Estimating the Period of a Cyclic Non‐Homogeneous Poisson Process

Abstract. Motivated by applications of Poisson processes for modelling periodic time‐varying phenomena, we study a semi‐parametric estimator of the period of cyclic intensity function of a non‐homogeneous Poisson process. There are no parametric assumptions on the intensity function which is treated as an infinite dimensional nuisance parameter. We propose a new family of estimators for the period of the intensity function, address the identifiability and consistency issues and present simulations which demonstrate good performance of the proposed estimation procedure in practice. We compare our method to competing methods on synthetic data and apply it to a real data set from a call center.     

Block‐threshold‐adapted Estimators via a Maxiset Approach

We study the maxiset performance of a large collection of block thresholding wavelet estimators, namely the horizontal block thresholding family. We provide sufficient conditions on the choices of rates and threshold values to ensure that the involved adaptive estimators obtain large maxisets. Moreover, we prove that any estimator of such a family reconstructs the Besov balls with a near‐minimax optimal rate that can be faster than the one of any separable thresholding estimator. Then, we identify, in particular cases, the best estimator of such a family, that is, the one associated with the largest maxiset. As a particularity of this paper, we propose a refined approach that models method‐dependent threshold values. By a series of simulation studies, we confirm the good performance of the best estimator by comparing it with the other members of its family.     

A Spline‐Based Semiparametric Maximum Likelihood Estimation Method for the Cox Model with Interval‐Censored Data

Abstract. We propose a spline‐based semiparametric maximum likelihood approach to analysing the Cox model with interval‐censored data. With this approach, the baseline cumulative hazard function is approximated by a monotone B‐spline function. We extend the generalized Rosen algorithm to compute the maximum likelihood estimate. We show that the estimator of the regression parameter is asymptotically normal and semiparametrically efficient, although the estimator of the baseline cumulative hazard function converges at a rate slower than root‐n. We also develop an easy‐to‐implement method for consistently estimating the standard error of the estimated regression parameter, which facilitates the proposed inference procedure for the Cox model with interval‐censored data. The proposed method is evaluated by simulation studies regarding its finite sample performance and is illustrated using data from a breast cosmesis study.     

A Non‐Parametric Estimator of the Spectral Density of a Continuous‐Time Gaussian Process Observed at Random Times

Abstract. In numerous applications data are observed at random times and an estimated graph of the spectral density may be relevant for characterizing and explaining phenomena. By using a wavelet analysis, one derives a non‐parametric estimator of the spectral density of a Gaussian process with stationary increments (or a stationary Gaussian process) from the observation of one path at random discrete times. For every positive frequency, this estimator is proved to satisfy a central limit theorem with a convergence rate depending on the roughness of the process and the moment of random durations between successive observations. In the case of stationary Gaussian processes, one can compare this estimator with estimators based on the empirical periodogram. Both estimators reach the same optimal rate of convergence, but the estimator based on wavelet analysis converges for a different class of random times. Simulation examples and an application to biological data are also provided.     

Structure Learning of Contextual Markov Networks using Marginal Pseudo‐likelihood

Markov networks are popular models for discrete multivariate systems where the dependence structure of the variables is specified by an undirected graph. To allow for more expressive dependence structures, several generalizations of Markov networks have been proposed. Here, we consider the class of contextual Markov networks which takes into account possible context‐specific independences among pairs of variables. Structure learning of contextual Markov networks is very challenging due to the extremely large number of possible structures. One of the main challenges has been to design a score, by which a structure can be assessed in terms of model fit related to complexity, without assuming chordality. Here, we introduce the marginal pseudo‐likelihood as an analytically tractable criterion for general contextual Markov networks. Our criterion is shown to yield a consistent structure estimator. Experiments demonstrate the favourable properties of our method in terms of predictive accuracy of the inferred models.     

Automated selection of post‐strata using a model‐assisted regression tree estimator

Despite having desirable properties, model‐assisted estimators are rarely used in anything but their simplest form to produce official statistics. This is due to the fact that the more complicated models are often ill suited to the available auxiliary data. Under a model‐assisted framework, we propose a regression tree estimator for a finite‐population total. Regression tree models are adept at handling the type of auxiliary data usually available in the sampling frame and provide a model that is easy to explain and justify. The estimator can be viewed as a post‐stratification estimator where the post‐strata are automatically selected by the recursive partitioning algorithm of the regression tree. We establish consistency of the regression tree estimator and a variance estimator, along with asymptotic normality of the regression tree estimator. We compare the performance of our estimator to other survey estimators using the United States Bureau of Labor Statistics Occupational Employment Statistics Survey data.     

Score estimation in the monotone single‐index model

We consider estimation in the single‐index model where the link function is monotone. For this model, a profile least‐squares estimator has been proposed to estimate the unknown link function and index. Although it is natural to propose this procedure, it is still unknown whether it produces index estimates that converge at the parametric rate. We show that this holds if we solve a score equation corresponding to this least‐squares problem. Using a Lagrangian formulation, we show how one can solve this score equation without any reparametrization. This makes it easy to solve the score equations in high dimensions. We also compare our method with the effective dimension reduction and the penalized least‐squares estimator methods, both available on CRAN as R packages, and compare with link‐free methods, where the covariates are elliptically symmetric.     

ROBUST ESTIMATION OF SMALL‐AREA MEANS AND QUANTILES

Small‐area estimation techniques have typically relied on plug‐in estimation based on models containing random area effects. More recently, regression M‐quantiles have been suggested for this purpose, thus avoiding conventional Gaussian assumptions, as well as problems associated with the specification of random effects. However, the plug‐in M‐quantile estimator for the small‐area mean can be shown to be the expected value of this mean with respect to a generally biased estimator of the small‐area cumulative distribution function of the characteristic of interest. To correct this problem, we propose a general framework for robust small‐area estimation, based on representing a small‐area estimator as a functional of a predictor of this small‐area cumulative distribution function. Key advantages of this framework are that it naturally leads to integrated estimation of small‐area means and quantiles and is not restricted to M‐quantile models. We also discuss mean squared error estimation for the resulting estimators, and demonstrate the advantages of our approach through model‐based and design‐based simulations, with the latter using economic data collected in an Australian farm survey.     

A general quantile residual life model for length‐biased right‐censored data

The quantile residual lifetime function provides comprehensive quantitative measures for residual life, especially when the distribution of the latter is skewed or heavy‐tailed and/or when the data contain outliers. In this paper, we propose a general class of semiparametric quantile residual life models for length‐biased right‐censored data. We use the inverse probability weighted method to correct the bias due to length‐biased sampling and informative censoring. Two estimating equations corresponding to the quantile regressions are constructed in two separate steps to obtain an efficient estimator. Consistency and asymptotic normality of the estimator are established. The main difficulty in implementing our proposed method is that the estimating equations associated with the quantiles are nondifferentiable, and we apply the majorize–minimize algorithm and estimate the asymptotic covariance using an efficient resampling method. We use simulation studies to evaluate the proposed method and illustrate its application by a real‐data example.     

Series estimation for single‐index models under constraints

In this paper, a semi‐parametric single‐index model is investigated. The link function is allowed to be unbounded and has unbounded support that answers a pending issue in the literature. Meanwhile, the link function is treated as a point in an infinitely many dimensional function space which enables us to derive the estimates for the index parameter and the link function simultaneously. This approach is different from the profile method commonly used in the literature. The estimator is derived from an optimisation with the constraint of identification condition for the index parameter, which addresses an important problem in the literature of single‐index models. In addition, making use of a property of Hermite orthogonal polynomials, an explicit estimator for the index parameter is obtained. Asymptotic properties for the two estimators of the index parameter are established. Their efficiency is discussed in some special cases as well. The finite sample properties of the two estimates are demonstrated through an extensive Monte Carlo study and an empirical example.     

Variance Estimation under Two‐Phase Sampling

We consider the variance estimation of the weighted likelihood estimator (WLE) under two‐phase stratified sampling without replacement. Asymptotic variance of the WLE in many semiparametric models contains unknown functions or does not have a closed form. The standard method of the inverse probability weighted (IPW) sample variances of an estimated influence function is then not available in these models. To address this issue, we develop the variance estimation procedure for the WLE in a general semiparametric model. The phase I variance is estimated by taking a numerical derivative of the IPW log likelihood. The phase II variance is estimated based on the bootstrap for a stratified sample in a finite population. Despite a theoretical difficulty of dependent observations due to sampling without replacement, we establish the (bootstrap) consistency of our estimators. Finite sample properties of our method are illustrated in a simulation study.     

An MSE‐reduced estimator for the response proportion in a two‐stage clinical trial

Two‐stage design is very useful in clinical trials for evaluating the validity of a specific treatment regimen. When the second stage is allowed to continue, the method used to estimate the response rate based on the results of both stages is critical for the subsequent design. The often‐used sample proportion has an evident upward bias. However, the maximum likelihood estimator or the moment estimator tends to underestimate the response rate. A mean‐square error weighted estimator is considered here; its performance is thoroughly investigated via Simon's optimal and minimax designs and Shuster's design. Compared with the sample proportion, the proposed method has a smaller bias, and compared with the maximum likelihood estimator, the proposed method has a smaller mean‐square error. Copyright © 2010 John Wiley & Sons, Ltd.     

New estimation and feature selection methods in mixture‐of‐experts models

We study estimation and feature selection problems in mixture‐of‐experts models. An $l_2$ ‐penalized maximum likelihood estimator is proposed as an alternative to the ordinary maximum likelihood estimator. The estimator is particularly advantageous when fitting a mixture‐of‐experts model to data with many correlated features. It is shown that the proposed estimator is root‐$n$ consistent, and simulations show its superior finite sample behaviour compared to that of the maximum likelihood estimator. For feature selection, two extra penalty functions are applied to the $l_2$ ‐penalized log‐likelihood function. The proposed feature selection method is computationally much more efficient than the popular all‐subset selection methods. Theoretically it is shown that the method is consistent in feature selection, and simulations support our theoretical results. A real‐data example is presented to demonstrate the method. The Canadian Journal of Statistics 38: 519–539; 2010 © 2010 Statistical Society of Canada     

Kernel‐based Generalized Cross‐validation in Non‐parametric Mixed‐effect Models

Abstract. Although generalized cross‐validation (GCV) has been frequently applied to select bandwidth when kernel methods are used to estimate non‐parametric mixed‐effect models in which non‐parametric mean functions are used to model covariate effects, and additive random effects are applied to account for overdispersion and correlation, the optimality of the GCV has not yet been explored. In this article, we construct a kernel estimator of the non‐parametric mean function. An equivalence between the kernel estimator and a weighted least square type estimator is provided, and the optimality of the GCV‐based bandwidth is investigated. The theoretical derivations also show that kernel‐based and spline‐based GCV give very similar asymptotic results. This provides us with a solid base to use kernel estimation for mixed‐effect models. Simulation studies are undertaken to investigate the empirical performance of the GCV. A real data example is analysed for illustration.     

Efficient estimation for case‐cohort studies

The author considers time‐to‐event data from case‐cohort designs. As existing methods are either inefficient or based on restrictive assumptions concerning the censoring mechanism, he proposes a semi‐parametrically efficient estimator under the usual assumptions for Cox regression models. The estimator in question is obtained by a one‐step Newton‐Raphson approximation that solves the efficient score equations with initial value obtained from an existing method. The author proves that the estimator is consistent, asymptotically efficient and normally distributed in the limit. He also resorts to simulations to show that the proposed estimator performs well in finite samples and that it considerably improves the efficiency of existing pseudo‐likelihood estimators when a correlate of the missing covariate is available. Although he focuses on the situation where covariates are discrete, the author also explores how the method can be applied to models with continuous covariates.     

Efficient Inference for Longitudinal Data Varying‐coefficient Regression Models

Informative identification of the within‐subject correlation is essential in longitudinal studies in order to forecast the trajectory of each subject and improve the validity of inferences. In this paper, we fit this correlation structure by employing a time adaptive autoregressive error process. Such a process can automatically accommodate irregular and possibly subject‐specific observations. Based on the fitted correlation structure, we propose an efficient two‐stage estimator of the unknown coefficient functions by using a local polynomial approximation. This procedure does not involve within‐subject covariance matrices and hence circumvents the instability of calculating their inverses. The asymptotic normality of resulting estimators is established. Numerical experiments were conducted to check the finite sample performance of our method and an example of an application involving a set of medical data is also illustrated.     

Empirical Uncertain Bayes Methods in Area‐level Models

Random effects model can account for the lack of fitting a regression model and increase precision of estimating area‐level means. However, in case that the synthetic mean provides accurate estimates, the prior distribution may inflate an estimation error. Thus, it is desirable to consider the uncertain prior distribution, which is expressed as the mixture of a one‐point distribution and a proper prior distribution. In this paper, we develop an empirical Bayes approach for estimating area‐level means, using the uncertain prior distribution in the context of a natural exponential family, which we call the empirical uncertain Bayes (EUB) method. The regression model considered in this paper includes the Poisson‐gamma and the binomial‐beta, and the normal‐normal (Fay–Herriot) model, which are typically used in small area estimation. We obtain the estimators of hyperparameters based on the marginal likelihood by using a well‐known expectation‐maximization algorithm and propose the EUB estimators of area means. For risk evaluation of the EUB estimator, we derive a second‐order unbiased estimator of a conditional mean squared error by using some techniques of numerical calculation. Through simulation studies and real data applications, we evaluate a performance of the EUB estimator and compare it with the usual empirical Bayes estimator.     

Iterative Bias Correction of the Cross‐Validation Criterion

Abstract. The cross‐validation (CV) criterion is known to be asecond‐order unbiased estimator of the risk function measuring the discrepancy between the candidate model and the true model, as well as the generalized information criterion (GIC) and the extended information criterion (EIC). In the present article, we show that the 2kth‐order unbiased estimator can be obtained using a linear combination from the leave‐one‐out CV criterion to the leave‐k‐out CV criterion. The proposed scheme is unique in that a bias smaller than that of a jackknife method can be obtained without any analytic calculation, that is, it is not necessary to obtain the explicit form of several terms in an asymptotic expansion of the bias. Furthermore, the proposed criterion can be regarded as a finite correction of a bias‐corrected CV criterion by using scalar coefficients in a bias‐corrected EIC obtained by the bootstrap iteration.     

A semi‐Markov model for binary longitudinal responses subject to misclassification

The authors propose a two‐state continuous‐time semi‐Markov model for an unobservable alternating binary process. Another process is observed at discrete time points that may misclassify the true state of the process of interest. To estimate the model's parameters, the authors propose a minimum Pearson chi‐square type estimating approach based on approximated joint probabilities when the true process is in equilibrium. Three consecutive observations are required to have sufficient degrees of freedom to perform estimation. The methodology is demonstrated on parasitic infection data with exponential and gamma sojourn time distributions.     

Parameter Estimation for Inhomogeneous Space‐Time Shot‐Noise Cox Point Processes

We consider the problem of parameter estimation for inhomogeneous space‐time shot‐noise Cox point processes. We explore the possibility of using a stepwise estimation method and dimensionality‐reducing techniques to estimate different parts of the model separately. We discuss the estimation method using projection processes and propose a refined method that avoids projection to the temporal domain. This remedies the main flaw of the method using projection processes – possible overlapping in the projection process of clusters, which are clearly separated in the original space‐time process. This issue is more prominent in the temporal projection process where the amount of information lost by projection is higher than in the spatial projection process. For the refined method, we derive consistency and asymptotic normality results under the increasing domain asymptotics and appropriate moment and mixing assumptions. We also present a simulation study that suggests that cluster overlapping is successfully overcome by the refined method.