Estimation of two order restricted normal means with unknown and possibly unequal variances

Estimation of each of and linear functions of two order restricted normal means is considered when variances are unknown and possibly unequal. We replace unknown variances with sample variances and construct isotonic regression estimators, which we call in our paper the plug-in estimators, to estimate ordered normal means. Under squared error loss, a necessary and sufficient condition is given for the plug-in estimators to improve upon the unrestricted maximum likelihood estimators uniformly. As for the estimation of linear functions of ordered normal means, we also show that when variances are known, the restricted maximum likelihood estimator always improves upon the unrestricted maximum likelihood estimator uniformly, but when variances are unknown, the plug-in estimator does not always improve upon the unrestricted maximum likelihood estimator uniformly.     

Estimation of odds ratios under order restrictions

A new method for estimating a set of odds ratios under an order restriction based on estimating equations is proposed. The method is applied to those of the conditional maximum likelihood estimators and the Mantel-Haenszel estimators. The estimators derived from the conditional likelihood estimating equations are shown to maximize the conditional likelihoods. It is also seen that the restricted estimators converge almost surely to the respective odds ratios when the respective sample sizes become large regularly. The restricted estimators are compared with the unrestricted maximum likelihood estimators by a Monte Carlo simulation. The simulation studies show that the restricted estimates improve the mean squared errors remarkably, while the Mantel-Haenszel type estimates are competitive with the conditional maximum likelihood estimates, being slightly worse.     

Testing for generalized linear mixed models with cluster correlated data under linear inequality constraints

Generalized linear mixed models (GLMMs) are often used for analyzing cluster correlated data, including longitudinal data and repeated measurements. Full unrestricted maximum likelihood (ML) approaches for inference on both fixed‐and random‐effects parameters in GLMMs have been extensively studied in the literature. However, parameter orderings or constraints may occur naturally in practice, and in such cases, the efficiency of a statistical method is improved by incorporating the parameter constraints into the ML estimation and hypothesis testing. In this paper, inference for GLMMs under linear inequality constraints is considered. The asymptotic properties of the constrained ML estimators and constrained likelihood ratio tests for GLMMs have been studied. Simulations investigated the empirical properties of the constrained ML estimators, compared to their unrestricted counterparts. An application to a recent survey on Canadian youth smoking patterns is also presented. As these survey data exhibit natural parameter orderings, a constrained GLMM has been considered for data analysis. The Canadian Journal of Statistics 40: 243–258; 2012 © 2012 Crown in the right of Canada     

Mixture Model Analysis of Partially Rank‐Ordered Set Samples: Age Groups of Fish from Length‐Frequency Data

We present a novel methodology for estimating the parameters of a finite mixture model (FMM) based on partially rank‐ordered set (PROS) sampling and use it in a fishery application. A PROS sampling design first selects a simple random sample of fish and creates partially rank‐ordered judgement subsets by dividing units into subsets of prespecified sizes. The final measurements are then obtained from these partially ordered judgement subsets. The traditional expectation–maximization algorithm is not directly applicable for these observations. We propose a suitable expectation–maximization algorithm to estimate the parameters of the FMMs based on PROS samples. We also study the problem of classification of the PROS sample into the components of the FMM. We show that the maximum likelihood estimators based on PROS samples perform substantially better than their simple random sample counterparts even with small samples. The results are used to classify a fish population using the length‐frequency data.     

Generalised quasi‐likelihood inference in a semi‐parametric binary dynamic mixed logit model

There exists a recent study where dynamic mixed‐effects regression models for count data have been extended to a semi‐parametric context. However, when one deals with other discrete data such as binary responses, the results based on count data models are not directly applicable. In this paper, we therefore begin with existing binary dynamic mixed models and generalise them to the semi‐parametric context. For inference, we use a new semi‐parametric conditional quasi‐likelihood (SCQL) approach for the estimation of the non‐parametric function involved in the semi‐parametric model, and a semi‐parametric generalised quasi‐likelihood (SGQL) approach for the estimation of the main regression, dynamic dependence and random effects variance parameters. A semi‐parametric maximum likelihood (SML) approach is also used as a comparison to the SGQL approach. The properties of the estimators are examined both asymptotically and empirically. More specifically, the consistency of the estimators is established and finite sample performances of the estimators are examined through an intensive simulation study.     

A novel weighted likelihood estimation with empirical Bayes flavor

We propose a novel approach to estimation, where a set of estimators of a parameter is combined into a weighted average to produce the final estimator. The weights are chosen to be proportional to the likelihood evaluated at the estimators. We investigate the method for a set of estimators obtained by using the maximum likelihood principle applied to each individual observation. The method can be viewed as a Bayesian approach with a data-driven prior distribution. We provide several examples illustrating the new method and argue for its consistency, asymptotic normality, and efficiency. We also conduct simulation studies to assess the performance of the estimators. This straightforward methodology produces consistent estimators comparable with those obtained by the maximum likelihood method. The method also approximates the distribution of the estimator through the “posterior” distribution.     

The role of reversals in order‐restricted inference

A statistical model is said to be an order‐restricted statistical model when its parameter takes its values in a closed convex cone C of the Euclidean space. In recent years, order‐restricted likelihood ratio tests and maximum likelihood estimators have been criticized on the grounds that they may violate a cone order monotonicity (COM) property, and hence reverse the cone order induced by C. The authors argue here that these reversals occur only in the case that C is an obtuse cone, and that in this case COM is an inappropriate requirement for likelihood‐based estimates and tests. They conclude that these procedures thus remain perfectly reasonable procedures for order‐restricted inference.     

Nonparametric Shrinkage Estimation for Aalen's Additive Hazards Model

Aalen's nonparametric additive model in which the regression coefficients are assumed to be unspecified functions of time is a flexible alternative to Cox's proportional hazards model when the proportionality assumption is in doubt. In this paper, we incorporate a general linear hypothesis into the estimation of the time‐varying regression coefficients. We combine unrestricted least squares estimators and estimators that are restricted by the linear hypothesis and produce James‐Stein‐type shrinkage estimators of the regression coefficients. We develop the asymptotic joint distribution of such restricted and unrestricted estimators and use this to study the relative performance of the proposed estimators via their integrated asymptotic distributional risks. We conduct Monte Carlo simulations to examine the relative performance of the estimators in terms of their integrated mean square errors. We also compare the performance of the proposed estimators with a recently devised LASSO estimator as well as with ridge‐type estimators both via simulations and data on the survival of primary billiary cirhosis patients.     

Robust-efficient fitting of mixed linear models: Methodology and theory

In many areas of application mixed linear models serve as a popular tool for analyzing highly complex data sets. For inference about fixed effects and variance components, likelihood-based methods such as (restricted) maximum likelihood estimators, (RE)ML, are commonly pursued. However, it is well-known that these fully efficient estimators are extremely sensitive to small deviations from hypothesized normality of random components as well as to other violations of distributional assumptions. In this article, we propose a new class of robust-efficient estimators for inference in mixed linear models. The new three-step estimation procedure provides truncated generalized least squares and variance components' estimators with hard-rejection weights adaptively computed from the data. More specifically, our data re-weighting mechanism first detects and removes within-subject outliers, then identifies and discards between-subject outliers, and finally it employs maximum likelihood procedures on the “clean” data. Theoretical efficiency and robustness properties of this approach are established.     

Shrinkage Estimation of the Memory Parameter in Stationary Gaussian Processes

The correct and efficient estimation of memory parameters in a stationary Gaussian processes is an important issue, since otherwise, forecasts based on the resulting time series would be misleading. On the other hand, if the memory parameters are suspected to fall in a smaller subspace through some hypothesis restrictions, it becomes a hard decision whether to use estimators based on the restricted spaces or to use unrestricted estimators over the full parameter space. In this article, we propose James-Stein-type estimators of the memory parameters of a stationary Gaussian times series process, which can efficiently incorporate the hypothetical restrictions. We show theoretically that the proposed estimators are more efficient than the usual unrestricted maximum likelihood estimators over the entire parameter space.     

Comment: The Future of Administrative Records in the Census Bureau's Demographic Activities

This article contains comments on “Bayesian Analysis of Stochastic Volatility Models,” by Jacquier, Poison, and Rossi. The Markov-chain Monte Carlo (MCMC) method proposed is compared empirically with a simulated maximum likelihood (SML) method. The MCMC and SML estimators yield very similar results, both when applied to actual data and in a Monte Carlo experiment.     

Estimation of a linear transformation and an associated distributional problem

A multivariate “errors in variables” regression model is proposed which generalizes a model previously considered by Gleser and Watson (1973). Maximum likelihood estimators [MLE's] for the parameters of this model are obtained, and the consistency properties of these estimators are investigated. Distribution of the MLE of the “error” variance is obtained in a simple case while the mean and the variance of the estimator are obtained in this case without appealing to the exact distribution.     

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.     

Improved two-parameter estimators for the negative binomial and Poisson regression models

Negative binomial regression (NBR) and Poisson regression (PR) applications have become very popular in the analysis of count data in recent years. However, if there is a high degree of relationship between the independent variables, the problem of multicollinearity arises in these models. We introduce new two-parameter estimators (TPEs) for the NBR and the PR models by unifying the two-parameter estimator (TPE) of Özkale and Kaç?ranlar [The restricted and unrestricted two-parameter estimators. Commun Stat Theory Methods. 2007;36:2707–2725]. These new estimators are general estimators which include maximum likelihood (ML) estimator, ridge estimator (RE), Liu estimator (LE) and contraction estimator (CE) as special cases. Furthermore, biasing parameters of these estimators are given and a Monte Carlo simulation is done to evaluate the performance of these estimators using mean square error (MSE) criterion. The benefits of the new TPEs are also illustrated in an empirical application. The results show that the new proposed TPEs for the NBR and the PR models are better than the ML estimator, the RE and the LE.     

Some aspects of maximum likelihood estimation of multistage cancer dose-response odels with application

Quantitative cancer dose-response models play an important role in cancer risk assessment. They also play a role in regulatory processes associated with potential occupational or environmental exposures. The multistage model is currently the most widely used cancer dose-response model. This paper describes the construction of the likelihood function in the special case of the multistage cancer dose-response models. The concavity of the likelihood function is also established. A criterion is developed to determine the degree of the polynomial portion of the multistage model. Finally, the restricted and unrestricted maximum likelihood estimators are considered and applied to some experimental data sets.     

Maximum Likelihood Estimations and EM Algorithms with Length-biased Data

Length-biased sampling has been well recognized in economics, industrial reliability, etiology applications, epidemiological, genetic and cancer screening studies. Length-biased right-censored data have a unique data structure different from traditional survival data. The nonparametric and semiparametric estimations and inference methods for traditional survival data are not directly applicable for length-biased right-censored data. We propose new expectation-maximization algorithms for estimations based on full likelihoods involving infinite dimensional parameters under three settings for length-biased data: estimating nonparametric distribution function, estimating nonparametric hazard function under an increasing failure rate constraint, and jointly estimating baseline hazards function and the covariate coefficients under the Cox proportional hazards model. Extensive empirical simulation studies show that the maximum likelihood estimators perform well with moderate sample sizes and lead to more efficient estimators compared to the estimating equation approaches. The proposed estimates are also more robust to various right-censoring mechanisms. We prove the strong consistency properties of the estimators, and establish the asymptotic normality of the semi-parametric maximum likelihood estimators under the Cox model using modern empirical processes theory. We apply the proposed methods to a prevalent cohort medical study. Supplemental materials are available online.     

Discriminant analysis of survey data

We consider the problem of the effect of sample designs on discriminant analysis. The selection of the learning sample is assumed to depend on the population values of auxiliary variables. Under a superpopulation model with a multivariate normal distribution, unbiasedness and consistency are examined for the conventional estimators (derived under the assumptions of simple random sampling), maximum likelihood estimators, probability-weighted estimators and conditionally unbiased estimators of parameters. Four corresponding sampled linear discriminant functions are examined. The rates of misclassification of these four discriminant functions and the effect of sample design on these four rates of misclassification are discussed. The performances of these four discriminant functions are assessed in a simulation study.     

PITMAN NEARNESS COMPARISONS OF ESTIMATES OF TWO ORDERED NORMAL MEANS

Maximum likelihood estimates of ordered means of two normal distributions having common variance have been shown to be better than the usual maximum likelihood estimates (i.e. corresponding sample means) with respect to Pitman Nearness criterion. The maximum likelihood estimate of common variance taking into consideration the order restriction of the means is shown to have smaller mean square error than the unrestricted maximum likelihood estimate of the common variance. These two estimators have also been compared with respect to Pitman Nearness criterion.     

Covariates in multipath change‐point problems: Modelling and consistency of the MLE

Although the single‐path change‐point problem has been extensively treated in the statistical literature, its multipath counterpart has largely been ignored. In the multipath change‐point setting, it is often of interest to assess the impact of covariates on the change point itself as well as on the parameters before and after the change point. This paper is concerned only with the inclusion of covariates in the change‐point distribution. This is achieved through the hazard of change. Maximum likelihood estimation is discussed and consistency of the maximum likelihood estimators established.     

ESTIMATION OF FAMILIAL CORRELATIONS UNDER AUTOREGRESSIVE CIRCULAR COVARIANCE

Circular covariance matrices play an important role in modeling phenomena in numerous epidemiological, communications and physical contexts. In this article, we propose a parsimonious, autoregressive type of circular covariance structure for modeling correlations between the “siblings” of a “family”. This structure, similar to AR(1) structure used in time series models, involves only two parameters. We derive the maximum likelihood estimators of these parameters, and discuss testing of hypotheses about the autoregressive parameter. Estimation of “parent-sib” correlation, namely, the interclass correlation, is also considered. Estimation of the parameters when there are unequal numbers of siblings in different families is also discussed.