Statistical inference of minimum BD estimators and classifiers for varying-dimensional models

Stochastic modeling for large-scale datasets usually involves a varying-dimensional model space. This paper investigates the asymptotic properties, when the number of parameters grows with the available sample size, of the minimum- estimators and classifiers under a broad and important class of Bregman divergence (), which encompasses nearly all of the commonly used loss functions in the regression analysis, classification procedures and machine learning literature. Unlike the maximum likelihood estimators which require the joint likelihood of observations, the minimum-BD estimators are useful for a range of models where the joint likelihood is unavailable or incomplete. Statistical inference tools developed for the class of large dimensional minimum- estimators and related classifiers are evaluated via simulation studies, and are illustrated by analysis of a real dataset.     

Estimating common vector parameters in interlaboratory studies

The primary goal of this work is to extend two methods of random effects models to multiparameter situation. These methods comprise the DerSimonian-Laird estimator, stemming from meta-analysis, and the Mandel-Paule algorithm widely used in interlaboratory studies. The maximum likelihood estimators are also discussed. Two methods of assessing the uncertainty of these estimators are given.     

Adaptive Bayes type estimators of ergodic diffusion processes from discrete observations

We consider adaptive Bayesian estimation of both drift and diffusion coefficient parameters for ergodic multidimensional diffusion processes based on sampled data. Under a general condition on the discretization step of the sampled data, three kinds of adaptive Bayes type estimators are proposed by applying adaptive maximum likelihood type methods of Uchida and Yoshida (Stoch Process Appl 122:2885–2924, 2012) to Bayesian procedures. We show asymptotic normality and convergence of moments for the adaptive Bayes type estimators by means of the Ibragimov–Has’minskii–Kutoyants program together with the polynomial type large deviation inequality for the statistical random field.     

Conditional and unconditional methods for selecting variables in linear mixed models

In the problem of selecting the explanatory variables in the linear mixed model, we address the derivation of the (unconditional or marginal) Akaike information criterion (AIC) and the conditional AIC (cAIC). The covariance matrices of the random effects and the error terms include unknown parameters like variance components, and the selection procedures proposed in the literature are limited to the cases where the parameters are known or partly unknown. In this paper, AIC and cAIC are extended to the situation where the parameters are completely unknown and they are estimated by the general consistent estimators including the maximum likelihood (ML), the restricted maximum likelihood (REML) and other unbiased estimators. We derive, related to AIC and cAIC, the marginal and the conditional prediction error criteria which select superior models in light of minimizing the prediction errors relative to quadratic loss functions. Finally, numerical performances of the proposed selection procedures are investigated through simulation studies.     

Estimating cumulative incidence functions when the life distributions are constrained

In competing risks studies, the Kaplan-Meier estimators of the distribution functions (DFs) of lifetimes and the corresponding estimators of cumulative incidence functions (CIFs) are used widely when no prior information is available for these distributions. In some cases better estimators of the DFs of lifetimes are available when they obey some inequality constraints, e.g., if two lifetimes are stochastically or uniformly stochastically ordered, or some functional of a DF obeys an inequality in an empirical likelihood estimation procedure. If the restricted estimator of a lifetime differs from the unrestricted one, then the usual estimators of the CIFs will not add up to the lifetime estimator. In this paper we show how to estimate the CIFs in this case. These estimators are shown to be strongly uniformly consistent. In all cases we consider, when the inequality constraints are strict the asymptotic properties of the restricted and the unrestricted estimators are the same, thus providing the asymptotic properties of the restricted estimators essentially “free of charge”. We give an example to illustrate our procedure.     

Maximum likelihood estimation for all-pass time series models

An autoregressive-moving average model in which all roots of the autoregressive polynomial are reciprocals of roots of the moving average polynomial and vice versa is called an all-pass time series model. All-pass models generate uncorrelated (white noise) time series, but these series are not independent in the non-Gaussian case. An approximate likelihood for a causal all-pass model is given and used to establish asymptotic normality for maximum likelihood estimators under general conditions. Behavior of the estimators for finite samples is studied via simulation. A two-step procedure using all-pass models to identify and estimate noninvertible autoregressive-moving average models is developed and used in the deconvolution of a simulated water gun seismogram.     

Drift parameter estimation in fractional diffusions driven by perturbed random walks

We estimate the drift parameter in a simple linear model driven by fractional Brownian motion. We propose maximum likelihood estimators (MLE) for the drift parameter construct by using a random walk approximation of the fractional Brownian motion.     

Wiener processes with random effects for degradation data

This article studies the maximum likelihood inference on a class of Wiener processes with random effects for degradation data. Degradation data are special case of functional data with monotone trend. The setting for degradation data is one on which n independent subjects, each with a Wiener process with random drift and diffusion parameters, are observed at possible different times. Unit-to-unit variability is incorporated into the model by these random effects. EM algorithm is used to obtain the maximum likelihood estimators of the unknown parameters. Asymptotic properties such as consistency and convergence rate are established. Bootstrap method is used for assessing the uncertainties of the estimators. Simulations are used to validate the method. The model is fitted to bridge beam data and corresponding goodness-of-fit tests are carried out. Failure time distributions in terms of degradation level passages are calculated and illustrated.     

Estimation of means of multivariate normal populations with order restriction

Multivariate isotonic regression theory plays a key role in the field of statistical inference under order restriction for vector valued parameters. Two cases of estimating multivariate normal means under order restricted set are considered. One case is that covariance matrices are known, the other one is that covariance matrices are unknown but are restricted by partial order. This paper shows that when covariance matrices are known, the estimator given by this paper always dominates unrestricted maximum likelihood estimator uniformly, and when covariance matrices are unknown, the plug-in estimator dominates unrestricted maximum likelihood estimator under the order restricted set of covariance matrices. The isotonic regression estimators in this paper are the generalizations of plug-in estimators in unitary case.     

Explicit estimators of parameters in the Growth Curve model with linearly structured covariance matrices

Estimation of parameters in the classical Growth Curve model, when the covariance matrix has some specific linear structure, is considered. In our examples maximum likelihood estimators cannot be obtained explicitly and must rely on optimization algorithms. Therefore explicit estimators are obtained as alternatives to the maximum likelihood estimators. From a discussion about residuals, a simple non-iterative estimation procedure is suggested which gives explicit and consistent estimators of both the mean and the linear structured covariance matrix.     

Estimation of Wishart mean matrices under simple tree ordering

For Wishart density functions, we study the risk dominance problems of the restricted maximum likelihood estimators of mean matrices with respect to the Kullback-Leibler loss function over restricted parameter space under the simple tree ordering set. The results are directly applied to the estimation of covariance matrices for the completely balanced multivariate multi-way random effects models without interactions.     

Estimation and confidence bands of a conditional survival function with censoring indicators missing at random

The nonparametric estimator of the conditional survival function proposed by Beran is a useful tool to evaluate the effects of covariates in the presence of random right censoring. However, censoring indicators of right censored data may be missing for different reasons in many applications. We propose some estimators of the conditional cumulative hazard and survival functions which allow to handle this situation. We also construct the likelihood ratio confidence bands for them and obtain their asymptotic properties. Simulation studies are used to evaluate the performances of the estimators and their confidence bands.     

Improving on the maximum likelihood estimators of the means in Poisson decomposable graphical models

In this article we study the simultaneous estimation of the means in Poisson decomposable graphical models. We derive some classes of estimators which improve on the maximum likelihood estimator under the normalized squared losses. Our estimators are based on the argument in Chou [Simultaneous estimation in discrete multivariate exponential families, Ann. Statist. 19 (1991) 314-328.] and shrink the maximum likelihood estimator depending on the marginal frequencies of variables forming a complete subgraph of the conditional independence graph.     

Estimation of location and scale parameters for the Burr XII distribution using generalized order statistics

The minimum variance linear unbiased estimators (MVLUE), the best linear invariant estimators (BLIE) and the maximum likelihood estimators (MLE) based on n-selected generalized order statistics are presented for the parameters of the Burr XII distribution.     

Asymptotic properties of particle filter-based maximum likelihood estimators for state space models

We study the asymptotic performance of approximate maximum likelihood estimators for state space models obtained via sequential Monte Carlo methods. The state space of the latent Markov chain and the parameter space are assumed to be compact. The approximate estimates are computed by, firstly, running possibly dependent particle filters on a fixed grid in the parameter space, yielding a pointwise approximation of the log-likelihood function. Secondly, extensions of this approximation to the whole parameter space are formed by means of piecewise constant functions or B-spline interpolation, and approximate maximum likelihood estimates are obtained through maximization of the resulting functions. In this setting we formulate criteria for how to increase the number of particles and the resolution of the grid in order to produce estimates that are consistent and asymptotically normal.     

High breakdown mixture discriminant analysis

Robust S-estimation is proposed for multivariate Gaussian mixture models generalizing the work of Hastie and Tibshirani (J. Roy. Statist. Soc. Ser. B 58 (1996) 155). In the case of Gaussian Mixture models, the unknown location and scale parameters are estimated by the EM algorithm. In the presence of outliers, the maximum likelihood estimators of the unknown parameters are affected, resulting in the misclassification of the observations. The robust S-estimators of the unknown parameters replace the non-robust estimators from M-step of the EM algorithm. The results were compared with the standard mixture discriminant analysis approach using the probability of misclassification criterion. This comparison showed a slight reduction in the average probability of misclassification using robust S-estimators as compared to the standard maximum likelihood estimators.     

Order restricted inference for sequential k-out-of-n systems

Sequential order statistics have been introduced to model sequential k-out-of-n systems which, as an extension of k-out-of-n systems, allow the failure of some components of the system to influence the remaining ones. Based on an independent sample of vectors of sequential order statistics, the maximum likelihood estimators of the model parameters of a sequential k-out-of-n system are derived under order restrictions. Special attention is paid to the simultaneous maximum likelihood estimation of the model parameters and the distribution parameters for a flexible location-scale family. Furthermore, order restricted hypothesis tests are considered for making the decision whether the usual k-out-of-n model or the general sequential k-out-of-n model is appropriate for a given data.     

Rates of convergence for minimum contrast estimators

Summary We shall present here a general study of minimum contrast estimators in a nonparametric setting (although our results are also valid in the classical parametric case) for independent observations. These estimators include many of the most popular estimators in various situations such as maximum likelihood estimators, least squares and other estimators of the regression function, estimators for mixture models or deconvolution... The main theorem relates the rate of convergence of those estimators to the entropy structure of the space of parameters. Optimal rates depending on entropy conditions are already known, at least for some of the models involved, and they agree with what we get for minimum contrast estimators as long as the entropy counts are not too large. But, under some circumstances (large entropies or changes in the entropy structure due to local perturbations), the resulting the rates are only suboptimal. Counterexamples are constructed which show that the phenomenon is real for non-parametric maximum likelihood or regression. This proves that, under purely metric assumptions, our theorem is optimal and that minimum contrast estimators happen to be suboptimal.     

MOMENT ESTIMATION FOR MULTIVARIATE EXTREME VALUE DISTRIBUTION

Moment estimation for multivariate extreme value distribution is described in this paper. Asymptotic covariance matrix of the estimators is given. The relative efficiencies of moment estimators as compared with the maximum likelihood and the stepwise estimators are computed. We show that when there is strong dependence between the variates, the generalized variance of moment estimators is much lower than the stepwise estimators. It becomes more obvious when the dimension increases.     

Asymptotic properties of computationally efficient alternative estimators for a class of multivariate normal models

Parameters of Gaussian multivariate models are often estimated using the maximum likelihood approach. In spite of its merits, this methodology is not practical when the sample size is very large, as, for example, in the case of massive georeferenced data sets. In this paper, we study the asymptotic properties of the estimators that minimize three alternatives to the likelihood function, designed to increase the computational efficiency. This is achieved by applying the information sandwich technique to expansions of the pseudo-likelihood functions as quadratic forms of independent normal random variables. Theoretical calculations are given for a first-order autoregressive time series and then extended to a two-dimensional autoregressive process on a lattice. We compare the efficiency of the three estimators to that of the maximum likelihood estimator as well as among themselves, using numerical calculations of the theoretical results and simulations.