On simultaneous confidence intervals for all contrasts in the means of the intraclass correlation model with missing data

In this paper, we consider simultaneous confidence intervals for all contrasts in the means when the observations are missing at random in the intraclass correlation model. An exact test statistic for the equality of the means and Scheffé, Bonferroni and Tukey types of simultaneous confidence intervals are given by an extension of Bhargava and Srivastava [On Tukey's confidence intervals for the contrasts in the means of the intraclass correlation model, J. Royal Statist. Soc. B35 (1973) 147-152] when the missing observations are of the monotone type. Finally, numerical results of simultaneous confidence intervals are presented.     

Exact inference on contrasts in means of intraclass correlation models with missing responses

Intraclass correlation models with missing data at random are considered. With a properly reduced model, a general method, which allows repeated observations with missing data in a non-monotone pattern, is proposed to construct exact test statistics and simultaneous confidence intervals for linear contrasts in the means. Simulation results are given to compare exact and asymptotic simultaneous confidence intervals. A real example is provided for the illustration of the proposed method.     

Exact confidence coefficients of simultaneous confidence intervals for multinomial proportions

Simultaneous confidence intervals for multinomial proportions are useful in many areas of science. Since 1964, approximate simultaneous 1-α confidence intervals have been proposed for multinomial proportions. Although at each point in the parameter space, these confidence sets have asymptotic 1-α coverage probability, the exact confidence coefficients of these simultaneous confidence intervals for a fixed sample size are unknown before.In this paper, we propose a procedure for calculating exact confidence coefficients for simultaneous confidence intervals of multinomial proportions for any fixed sample size. With this methodology, exact confidence coefficients can be clearly derived, and the point at which the infimum of the coverage probability occurs can be clearly identified.     

Oracally efficient spline smoothing of nonlinear additive autoregression models with simultaneous confidence band

Under weak conditions of smoothness and mixing, we propose spline-backfitted spline (SBS) estimators of the component functions for a nonlinear additive autoregression model that is both computationally expedient for analyzing high dimensional large time series data, and theoretically reliable as the estimator is oracally efficient and comes with asymptotically simultaneous confidence band. Simulation evidence strongly corroborates with the asymptotic theory.     

Statistical inference in a panel data semiparametric regression model with serially correlated errors

We consider a panel data semiparametric partially linear regression model with an unknown vector β of regression coefficients, an unknown nonparametric function g(·) for nonlinear component, and unobservable serially correlated errors. The correlated errors are modeled by a vector autoregressive process which involves a constant intraclass correlation. Applying the pilot estimators of β and g(·), we construct estimators of the autoregressive coefficients, the intraclass correlation and the error variance, and investigate their asymptotic properties. Fitting the error structure results in a new semiparametric two-step estimator of β, which is shown to be asymptotically more efficient than the usual semiparametric least squares estimator in terms of asymptotic covariance matrix. Asymptotic normality of this new estimator is established, and a consistent estimator of its asymptotic covariance matrix is presented. Furthermore, a corresponding estimator of g(·) is also provided. These results can be used to make asymptotically efficient statistical inference. Some simulation studies are conducted to illustrate the finite sample performances of these proposed estimators.     

Empirical likelihood inference in partially linear single-index models for longitudinal data

The empirical likelihood method is especially useful for constructing confidence intervals or regions of parameters of interest. Yet, the technique cannot be directly applied to partially linear single-index models for longitudinal data due to the within-subject correlation. In this paper, a bias-corrected block empirical likelihood (BCBEL) method is suggested to study the models by accounting for the within-subject correlation. BCBEL shares some desired features: unlike any normal approximation based method for confidence region, the estimation of parameters with the iterative algorithm is avoided and a consistent estimator of the asymptotic covariance matrix is not needed. Because of bias correction, the BCBEL ratio is asymptotically chi-squared, and hence it can be directly used to construct confidence regions of the parameters without any extra Monte Carlo approximation that is needed when bias correction is not applied. The proposed method can naturally be applied to deal with pure single-index models and partially linear models for longitudinal data. Some simulation studies are carried out and an example in epidemiology is given for illustration.     

The proportional hazards model for survey data from independent and clustered super-populations

Data from most complex surveys are subject to selection bias and clustering due to the sampling design. Results developed for a random sample from a super-population model may not apply. Ignoring the survey sampling weights may cause biased estimators and erroneous confidence intervals. In this paper, we use the design approach for fitting the proportional hazards (PH) model and prove formally the asymptotic normality of the sample maximum partial likelihood (SMPL) estimators under the PH model for both stochastically independent and clustered failure times. In the first case, we use the central limit theorem for martingales in the joint design-model space, and this enables us to obtain results for a general multistage sampling design under mild and easily verifiable conditions. In the case of clustered failure times, we require asymptotic normality in the sampling design space directly, and this holds for fewer sampling designs than in the first case. We also propose a variance estimator of the SMPL estimator. A key property of this variance estimator is that we do not have to specify the second-stage correlation model.     

Empirical likelihood for semiparametric regression model with missing response data

A bias-corrected technique for constructing the empirical likelihood ratio is used to study a semiparametric regression model with missing response data. We are interested in inference for the regression coefficients, the baseline function and the response mean. A class of empirical likelihood ratio functions for the parameters of interest is defined so that undersmoothing for estimating the baseline function is avoided. The existing data-driven algorithm is also valid for selecting an optimal bandwidth. Our approach is to directly calibrate the empirical log-likelihood ratio so that the resulting ratio is asymptotically chi-squared. Also, a class of estimators for the parameters of interest is constructed, their asymptotic distributions are obtained, and consistent estimators of asymptotic bias and variance are provided. Our results can be used to construct confidence intervals and bands for the parameters of interest. A simulation study is undertaken to compare the empirical likelihood with the normal approximation-based method in terms of coverage accuracies and average lengths of confidence intervals. An example for an AIDS clinical trial data set is used for illustrating our methods.     

Testing for positive evidence of equally likely outcomes

Goodness-of-fit tests allow one to conclude that k possible outcomes are not equally likely. In this paper, we develop an exact equivalence test that allows one to conclude that k possible outcomes are approximately equally likely. We show that the power properties of the test compare favorably to those of possible alternative tests, and we develop an associated simultaneous confidence interval procedure. We apply the test to data sets on the digits of π, winning roulette numbers, and winning numbers from the Pennsylvania Lottery.     

平衡单向分类模型中几种同时置信区间的比较

在平衡单向分类模型中,通过对对照的三种同时置信区间长度表达式的分析,可以看到,在不同情况下,三种区间的优劣.     

Estimation for parameters of interest in random effects growth curve models

In this paper, we consider the general growth curve model with multivariate random effects covariance structure and provide a new simple estimator for the parameters of interest. This estimator is not only convenient for testing the hypothesis on the corresponding parameters, but also has higher efficiency than the least-square estimator and the improved two-stage estimator obtained by Rao under certain conditions. Moreover, we obtain the necessary and sufficient condition for the new estimator to be identical to the best linear unbiased estimator. Examples of its application are given.     

On the comparison of the pre-test and shrinkage phi-divergence test estimators for the symmetry model of categorical data

The estimation problem of the parameters in a symmetry model for categorical data has been considered for many authors in the statistical literature (for example, Bowker (1948) [1], Ireland et al. (1969) [2], Quade and Salama (1975) [3], Cressie and Read (1988) [4], Menéndez et al. (2005) [5]) without using uncertain prior information. It is well known that many new and interesting estimators, using uncertain prior information, have been studied by a host of researchers in different statistical models, and many papers have been published on this topic (see Saleh (2006) [9] and references therein). In this paper, we consider the symmetry model of categorical data and we study, for the first time, some new estimators when non-sample information about the symmetry of the probabilities is considered. The decision to use a “restricted” estimator or an “unrestricted” estimator is based on the outcome of a preliminary test, and then a shrinkage technique is used. It is interesting to note that we present a unified study in the sense that we consider not only the maximum likelihood estimator and likelihood ratio test or chi-square test statistic but we consider minimum phi-divergence estimators and phi-divergence test statistics. Families of minimum phi-divergence estimators and phi-divergence test statistics are wide classes of estimators and test statistics that contain as a particular case the maximum likelihood estimator, likelihood ratio test and chi-square test statistic. In an asymptotic set-up, the biases and the risk under the squared loss function for the proposed estimators are derived and compared. A numerical example clarifies the content of the paper.     

Fiducial inference on the largest mean of a multivariate normal distribution

Inference on the largest mean of a multivariate normal distribution is a surprisingly difficult and unexplored topic. Difficulties arise when two or more of the means are simultaneously the largest mean. Our proposed solution is based on an extension of R.A. Fisher’s fiducial inference methods termed generalized fiducial inference. We use a model selection technique along with the generalized fiducial distribution to allow for equal largest means and alleviate the overestimation that commonly occurs. Our proposed confidence intervals for the largest mean have asymptotically correct frequentist coverage and simulation results suggest that they possess promising small sample empirical properties. In addition to the theoretical calculations and simulations we also applied this approach to the air quality index of the four largest cities in the northeastern United States (Baltimore, Boston, New York, and Philadelphia).     

Parameter estimation under ambiguity and contamination with the spurious model

Recently, we proposed variants as a statistical model for treating ambiguity. If data are extracted from an object with a machine then it might not be able to give a unique safe answer due to ambiguity about the correct interpretation of the object. On the other hand, the machine is often able to produce a finite number of alternative feature sets (of the same object) that contain the desired one. We call these feature sets variants of the object. Data sets that contain variants may be analyzed by means of statistical methods and all chapters of multivariate analysis can be seen in the light of variants. In this communication, we focus on point estimation in the presence of variants and outliers. Besides robust parameter estimation, this task requires also selecting the regular objects and their valid feature sets (regular variants). We determine the mixed MAP-ML estimator for a model with spurious variants and outliers as well as estimators based on the integrated likelihood. We also prove asymptotic results which show that the estimators are nearly consistent.The problem of variant selection turns out to be computationally hard; therefore, we also design algorithms for efficient approximation. We finally demonstrate their efficacy with a simulated data set and a real data set from genetics.     

Multiple imputations and the missing censoring indicator model

Semiparametric random censorship (SRC) models (Dikta, 1998) provide an attractive framework for estimating survival functions when censoring indicators are fully or partially available. When there are missing censoring indicators (MCIs), the SRC approach employs a model-based estimate of the conditional expectation of the censoring indicator given the observed time, where the model parameters are estimated using only the complete cases. The multiple imputations approach, on the other hand, utilizes this model-based estimate to impute the missing censoring indicators and form several completed data sets. The Kaplan-Meier and SRC estimators based on the several completed data sets are averaged to arrive at the multiple imputations Kaplan-Meier (MIKM) and the multiple imputations SRC (MISRC) estimators. While the MIKM estimator is asymptotically as efficient as or less efficient than the standard SRC-based estimator that involves no imputations, here we investigate the performance of the MISRC estimator and prove that it attains the benchmark variance set by the SRC-based estimator. We also present numerical results comparing the performances of the estimators under several misspecified models for the above mentioned conditional expectation.     

Stein's phenomenon in estimation of means restricted to a polyhedral convex cone

This paper treats the problem of estimating the restricted means of normal distributions with a known variance, where the means are restricted to a polyhedral convex cone which includes various restrictions such as positive orthant, simple order, tree order and umbrella order restrictions. In the context of the simultaneous estimation of the restricted means, it is of great interest to investigate decision-theoretic properties of the generalized Bayes estimator against the uniform prior distribution over the polyhedral convex cone. In this paper, the generalized Bayes estimator is shown to be minimax. It is also proved that it is admissible in the one- or two-dimensional case, but is improved on by a shrinkage estimator in the three- or more-dimensional case. This means that the so-called Stein phenomenon on the minimax generalized Bayes estimator can be extended to the case where the means are restricted to the polyhedral convex cone. The risk behaviors of the estimators are investigated through Monte Carlo simulation, and it is revealed that the shrinkage estimator has a substantial risk reduction.     

A class of proper priors for Bayesian simultaneous prediction of independent Poisson observables

Simultaneous prediction and parameter inference for the independent Poisson observables model are considered. A class of proper prior distributions for Poisson means is introduced. Bayesian predictive densities and estimators based on priors in the introduced class dominate the Bayesian predictive density and estimator based on the Jeffreys prior under Kullback-Leibler loss.     

The restricted EM algorithm under inequality restrictions on the parameters

One of the most powerful algorithms for maximum likelihood estimation for many incomplete-data problems is the EM algorithm. The restricted EM algorithm for maximum likelihood estimation under linear restrictions on the parameters has been handled by Kim and Taylor (J. Amer. Statist. Assoc. 430 (1995) 708-716). This paper proposes an EM algorithm for maximum likelihood estimation under inequality restrictions A₀β?0, where β is the parameter vector in a linear model W=Xβ+ε and ε is an error variable distributed normally with mean zero and a known or unknown variance matrix Σ>0. Some convergence properties of the EM sequence are discussed. Furthermore, we consider the consistency of the restricted EM estimator and a related testing problem.     

Simultaneous estimation of Poisson means

In the simultaneous estimation of means from independent Poisson distributions, an estimator is developed which incorporates a prior mean and variance for each Poisson mean estimated. This estimator possesses substantially smaller risk than the usual estimator in a region of the parameter space and seems superior to other estimators proposed to estimate p Poisson means. It is indicated through two asymptotic results that, unlike the conjugate Bayes estimator, the risk of the estimator does not greatly exceed the risk of the usual estimator outside of the region of risk improvement.     

Evaluation of reproducibility for paired functional data

Evaluation of reproducibility is important in assessing whether a new method or instrument can reproduce the results from a traditional gold standard approach. In this paper, we propose a measure to assess measurement agreement for functional data which are frequently encountered in medical research and many other research fields. Formulae to compute the standard error of the proposed estimator and confidence intervals for the proposed measure are derived. The estimators and the coverage probabilities of the confidence intervals are empirically tested for small-to-moderate sample sizes via Monte Carlo simulations. A real data example in physiology study is used to illustrate the proposed statistical inference procedures.