Guaranteed nonlinear parameter estimation in knowledge-based models

Knowledge-based models are ubiquitous in pure and applied sciences. They often involve unknown parameters to be estimated from experimental data. This is usually much more difficult than for black-box models, only intended to mimic a given input–output behavior. The output of knowledge-based models is almost always nonlinear in their parameters, so that linear least squares cannot be used, and analytical solutions for the model equations are seldom available. Moreover, since the parameters have some physical meaning, it is not enough to find some numerical values of these quantities that are such that the model fits the data reasonably well. One would like, for instance, to make sure that the parameters to be estimated are identifiable. If this is not the case, all equivalent solutions should be provided. The uncertainty in the parameters resulting from the measurement noise and approximate nature of the model should also be characterized. This paper describes how guaranteed methods based on interval analysis may contribute to these tasks. Examples in linear and nonlinear compartmental modeling, widely used in biology, are provided.     

Robust discrimination under a hierarchy on the scatter matrices

Under normality, Flury and Schmid [Quadratic discriminant functions with constraints on the covariances matrices: some asymptotic results, J. Multivariate Anal. 40 (1992) 244-261] investigated the asymptotic properties of the quadratic discrimination procedure under hierarchical models for the scatter matrices, that is: (i) arbitrary scatter matrices, (ii) common principal components, (iii) proportional scatter matrices and (iv) identical matrices. In this paper, we study the properties of robust quadratic discrimination rules based on robust estimates of the involved parameters. Our analysis is based on the partial influence functions of the functionals related to these parameters and allows to derive the asymptotic variances of the estimated coefficients under models (i)-(iv). From them, we conclude that the asymptotic variances verify the same order relations as those obtained by Flury and Schmid [Quadratic discriminant functions with constraints on the covariances matrices: some asymptotic results, J. Multivariate Anal. 40 (1992) 244-261] for the classical estimators. We also perform a Monte Carlo study for different sample sizes and different hierarchies which shows the advantage of using robust procedures over classical ones, when anomalous data are present. It also confirms that better rates of misclassification can be achieved if a more parsimonious model among all the correct ones is used instead of the standard quadratic discrimination.     

Tail dependence of random variables from ARCH and heavy-tailed bilinear models

Discussed in this paper is the dependent structure in the tails of distributions of random variables from some heavy-tailed stationary nonlinear time series. One class of models discussed is the first-order autoregressive conditional heteroscedastic (ARCH) process introduced by Engle (1982). The other class is the simple first-order bilinear models driven by heavy-tailed innovations. We give some explicit formulas for the asymptotic values of conditional probabilities used for measuring the tail dependence between two random variables from these models. Our results have significant meanings in finance.     

Innovations algorithm asymptotics for periodically stationary time series with heavy tails

The innovations algorithm can be used to obtain parameter estimates for periodically stationary time series models. In this paper we compute the asymptotic distribution for these estimates in the case where the underlying noise sequence has infinite fourth moment but finite second moment. In this case, the sample covariances on which the innovations algorithm are based are known to be asymptotically stable. The asymptotic results developed here are useful to determine which model parameters are significant. In the process, we also compute the asymptotic distributions of least squares estimates of parameters in an autoregressive model.     

Likelihood-based kernel estimation in semiparametric errors-in-covariables models with validation data

We present methods to handle error-in-variables models. Kernel-based likelihood score estimating equation methods are developed for estimating conditional density parameters. In particular, a semiparametric likelihood method is proposed for sufficiently using the information in the data. The asymptotic distribution theory is derived. Small sample simulations and a real data set are used to illustrate the proposed estimation methods.     

Asymptotic Properties of Quasi-Maximum Likelihood Estimators for ARMA Models with Time-Dependent Coefficients

For about thirty years, time series models with time-dependent coefficients have sometimes been considered as an alternative to models with constant coefficients or non-linear models. Analysis based on models with time-dependent models has long suffered from the absence of an asymptotic theory except in very special cases. The purpose of this paper is to provide such a theory without using a locally stationary spectral representation and time rescaling. We consider autoregressive-moving average (ARMA) models with time-dependent coefficients and a heteroscedastic innovation process. The coefficients and the innovation variance are deterministic functions of time which depend on a finite number of parameters. These parameters are estimated by maximising the Gaussian likelihood function. Deriving conditions for consistency and asymptotic normality and obtaining the asymptotic covariance matrix are done using some assumptions on the functions of time in order to attenuate non-stationarity, mild assumptions for the distribution of the innovations, and also a kind of mixing condition. Theorems from the theory of martingales and mixtingales are used. Some simulation results are given and both theoretical and practical examples are treated. Received 2004; Final version 23 December 2004     

Numerical investigation of bifurcations of equilibria and Hopf bifurcations in disease transmission models

One of the general SIRS disease transmission model is considered under the assumptions that the size of the population varies, the incidence rate is nonlinear, and the recovered (removed) class may also be directly reinfected. A combination of analytical and numerical techniques is used to show that (for some parameters) the bifurcations of equilibria can occur and also asymptotically orbitally stable periodic solutions with asymptotic phase can arise through Hopf bifurcations. The investigation is based on computer simulation of bifurcation manifolds in the parameter space. Hopf bifurcations are investigated on the base of center manifold theory by the computation of bifurcation parameters and the approximation of Hopf-bifurcating cycles by bifurcation formulas. This method finds the limit cycle to a good approximation and also its stability. For computer simulations the necessary computer oriented algorithms were developed and encoded by C++. Some results of computer simulations are presented and numerical evidence of existence of bifurcations of equilibria and Hopf bifurcations for the considered model is provided.     

Asymptotic Theory for Relative-Risk Models with Missing Time-Dependent Covariates

Relative-risk models are often used to characterize the relationship between survival time and time-dependent covariates. When the covariates are observed, the estimation and asymptotic theory for parameters of interest are available; challenges remain when missingness occurs. A popular approach at hand is to jointly model survival data and longitudinal data. This seems efficient, in making use of more information, but the rigorous theoretical studies have long been ignored. For both additive risk models and relative-risk models, we consider the missing data nonignorable. Under general regularity conditions, we prove asymptotic normality for the nonparametric maximum likelihood estimators.     

Estimating Functions for Nonlinear Time Series Models

This paper discusses the problem of estimation for two classes of nonlinear models, namely random coefficient autoregressive (RCA) and autoregressive conditional heteroskedasticity (ARCH) models. For the RCA model, first assuming that the nuisance parameters are known we construct an estimator for parameters of interest based on Godambe's asymptotically optimal estimating function. Then, using the conditional least squares (CLS) estimator given by Tjøstheim (1986, Stochastic Process. Appl., 21, 251–273) and classical moment estimators for the nuisance parameters, we propose an estimated version of this estimator. These results are extended to the case of vector parameter. Next, we turn to discuss the problem of estimating the ARCH model with unknown parameter vector. We construct an estimator for parameters of interest based on Godambe's optimal estimator allowing that a part of the estimator depends on unknown parameters. Then, substituting the CLS estimators for the unknown parameters, the estimated version is proposed. Comparisons between the CLS and estimated optimal estimator of the RCA model and between the CLS and estimated version of the ARCH model are given via simulation studies.     

Development of the collocations and least squares method

We propose and implement new, more general versions of the method of collocations and least squares (the CLS method) and, for a system of linear algebraic equations, an orthogonal method for accelerating the convergence of an iterative solution. The use of the latter method and the proper choice of values of control parameters, based on the results of investigating the dependence of the properties of the CLS method on these parameters, as well as some other improvements of the CLS method suggested in this paper, allow one to solve numerically problems for Navier-Stokes equations in a reasonable time using a single-processor computer even for grids as large as 1280 × 1280. In this case, the total number of unknown variables is ～ 25 · 10⁶. The numerical results for the problem of the lid-driven cavity flow of a viscous fluid are in good agreement with known results of other authors, including those obtained by means of schemes of higher approximation order with a small artificial viscosity. This and some other facts prove that the new versions of the CLS method make it possible to obtain an approximate solution with high accuracy.     

强相依数据的函数系数部分线性模型的估计

本文讨论在数据是强相依的情况下函数系数部分线性模型的估计．首先,采用局部线性方法,给出该模型函数项函数的估计;然后,使用两阶段方法给出系数函数的估计．并且讨论了函数项函数估计的渐近正态性,以及系数函数估计的弱相合性和渐近正态性.模拟研究显示,这些估计是较为理想的．     

On the parametrization of autoregressive models by partial autocorrelations

One of the difficulties that arise in the statistical analysis of autoregressive schemes is the very complex nature of the domain of the regression parameters. In the present paper we study an alternative parametrization of autoregressive models of finite order, namely the parametrization by the partial autocorrelations. These are shown to vary freely from −1 to +1 and to be in a one-to-one, continuously differentiable correspondence with the regression parameters. Properties of the asymptotic normal distribution of the maximum likelihood estimates are discussed, and we present a new deduction of Quenouille's result on the asymptotic independence of some of the estimated partial autocorrelations.     

Geometric approach to global asymptotic stability for the SEIRS models in epidemiology

In this paper, we present a more general criterion for the global asymptotic stability of equilibria for nonlinear autonomous differential equations based on the geometric criterion developed by Li and Muldowney. By applying this criterion, we obtain some results for the global asymptotic stability of SEIRS models with constant recruitment and varying total population size. Based on these results, we give a complete affirmative answer to Liu–Hethcote–Levin conjecture. Furthermore, an affirmative answer to Li–Graef–Wang–Karsai’s problem for SEIR model with permanent immunity and varying total population size is given.     

关于两总体均值差的一种半参数假设检验方法

近年来, 已有一些在半参数密度函数比模型下建立半参数统计分析方法的报道, 这些方法往往比参数方法稳健, 比非参数方法有效. 在本文里, 我们提出一种半参数的假设检验方法用于对两总体均值差进行假设检验. 该方法主要建立在对两总体均值差进行半参数估计的基础上. 我们报告了一些理论和统计模拟的结果, 得出该方法在数据符合正态性假设时, 比常用的参数和非参数方法略好; 而在数据不符合正态性假设时, 它的优势就非常明显. 我们还将提出的方法用到了两组真实数据的分析上.     

Transitions in a logistic population model incorporating an Allee effect

In some species, the population may decline to zero; that is, the species becomes extinct if the population falls below a given threshold. This phenomenon is well known as an Allee effect. In most Allee models, the model parameters are constants, and the population tends either to a nonzero limiting state (survival) or to zero (extinction). However, when environmental changes occur, these parameters may be slowly varying functions of time. Then, application of multitiming techniques allows us to construct approximations to the evolving population in cases where the population survives to a slowly varying surviving state and those where the population declines to zero. Here, we investigate the solution of a logistic population model exhibiting an Allee effect, when the carrying capacity and the limiting density interchange roles, via a transition point. We combine multiscaling analysis with local asymptotic analysis at the transition point to obtain an overall expression for the evolution of the population. We show that this shows excellent agreement with the results of numerical computations. Copyright © 2015 John Wiley & Sons, Ltd.     

Exponential Mixture Models with Long-Term Survivors and Covariates

Suppose a population contains individuals who may be subject to failure with exponentially distributed failure times, or else are "immune" to failure. We do not know which individuals are immune but we can infer their presence in a data set if many of the largest failure times are censored. We also have explanatory vectors containing covariate information on each individual. Models for data with such immune or "cured" individuals are of great interest in medical and criminological statistics, for example. In this paper we provide sufficient conditions for the existence, consistency, and asymptotic normality of maximum likelihood estimators for the parameters in a useful parameterization of these models. The theory is then applied to derive the asymptotic properties of the likelihood ratio test for a difference between immune proportions in a "one-way" classification. A procedure for testing the "boundary" hypothesis, that there are in fact no immunes present in data with a one-way classification, is also discussed.     

Stochastic modeling of carcinogenesis by state space models: a new approach

In this paper, we have developed some state space models for carcinogenesis involving multievent models and multiple pathways models. In these state space models, the stochastic system models are stochastic models of carcinogenesis expressed in terms of stochastic differential equations, whereas the observation models are statistical models based on the observed number of detectable preneoplastic lesions per individual over time and the observed number of detectable cancer tumors per individual over time. In this paper, we have applied some of the theories to some animal papillomas data from some initiation-promotion experiments on skin cancer in mice to estimate some unknown parameters. For this data set we have obtained excellent fit by a model with three piece-wise intervals.     

On the Asymptotic Stability of the Intrinsic and Fractional Bayes Factors for Testing Some Diffusion Models

Random processes, from which a single sample path data are available on a fine time scale, abound in many areas including finance and genetics. An effective way to model such data is to consider a suitable continuous-time-scale analog, X _t say, for the underlying process. We consider three diffusion models for the process X _t and address model selection under improper priors. Specifically, fractional and intrinsic Bayes factors (FBF and IBF) for model selection are considered. Here, we focus on the asymptotic stability of the IBF's and FBF's for comparing these models. Specifically, we propose to employ certain novel transformations of the data in order to ensure the asymptotic stability of the IBF's. While we use different transformations for pairwise comparisons of the models, we also show that a single common transformation can be used when simultaneously comparing all three models. We then demonstrate that, when FBF's are used to compare these models, we may have to employ different, model-specific training fractions in order to achieve asymptotic stability of the FBF's.     

Asymptotics for testing hypothesis in some multivariate variance components model under non-normality

We consider the problem of deriving the asymptotic distribution of the three commonly used multivariate test statistics, namely likelihood ratio, Lawley-Hotelling and Bartlett-Nanda-Pillai statistics, for testing hypotheses on the various effects (main, nested or interaction) in multivariate mixed models. We derive the distributions of these statistics, both in the null as well as non-null cases, as the number of levels of one of the main effects (random or fixed) goes to infinity. The robustness of these statistics against departure from normality will be assessed.Essentially, in the asymptotic spirit of this paper, both the hypothesis and error degrees of freedom tend to infinity at a fixed rate. It is intuitively appealing to consider asymptotics of this type because, for example, in random or mixed effects models, the levels of the main random factors are assumed to be a random sample from a large population of levels.For the asymptotic results of this paper to hold, we do not require any distributional assumption on the errors. That means the results can be used in real-life applications where normality assumption is not tenable.As it happens, the asymptotic distributions of the three statistics are normal. The statistics have been found to be asymptotically null robust against the departure from normality in the balanced designs. The expressions for the asymptotic means and variances are fairly simple. That makes the results an attractive alternative to the standard asymptotic results. These statements are favorably supported by the numerical results.     

Linear relations in time series models. II

In this paper an asymptotic theory is developed for a new time series model which was introduced in a previous paper [5]. An algorithm for computing estimates of the parameters of this time series model is given, and it is shown that these estimators are asymptotically efficient in the sense that they have the same asymptotic distribution as the maximum likelihood estimators.