LIMIT THEOREMS FOR SUBCRITICAL AGE-DEPENDENT BRANCHING PROCESSES WITH TWO TYPES OF IMMIGRATION

ABSTRACT

For the classical subcritical age-dependent branching process the effect of the following two-type immigration pattern is studied. At a sequence of renewal epochs a random number of immigrants enters the population. Each subpopulation stemming from one of these immigrants or one of the ancestors is revived by new immigrants and their offspring whenever it dies out, possibly after an additional delay period. All individuals have the same lifetime distribution and produce offspring according to the same reproduction law. This is the Bellman-Harris process with immigration at zero and immigration of renewal type (BHPIOR). We prove a strong law of large numbers and a central limit theorem for such processes. Similar conclusions are obtained for their discrete-time counterparts (lifetime per individual equals one), called Galton-Watson processes with immigration at zero and immigration of renewal type (GWPIOR). Our approach is based on the theory of regenerative processes, renewal theory and occupation measures and is quite different from those in earlier related work using analytic tools.     

Modern Statistics for Spatial Point Processes*

Abstract. We summarize and discuss the current state of spatial point process theory and directions for future research, making an analogy with generalized linear models and random effect models, and illustrating the theory with various examples of applications. In particular, we consider Poisson, Gibbs and Cox process models, diagnostic tools and model checking, Markov chain Monte Carlo algorithms, computational methods for likelihood-based inference, and quick non-likelihood approaches to inference.     

On nonlinear models for time series

The paper is a review of nonlinear processes used in time series analysis and presents some new original results about stationary distribution of a nonlinear autoregres-sive process of the first order. The following models are considered: nonlinear autoregessive processes, threshold AR processes, threshold MA processes, bilinear models, auto-regressive models with random parameters including double stochastic models, exponential AR models, generalized threshold models and smooth transition autoregressive models, Some tests for linearity of processes are also presented.     

Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics

Non-Gaussian processes of Ornstein–Uhlenbeck (OU) type offer the possibility of capturing important distributional deviations from Gaussianity and for flexible modelling of dependence structures. This paper develops this potential, drawing on and extending powerful results from probability theory for applications in statistical analysis. Their power is illustrated by a sustained application of OU processes within the context of finance and econometrics. We construct continuous time stochastic volatility models for financial assets where the volatility processes are superpositions of positive OU processes, and we study these models in relation to financial data and theory.     

Basic structure of the asymptotic theory in dynamic nonlineaerco nometric models, part i: consistency and approximation concepts

This is the first of two papers that provide an expository discussion of the basic structure of the asymptotic theory of M-estimators in dynamic nonlinear models and a revlew of the literature. In this paper we discuss consistency,uniform laws of large numbers and develop a new framework for laws of large numbers for dependent and heterogeneous processes, encompassing the theory of stochastically stable as well as near epoch dependent processes. This framework results in simplified catalogues of sufficient conditions for consistency. The second paper, Potscher and Prucha (1990b), deals with asymptotic distribution theory.     

Basic structure of the asymptotic theory in dynamic nonlineaerco nometric models,part i: consistency and approximation concepts

This is the first of two papers that provide an expository discussion of the basic structure of the asymptotic theory of M-estimators in dynamic nonlinear models and a revlew of the literature. In this paper we discuss consistency,uniform laws of large numbers and develop a new framework for laws of large numbers for dependent and heterogeneous processes, encompassing the theory of stochastically stable as well as near epoch dependent processes. This framework results in simplified catalogues of sufficient conditions for consistency. The second paper, Potscher and Prucha (1990b), deals with asymptotic distribution theory.     

Sequential estimation for a family of counting processes in the nuisance parameter case

The paper deals with the problem of sequential estimation for stochastic processes in the presence of a nuisance parameter. Using the approach to estimation through estimating equations, optimum estimating functions based on a random observation time are investigated in some models for processes appearing in reliability systems theory.     

Reliability of semi-Markov missions

We consider a device that is designed to perform missions consisting of a random sequence of phases or stages with random durations. The mission process is described by a Markov renewal process and the system is a complex one consisting of a number of components whose lifetimes depend on the phases of the mission. We discuss models and tools to compute system, mission, and phase reliabilities using Markov renewal theory. A simplified model involving a mission-based system with maximal repair is analyzed first, and the results are then extended to the case where there is no repair using intrinsic aging concepts. Our objective is to focus on computation of system reliability for these two possible extreme cases.     

Bivariate degradation modelling with marginal heterogeneous stochastic processes

In this paper, we consider that the degradation of two performance characteristics of a product can be modelled by stochastic processes and jointly by copula functions, but different stochastic processes govern the behaviour of each performance characteristic (PC) degradation. Different heterogeneous and homogeneous models are presented considering copula functions and different combinations of the most used stochastic processes in degradation analysis as marginal distributions. This is an important aspect to consider because the behaviour of the degradation of each PC may be different in its nature. As the joint distributions of the proposed models result in complex distributions, the estimation of the parameters of interest is performed via MCMC. A simulation study is performed to compare heterogeneous and homogeneous models. In addition, the proposed models are implemented to crack propagation data of two terminals of an electronic device, and some insights are provided about the product reliability under heterogeneous models.     

Nonlinear GCV and quasi-GCV for shrinkage models

The generalized cross-validation (GCV) method has been a popular technique for the selection of tuning parameters for smoothing and penalty, and has been a standard tool to select tuning parameters for shrinkage models in recent works. Its computational ease and robustness compared to the cross-validation method makes it competitive for model selection as well. It is well known that the GCV method performs well for linear estimators, which are linear functions of the response variable, such as ridge estimator. However, it may not perform well for nonlinear estimators since the GCV emphasizes linear characteristics by taking the trace of the projection matrix. This paper aims to explore the GCV for nonlinear estimators and to further extend the results to correlated data in longitudinal studies. We expect that the nonlinear GCV and quasi-GCV developed in this paper will provide similar tools for the selection of tuning parameters in linear penalty models and penalized GEE models.     

Locally optimal designs for binary dose‐response models

In this article, we consider the problem of seeking locally optimal designs for nonlinear dose‐response models with binary outcomes. Applying the theory of Tchebycheff Systems and other algebraic tools, we show that the locally D‐, A‐, and c‐optimal designs for three binary dose‐response models are minimally supported in finite, closed design intervals. The methods to obtain such designs are presented along with examples. The efficiencies of these designs are also discussed. The Canadian Journal of Statistics 46: 336–354; 2018 © 2018 Statistical Society of Canada     

Inference for Observations of Integrated Diffusion Processes

Abstract.  Estimation of parameters in diffusion models is investigated when the observations are integrals over intervals of the process with respect to some weight function. This type of observations can, for example, be obtained when the process is observed after passage through an electronic filter. Another example is provided by the ice-core data on oxygen isotopes used to investigate paleo-temperatures. Finally, such data play a role in connection with the stochastic volatility models of finance. The integrated process is not a Markov process. Therefore, prediction-based estimating functions are applied to estimate parameters in the underlying diffusion model. The estimators are shown to be consistent and asymptotically normal. The theory developed in the paper also applies to integrals of processes other than diffusions. The method is applied to inference based on integrated data from Ornstein–Uhlenbeck processes and from the Cox–Ingersoll–Ross model, for both of which an explicit optimal estimating function is found.     

Estimation of semi-varying coefficient models with nonstationary regressors

We study a semivarying coefficient model where the regressors are generated by the multivariate unit root I(1) processes. The influence of the explanatory vectors on the response variable satisfies the semiparametric partially linear structure with the nonlinear component being functional coefficients. A semiparametric estimation methodology with the first-stage local polynomial smoothing is applied to estimate both the constant coefficients in the linear component and the functional coefficients in the nonlinear component. The asymptotic distribution theory for the proposed semiparametric estimators is established under some mild conditions, from which both the parametric and nonparametric estimators are shown to enjoy the well-known super-consistency property. Furthermore, a simulation study is conducted to investigate the finite sample performance of the developed methodology and results.     

The marked empirical process to test nonlinear time series against a large class of alternatives when the random vectors are nonstationary and absolutely regular

In this paper, we propose a method for testing absolutely regular and possibly nonstationary nonlinear time-series, with application to general AR-ARCH models. Our test statistic is based on a marked empirical process of residuals which is shown to converge to a Gaussian process with respect to the Skohorod topology. This testing procedure was first introduced by Stute [Nonparametric model checks for regression, Ann. Statist. 25 (1997), pp. 613–641] and then widely developed by Ngatchou-Wandji [Weak convergence of some marked empirical processes: Application to testing heteroscedasticity, J. Nonparametr. Stat. 14 (2002), pp. 325–339; Checking nonlinear heteroscedastic time series models, J. Statist. Plann. Inference 133 (2005), pp. 33–68; Local power of a Cramer-von Mises type test for parametric autoregressive models of order one, Compt. Math. Appl. 56(4) (2008), pp. 918–929] under more general conditions. Applications to general AR-ARCH models are given.     

Censored regression with the multistate accelerated sojourn times model

Many disease processes are characterized by two or more successive health states, and it is often of interest and importance to assess state-specific covariate effects. However, with incomplete follow-up data such inference has not been satisfactorily addressed in the literature. We model the logarithm-transformed sojourn time in each state as linearly related to the covariates; however, neither the distributional form of the error term nor the dependence structure of the states needs to be specified. We propose a regression procedure to accommodate incomplete follow-up data. Asymptotic theory is presented, along with some tools for goodness-of-fit diagnostics. Simulation studies show that the proposal is reliable for practical use. We illustrate it by application to a cancer clinical trial.     

Flowgraph Models for Generalized Phase Type Distributions Having Non-Exponential Waiting Times

ABSTRACT. Aalen (1995) introduced phase type distributions based on Markov processes for modelling disease progression in survival analysis. For tractability and to maintain the Markov property, these use exponential waiting times for transitions between states. This article extends the work of Aalen (1995) by generalizing these models to semi-Markov processes with non-exponential waiting times. The generalization allows more realistic modelling of the stages of a disease where the Markov property and exponential waiting times may not hold. Flowgraph models are introduced to provide a closed form for the distributions in situations involving non-exponential waiting times. Flowgraph models work where traditional methods of stochastic processes are intractable. Saddlepoint approximations are used in the analysis. Together, generalized phase type distributions, flowgraphs, and saddlepoint approximations create exciting and innovative prospects for the analysis of survival data.     

Bayesian prediction of waiting times in stochastic models

The authors show how saddlepoint techniques lead to highly accurate approximations for Bayesian predictive densities and cumulative distribution functions in stochastic model settings where the prior is tractable, but not necessarily the likelihood or the predictand distribution. They consider more specifically models involving predictions associated with waiting times for semi‐Markov processes whose distributions are indexed by an unknown parameter θ. Bayesian prediction for such processes when they are not stationary is also addressed and the inverse‐Gaussian based saddlepoint approximation of Wood, Booth & Butler (1993) is shown to accurately deal with the nonstationarity whereas the normal‐based Lugannani & Rice (1980) approximation cannot, Their methods are illustrated by predicting various waiting times associated with M/M/q and M/G/1 queues. They also discuss modifications to the matrix renewal theory needed for computing the moment generating functions that are used in the saddlepoint methods.     

Fast regression surrogates for computer models with time-dependent outputs

The study of physical processes is often aided by computer models or codes. Computer models that simulate such processes are sometimes computationally intensive and therefore not very efficient exploratory tools. In this paper, we address computer models characterized by temporal dynamics and propose new statistical correlation structures aimed at modelling their time dependence. These correlations are embedded in regression models with input-dependent design matrix and input-correlated errors that act as fast statistical surrogates for the computationally intensive dynamical codes. The methods are illustrated with an automotive industry application involving a road load data acquisition computer model.     

A nonparametric test of conditional autoregressive heteroscedasticity for threshold autoregressive models

Threshold autoregressive models are widely used in time‐series applications. When building or using such a model, it is important to know whether conditional heteroscedasticity exists. The authors propose a nonparametric test of this hypothesis. They develop the large‐sample theory of a test of nonlinear conditional heteroscedasticity adapted to nonlinear autoregressive models and study its finite‐sample properties through simulations. They also provide percentage points for carrying out this test, which is found to have very good power overall.     

Non‐stationary Cross‐Covariance Models for Multivariate Processes on a Globe

Abstract. In geophysical and environmental problems, it is common to have multiple variables of interest measured at the same location and time. These multiple variables typically have dependence over space (and/or time). As a consequence, there is a growing interest in developing models for multivariate spatial processes, in particular, the cross‐covariance models. On the other hand, many data sets these days cover a large portion of the Earth such as satellite data, which require valid covariance models on a globe. We present a class of parametric covariance models for multivariate processes on a globe. The covariance models are flexible in capturing non‐stationarity in the data yet computationally feasible and require moderate numbers of parameters. We apply our covariance model to surface temperature and precipitation data from an NCAR climate model output. We compare our model to the multivariate version of the Matérn cross‐covariance function and models based on coregionalization and demonstrate the superior performance of our model in terms of AIC (and/or maximum loglikelihood values) and predictive skill. We also present some challenges in modelling the cross‐covariance structure of the temperature and precipitation data. Based on the fitted results using full data, we give the estimated cross‐correlation structure between the two variables.