On the estimation of serial correlation in Markov-dependent production processes

In this paper, we present a study about the estimation of the serial correlation for Markov chain models which is used often in the quality control of autocorrelated processes. Two estimators, non-parametric and multinomial, for the correlation coefficient are discussed. They are compared with the maximum likelihood estimator [U.N. Bhat and R. Lal, Attribute control charts for Markov dependent production process, IIE Trans. 22 (2) (1990), pp. 181–188.] by using some theoretical facts and the Monte Carlo simulation under several scenarios that consider large and small correlations as well a range of fractions (p) of non-conforming items. The theoretical results show that for any value of p≠0.5 and processes with autocorrelation higher than 0.5, the multinomial is more precise than maximum likelihood. However, the maximum likelihood is better when the autocorrelation is smaller than 0.5. The estimators are similar for p=0.5. Considering the average of all simulated scenarios, the multinomial estimator presented lower mean error values and higher precision, being, therefore, an alternative to estimate the serial correlation. The performance of the non-parametric estimator was reasonable only for correlation higher than 0.5, with some improvement for p=0.5.     

Goodness-of-fit Tests for Semi-Markov and Markov Survival Models with One Intermediate State

Survival data with one intermediate state are described by semi-Markov and Markov models for counting processes whose intensities are defined in terms of two stopping times T ₁< T ₂. Problems of goodness-of-fit for these models are studied. The test statistics are proposed by comparing Nelson–Aalen estimators for data stratified according to T ₁. Asymptotic distributions of these statistics are established in terms of the weak convergence of some random fields. Asymptotic consistency of these test statistics is also established. Simulation studies are included to indicate their numerical performance.     

Monte Carlo integration with Markov chain

There are two conceptually distinct tasks in Markov chain Monte Carlo (MCMC): a sampler is designed for simulating a Markov chain and then an estimator is constructed on the Markov chain for computing integrals and expectations. In this article, we aim to address the second task by extending the likelihood approach of Kong et al. for Monte Carlo integration. We consider a general Markov chain scheme and use partial likelihood for estimation. Basically, the Markov chain scheme is treated as a random design and a stratified estimator is defined for the baseline measure. Further, we propose useful techniques including subsampling, regulation, and amplification for achieving overall computational efficiency. Finally, we introduce approximate variance estimators for the point estimators. The method can yield substantially improved accuracy compared with Chib's estimator and the crude Monte Carlo estimator, as illustrated with three examples.     

Rates of convergence of autocorrelation estimates for periodically correlated autoregressive Hilbertian processes

Autoregressive Hilbertian (ARH) processes are of great importance in the analysis of functional time series data and estimation of the autocorrelation operators attracts the attention of various researchers. In this paper, we study estimators of the autocorrelation operators of periodically correlated autoregressive Hilbertian processes of order one (PCARH(1)), which is an extension of ARH(1) processes. The estimation method is based on the spectral decomposition of the covariance operator and considers two main cases: known and unknown eigenvectors. We show the consistency in the mean integrated quadratic sense of the estimators of the autocorrelation operators and present upper bounds for the corresponding rates.     

Estimating models based on Markov jump processes given fragmented observation series

We consider the problem of estimating the rate matrix governing a finite-state Markov jump process given a number of fragmented time series. We propose to concatenate the observed series and to employ the emerging non-Markov process for estimation. We describe the bias arising if standard methods for Markov processes are used for the concatenated process, and provide a post-processing method to correct for this bias. This method applies to discrete-time Markov chains and to more general models based on Markov jump processes where the underlying state process is not observed directly. This is demonstrated in detail for a Markov switching model. We provide applications to simulated time series and to financial market data, where estimators resulting from maximum likelihood methods and Markov chain Monte Carlo sampling are improved using the presented correction.     

The discretely observed immigration-death process: Likelihood inference and spatiotemporal applications

ABSTRACT

We consider a stochastic process, the homogeneous spatial immigration-death (HSID) process, which is a spatial birth-death process with as building blocks (i) an immigration-death (ID) process (a continuous-time Markov chain) and (ii) a probability distribution assigning iid spatial locations to all events. For the ID process, we derive the likelihood function, reduce the likelihood estimation problem to one dimension, and prove consistency and asymptotic normality for the maximum likelihood estimators (MLEs) under a discrete sampling scheme. We additionally prove consistency for the MLEs of HSID processes. In connection to the growth-interaction process, which has a HSID process as basis, we also fit HSID processes to Scots pine data.     

Bayesian estimation under a mixture of the Burr type XII distribution and its reciprocal

The Bayesian estimation for the parameters of the finite mixture of the Burr type XII distribution with its reciprocal are obtained based on the type-I censored data. The Bayes estimators are computed based on squared error and Linex loss functions and using the idea of Markov chain Monte Carlo algorithm. Based on the Monte Carlo simulation, Bayes estimators are compared with their corresponding maximum-likelihood estimators.     

Choice between Semi-parametric Estimators of Markov and Non-Markov Multi-state Models from Coarsened Observations

Abstract.  We consider models based on multivariate counting processes, including multi-state models. These models are specified semi-parametrically by a set of functions and real parameters. We consider inference for these models based on coarsened observations, focusing on families of smooth estimators such as produced by penalized likelihood. An important issue is the choice of model structure, for instance, the choice between a Markov and some non-Markov models. We define in a general context the expected Kullback–Leibler criterion and we show that the likelihood-based cross-validation (LCV) is a nearly unbiased estimator of it. We give a general form of an approximate of the leave-one-out LCV. The approach is studied by simulations, and it is illustrated by estimating a Markov and two semi-Markov illness–death models with application on dementia using data of a large cohort study.     

Large Deviations for Empirical Estimators of the Stationary Distribution of a Semi-Markov Process with Finite State Space

We prove the large deviation principle for empirical estimators of stationary distributions of semi-Markov processes with finite state space, irreducible embedded Markov chain, and finite mean sojourn time in each state. We consider on/off Gamma sojourn processes as an illustrative example, and, in particular, continuous time Markov chains with two states. In the second case, we compare the rate function in this article with the known rate function concerning another family of empirical estimators of the stationary distribution.     

Estimation of the Entropy Rate of a Countable Markov Chain

We consider here ergodic homogeneous Markov chains with countable state spaces. The entropy rate of the chain is an explicit function of its transition and stationary distributions. We construct estimators for this entropy rate and for the entropy of the stationary distribution of the chain, in the parametric and nonparametric cases. We study estimation from one sample with long length and from many independent samples with given length. In the parametric case, the estimators are deduced by plug-in from the maximum likelihood estimator of the parameter. In the nonparametric case, the estimators are deduced by plug-in from the empirical estimators of the transition and stationary distributions. They are proven to have good asymptotic properties.     

Smoothed nonparametric estimation in window censored semi-Markov processes

Consider a process that jumps among a finite set of states, with random times spent in between. In semi-Markov processes transitions follow a Markov chain and the sojourn distributions depend only on the connecting states. Suppose that the process started far in the past, achieving stationary. We consider non-parametric estimation by modelling the log-hazard of the sojourn times through linear splines; and we obtain maximum penalized likelihood estimators when data consist of several i.i.d. windows. We prove consistency using Grenander's method of sieves.     

Nonparametric estimation of waiting time distributions in a Markov model based on current status data

We construct nonparametric estimators of state waiting time distribution functions in a Markov multistate model using current status data. This is a particularly difficult problem since neither the entry nor the exit times of a given state are directly observed. These estimators are obtained, using the Markov property, from estimators of counting processes of state entry and exit times, as well as, the size of “at risk” sets of state entry and transitions out of that state. Consistency of our estimators is established. Finite-sample behavior of our estimators is studied by simulation, in which we show that our estimators based on current status data compare well with those based on complete data. We also illustrate our method using a pubertal development data set obtained from the NHANES III [1997. NHANES III Reference Manuals and Reports (CD-ROM). Analytic and Reporting Guidelines: The Third National Health and Nutrition Examination Survey (1988–94). National Center for Health Statistics, Centers for Disease Control and Prevention, Hyattsville, MD] study.     

Efficient sequential estimation in multiplicative counting processes

The problem of efficient sequential estimation is counting processes with multiplicative intensity processes is considered. A sequential version of Cramér-Rao type information inequality is obtained and all the 'efficient' triples (S, f, g) are characterized: the variance of an unbiased estimator f for g attains the lower bound under a sampling plan S. Applications to Poisson processes, Markov processes, birth and death processes and Markov branching processes with immigration are also considered.     

On Minimum Distance Estimation in Recurrent Markov Step Processes. I

Consider a Markov step process X=(X_t)_t≥0 whose generator depends on an unknown d -dimensional parameter ϑ. We look at certain empirical measures for recurrent Markov step processes and their a.s. convergence; based on this, we introduce a class of minimum distance estimators. For broad families of sequential observation schemes (at stage n, the trajectory of X is observed up to time S_n, (S_n)_n a sequence of stopping times increasing to ∞), we formulate a stochastic expansion of the suitably rescaled estimation error; for a particular scheme, asymptotic normality is obtained as n →∞. A minimax property under misspecification of the model (in the sense that the true probability law is contiguous to the parametric model but not contained in it) is given.     

Efficient Monte Carlo simulation via the generalized splitting method

We describe a new Monte Carlo algorithm for the consistent and unbiased estimation of multidimensional integrals and the efficient sampling from multidimensional densities. The algorithm is inspired by the classical splitting method and can be applied to general static simulation models. We provide examples from rare-event probability estimation, counting, and sampling, demonstrating that the proposed method can outperform existing Markov chain sampling methods in terms of convergence speed and accuracy.     

Sur l'estimation du risque encouru en reconstruisant une forme aléatoire

The problem of estimating the risk corresponding to the reconstruction of a random pattern is reviewed. It is shown that for a particular but important model, the problem is reduced to the estimation of two parameters closely related to those appearing in a two-state Markov chain, which is of independent interest. The estimation of the Markov chain's parameters is studied from the decision-theoretic point of view. Estimators which are better than others previously considered are obtained and adapted to the estimation of the corresponding risk. Examples are are analyzed; even if a very empirical method is used to give values to the parameters of an a priorilaw, some good estimators of the risk are obtained.     

Modeling statistical dependence of Markov chains via copula models

Conditional probability distributions have been commonly used in modeling Markov chains. In this paper we consider an alternative approach based on copulas to investigate Markov-type dependence structures. Based on the realization of a single Markov chain, we estimate the parameters using one- and two-stage estimation procedures. We derive asymptotic properties of the marginal and copula parameter estimators and compare performance of the estimation procedures based on Monte Carlo simulations. At low and moderate dependence structures the two-stage estimation has comparable performance as the maximum likelihood estimation. In addition we propose a parametric pseudo-likelihood ratio test for copula model selection under the two-stage procedure. We apply the proposed methods to an environmental data set.     

Bayesian analysis of the scatterometer wind retrieval inverse problem: some new approaches

Summary.  The retrieval of wind vectors from satellite scatterometer observations is a non-linear inverse problem. A common approach to solving inverse problems is to adopt a Bayesian framework and to infer the posterior distribution of the parameters of interest given the observations by using a likelihood model relating the observations to the parameters, and a prior distribution over the parameters. We show how Gaussian process priors can be used efficiently with a variety of likelihood models, using local forward (observation) models and direct inverse models for the scatterometer. We present an enhanced Markov chain Monte Carlo method to sample from the resulting multimodal posterior distribution. We go on to show how the computational complexity of the inference can be controlled by using a sparse, sequential Bayes algorithm for estimation with Gaussian processes. This helps to overcome the most serious barrier to the use of probabilistic, Gaussian process methods in remote sensing inverse problems, which is the prohibitively large size of the data sets. We contrast the sampling results with the approximations that are found by using the sparse, sequential Bayes algorithm.     

Maximum likelihood estimation for reflected Ornstein-Uhlenbeck processes

In this paper, we investigate the maximum likelihood estimation for the reflected Ornstein-Uhlenbeck (ROU) processes based on continuous observations. Both the cases with one-sided barrier and two-sided barriers are considered. We derive the explicit formulas for the estimators, and then prove their strong consistency and asymptotic normality. Moreover, the bias and mean square errors are represented in terms of the solutions to some PDEs with homogeneous Neumann boundary conditions. We also illustrate the asymptotic behavior of the estimators through a simulation study.     

Parametric estimation for ARFIMA models via spectral methods

Given a fractional integrated, autoregressive, moving average,ARFIMA (p, d, q) process, the simultaneous estimation of the short and long memory parameters can be achieved by maximum likelihood estimators. In this paper, following a two-step algorithm, the coefficients are estimated combining the maximum likelihood estimators with the general orthogonal decomposition of stochastic processes. In particular, the principal component analysis of stochastic processes is exploited to estimate the short memory parameters, which are plugged into the maximum likelihood function to obtain the fractional differencingd.