deconvolution of oil well test data using Gaussian processes

We use Bayesian methods to infer an unobserved function that is convolved with a known kernel. Our method is based on the assumption that the function of interest is a Gaussian process and, assuming a particular correlation structure, the resulting convolution is also a Gaussian process. This fact is used to obtain inferences regarding the unobserved process, effectively providing a deconvolution method. We apply the methodology to the problem of estimating the parameters of an oil reservoir from well-test pressure data. Here, the unknown process describes the structure of the well. Applications to data from Mexican oil wells show very accurate results.     

Modelling cell generation times by using the tempered stable distribution

Summary.  We show that the family of tempered stable distributions has considerable potential for modelling cell generation time data. Several real examples illustrate how these distributions can improve on currently assumed models, including the gamma and inverse Gaussian distributions which arise as special cases. Our applications concentrate on the generation times of oligodendrocyte progenitor cells and the yeast Saccharomyces cerevisiae . Numerical inversion of the Laplace transform of the probability density function provides fast and accurate approximations to the tempered stable density, for which no closed form generally exists. We also show how the asymptotic population growth rate is easily calculated under a tempered stable model.     

Measuring the association of stationary point processes using spectral analysis techniques

In this work we focus on relationships between stationary point process using spectral analysis techniques. The evaluation of these relationships is accomplished with the help of the product ratio of association (PRA), which is based on the cumulant densities of the point processes. The estimation procedure is obtained by smoothing the periodogram statistic, a function of the frequency domain. It is proved that the asymptotic distribution of the square root of the estimated PRA is Normal with a constant variance. Statistical tests for hypotheses concerning the independence of two point processes and the characterization of a Poisson process are proposed. Furthermore, approximate 95% pointwise confidence interval can be obtained for the estimated PRA. These results can be applied on stochastic systems involving as input and output stationary point processes. An illustrative example from the framework of neurophysiology is presented.     

Modelling Poisson marked point processes using bivariate mixture transition distributions

A class of bivariate continuous-discrete distributions is proposed to fit Poisson dynamic models in a single unified framework via bivariate mixture transition distributions (BMTDs). Potential advantages of this class over the current models include its ability to capture stretches, bursts and nonlinear patterns characterized by Internet network traffic, high-frequency financial data and many others. It models the inter-arrival times and the number of arrivals (marks) in a single unified model which benefits from the dependence structure of the data. The continuous marginal distributions of this class include as special cases the exponential, gamma, Weibull and Rayleigh distributions (for the inter-arrival times), whereas the discrete marginal distributions are geometric and negative binomial. The conditional distributions are Poisson and Erlang. Maximum-likelihood estimation is discussed and parameter estimates are obtained using an expectation–maximization algorithm, while the standard errors are estimated using the missing information principle. It is shown via real data examples that the proposed BMTD models appear to capture data features better than other competing models.     

Obtaining prediction intervals for FARIMA processes using the sieve bootstrap

The sieve bootstrap (SB) prediction intervals for invertible autoregressive moving average (ARMA) processes are constructed using resamples of residuals obtained by fitting a finite degree autoregressive approximation to the time series. The advantage of this approach is that it does not require the knowledge of the orders, p and q, associated with the ARMA(p, q) model. Up until recently, the application of this method has been limited to ARMA processes whose autoregressive polynomials do not have fractional unit roots. The authors, in a 2012 publication, introduced a version of the SB suitable for fractionally integrated autoregressive moving average (FARIMA (p,d,q)) processes with 0<d<0.5 and established its asymptotic validity. Herein, we study the finite sample properties this new method and compare its performance against an older method introduced by Bisaglia and Grigoletto in 2001. The sieve bootstrap (SB) method is a numerically simpler alternative to the older method which requires the estimation of p, d, and q at every bootstrap step. Monte-Carlo simulation studies, carried out under the assumption of normal, mixture of normals, and exponential distributions for the innovations, show near nominal coverages for short-term and long-term SB prediction intervals under most situations. In addition, the sieve bootstrap method yields better coverage and narrower intervals compared to the Bisaglia–Grigoletto method in some situations, especially when the error distribution is a mixture of normals.