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The three-parameter cluster model is a combinatorial stochastic process that generates categorical response sequences by randomly perturbing a fixed clustering parameter. This clear relationship between the observed data and the underlying clustering is particularly attractive in cluster analysis, in which supervised learning is a common goal and missing data is a familiar issue. The model is well equipped for this task, as it can handle missing data, perform out-of-sample inference, and accommodate both independent and dependent data sequences. Moreover, its clustering parameter lies in the unrestricted space of partitions, so that the number of clusters need not be specified beforehand. We establish these and other theoretical properties and also demonstrate the model on datasets from epidemiology, genetics, political science, and legal studies. [ABSTRACT FROM AUTHOR]

This article introduces false discovery rate regression, a method for incorporating covariate information into large-scale multiple-testing problems. FDR regression estimates a relationship between test-level covariates and the prior probability that a given observation is a signal. It then uses this estimated relationship to inform the outcome of each test in a way that controls the overall false discovery rate at a prespecified level. This poses many subtle issues at the interface between inference and computation, and we investigate several variations of the overall approach. Simulation evidence suggests that: (1) when covariate effects are present, FDR regression improves power for a fixed false-discovery rate; and (2) when covariate effects are absent, the method is robust, in the sense that it does not lead to inflated error rates. We apply the method to neural recordings from primary visual cortex. The goal is to detect pairs of neurons that exhibit fine-time-scale interactions, in the sense that they fire together more often than expected due to chance. Our method detects roughly 50% more synchronous pairs versus a standard FDR-controlling analysis. The companion R package FDRreg implements all methods described in the article. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

Envelopes were recently proposed by Cook, Li and Chiaromonte as a method for reducing estimative and predictive variations in multivariate linear regression. We extend their formulation, proposing a general definition of an envelope and a general framework for adapting envelope methods to any estimation procedure. We apply the new envelope methods to weighted least squares, generalized linear models and Cox regression. Simulations and illustrative data analysis show the potential for envelope methods to significantly improve standard methods in linear discriminant analysis, logistic regression and Poisson regression. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

We consider a specific situation of correlated data where multiple outcomes are repeatedly measured on each member of a couple. Such multivariate longitudinal data from couples may exhibit multi-faceted correlations that can be further complicated if there are polygamous partnerships. An example is data from cohort studies on human papillomavirus (HPV) transmission dynamics in heterosexual couples. HPV is a common sexually transmitted disease with 14 known oncogenic types causing anogenital cancers. The binary outcomes on the multiple types measured in couples over time may introduce inter-type, intra-couple, and temporal correlations. Simple analysis using generalized estimating equations or random effects models lacks interpretability and cannot fully use the available information. We developed a hybrid modeling strategy using Markov transition models together with pairwise composite likelihood for analyzing such data. The method can be used to identify risk factors associated with HPV transmission and persistence, estimate difference in risks between male-to-female and female-to-male HPV transmission, compare type-specific transmission risks within couples, and characterize the inter-type and intra-couple associations. Applying the method to HPV couple data collected in a Ugandan male circumcision (MC) trial, we assessed the effect of MC and the role of gender on risks of HPV transmission and persistence. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

This article shows that a probabilistic version of the classical forward-stepwise variable inclusion procedure can serve as a general data-augmentation scheme for model space distributions in (generalized) linear models. This latent variable representation takes the form of a Markov process, thereby allowing information propagation algorithms to be applied for sampling from model space posteriors. In particular, We propose a sequential Monte Carlo method for achieving effective unbiased Bayesian model averaging in high-dimensional problems, using proposal distributions constructed using local information propagation. The method—called LIPS for local information propagation based sampling—is illustrated using real and simulated examples with dimensionality ranging from 15 to 1000, and its performance in estimating posterior inclusion probabilities and in out-of-sample prediction is compared to those of several other methods—namely, MCMC, BAS, iBMA, and LASSO. In addition, it is shown that the latent variable representation can also serve as a modeling tool for specifying model space priors that account for knowledge regarding model complexity and conditional inclusion relationships. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

This article develops a systematic procedure of statistical inference for the auto-regressive moving average (ARMA) model with unspecified and heavy-tailed heteroscedastic noises. We first investigate the least absolute deviation estimator (LADE) and the self-weighted LADE for the model. Both estimators are shown to be strongly consistent and asymptotically normal when the noise has a finite variance and infinite variance, respectively. The rates of convergence of the LADE and the self-weighted LADE aren− 1/2, which is faster than those of least-square estimator (LSE) for the ARMA model when the tail index of generalized auto-regressive conditional heteroskedasticity (GARCH) noises is in (0, 4], and thus they are more efficient in this case. Since their asymptotic covariance matrices cannot be estimated directly from the sample, we develop the random weighting approach for statistical inference under this nonstandard case. We further propose a novel sign-based portmanteau test for model adequacy. Simulation study is carried out to assess the performance of our procedure and one real illustrating example is given. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

The kernel density estimator (KDE) suffers boundary biases when applied to densities on bounded supports, which are assumed to be the unit interval. Transformations mapping the unit interval to the real line can be used to remove boundary biases. However, this approach may induce erratic tail behaviors when the estimated density of transformed data is transformed back to its original scale. We propose a modified, transformation-based KDE that employs a tapered and tilted back-transformation. We derive the theoretical properties of the new estimator and show that it asymptotically dominates the naive transformation based estimator while maintains its simplicity. We then propose three automatic methods of smoothing parameter selection. Our Monte Carlo simulations demonstrate the good finite sample performance of the proposed estimator, especially for densities with poles near the boundaries. An example with real data is provided. [ABSTRACT FROM AUTHOR]

Genomic imprinting is a known aspect of the etiology of many diseases. The imprinting phenomenon depicts differential expression levels of the allele depending on its parental origin. When the parental origin is unknown, the expression level has a finite normal mixture distribution. In such applications, a random sample of expression levels consists of three subsamples according to the number of minor alleles an individual possesses, of which one is the mixture and the other two are homogeneous. This understanding leads to a likelihood ratio test (LRT) for the presence of imprinting. Because of the nonregularity of the finite mixture model, the classical asymptotic conclusions on likelihood-based inference are not applicable. We show that the maximum likelihood estimator of the mixing distribution remains consistent. More interestingly, thanks to the homogeneous subsamples, the LRT statistic has an elegant and rather distinct 0.5χ21+ 0.5χ22null limiting distribution. Simulation studies confirm that the limiting distribution provides precise approximations of the finite sample distributions under various parameter settings. The LRT is applied to expression data. Our analyses provide evidence for imprinting for a number of isoform expressions. [ABSTRACT FROM AUTHOR]

Dynamic treatment regimes (DTRs) are sequential decision rules for individual patients that can adapt over time to an evolving illness. The goal is to accommodate heterogeneity among patients and find the DTR which will produce the best long-term outcome if implemented. We introduce two new statistical learning methods for estimating the optimal DTR, termed backward outcome weighted learning (BOWL), and simultaneous outcome weighted learning (SOWL). These approaches convert individualized treatment selection into an either sequential or simultaneous classification problem, and can thus be applied by modifying existing machine learning techniques. The proposed methods are based on directly maximizing over all DTRs a nonparametric estimator of the expected long-term outcome; this is fundamentally different than regression-based methods, for example,Q-learning, which indirectly attempt such maximization and rely heavily on the correctness of postulated regression models.  We prove that the resulting rules are consistent, and provide finite sample bounds for the errors using the estimated rules. Simulation results suggest the proposed methods produce superior DTRs compared withQ-learning especially in small samples. We illustrate the methods using data from a clinical trial for smoking cessation. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

Diagnostic classification models (DMCs) have recently gained prominence in educational assessment, psychiatric evaluation, and many other disciplines. Central to the model specification is the so-calledQ-matrix that provides a qualitative specification of the item-attribute relationship. In this article, we develop theories on the identifiability for theQ-matrix under the DINA and the DINO models. We further propose an estimation procedure for theQ-matrix through the regularized maximum likelihood. The applicability of this procedure is not limited to the DINA or the DINO model and it can be applied to essentially allQ-matrix based DMCs. Simulation studies show that the proposed method admits high probability recovering the trueQ-matrix. Furthermore, two case studies are presented. The first case is a dataset on fraction subtraction (educational application) and the second case is a subsample of the National Epidemiological Survey on Alcohol and Related Conditions concerning the social anxiety disorder (psychiatric application). [ABSTRACT FROM AUTHOR]

We develop a test statistic for testing the equality of two population mean vectors in the “large-p-small-n” setting. Such a test must surmount the rank-deficiency of the sample covariance matrix, which breaks down the classic HotellingT2test. The proposed procedure, called the generalized component test, avoids full estimation of the covariance matrix by assuming that thepcomponents admit a logical ordering such that the dependence between components is related to their displacement. The test is shown to be competitive with other recently developed methods under ARMA and long-range dependence structures and to achieve superior power for heavy-tailed data. The test does not assume equality of covariance matrices between the two populations, is robust to heteroscedasticity in the component variances, and requires very little computation time, which allows its use in settings with very largep. An analysis of mitochondrial calcium concentration in mouse cardiac muscles over time and of copy number variations in a glioblastoma multiforme dataset from The Cancer Genome Atlas are carried out to illustrate the test. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

The regression problem involving functional predictors has many important applications and a number of functional regression methods have been developed. However, a common complication in functional data analysis is one of sparsely observed curves, that is predictors that are observed, with error, on a small subset of the possible time points. Such sparsely observed data induce anerrors-in-variablesmodel, where one must account for measurement error in the functional predictors. Faced with sparsely observed data, most current functional regression methods simply estimate the unobserved predictors and treat them as fully observed; thus failing to account for the extra uncertainty from the measurement error. We propose a new functional errors-in-variables approach, sparse index model functional estimation (SIMFE), which uses a functional index model formulation to deal with sparsely observed predictors. SIMFE has several advantages over more traditional methods. First, the index model implements a nonlinear regression and uses an accurate supervised method to estimate the lower dimensional space into which the predictors should be projected. Second, SIMFE can be applied to both scalar and functional responses and multiple predictors. Finally, SIMFE uses a mixed effects model to effectively deal with very sparsely observed functional predictors and to correctly model the measurement error. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

In recent years, the ultrahigh-dimensional linear regression problem has attracted enormous attention from the research community. Under the sparsity assumption, most of the published work is devoted to the selection and estimation of the predictor variables with nonzero coefficients. This article studies a different but fundamentally important aspect of this problem: uncertainty quantification for parameter estimates and model choices. To be more specific, this article proposes methods for deriving a probability density function on the set of all possible models, and also for constructing confidence intervals for the corresponding parameters. These proposed methods are developed using the generalized fiducial methodology, which is a variant of Fisher’s controversial fiducial idea. Theoretical properties of the proposed methods are studied, and in particular it is shown that statistical inference based on the proposed methods will have correct asymptotic frequentist property. In terms of empirical performance, the proposed methods are tested by simulation experiments and an application to a real dataset. Finally, this work can also be seen as an interesting and successful application of Fisher’s fiducial idea to an important and contemporary problem. To the best of the authors’ knowledge, this is the first time that the fiducial idea is being applied to a so-called “largepsmalln” problem. A connection to objective Bayesian model selection is also discussed. [ABSTRACT FROM AUTHOR]

Motivated by the problem of selecting representative portfolios for backtesting counterparty credit risks, we propose a matching quantiles estimation (MQE) method for matching a target distribution by that of a linear combination of a set of random variables. An iterative procedure based on the ordinary least-squares estimation (OLS) is proposed to compute MQE. MQE can be easily modified by adding a LASSO penalty term if a sparse representation is desired, or by restricting the matching within certain range of quantiles to match a part of the target distribution. The convergence of the algorithm and the asymptotic properties of the estimation, both with or without LASSO, are established. A measure and an associated statistical test are proposed to assess the goodness-of-match. The finite sample properties are illustrated by simulation. An application in selecting a counterparty representative portfolio with a real dataset is reported. The proposed MQE also finds applications in portfolio tracking, which demonstrates the usefulness of combining MQE with LASSO. [ABSTRACT FROM AUTHOR]

This article examines the problem of estimation in a quantile regression model when observations are missing at random under independent and nonidentically distributed errors. We consider three approaches of handling this problem based on nonparametric inverse probability weighting, estimating equations projection, and a combination of both. An important distinguishing feature of our methods is their ability to handle missing response and/or partially missing covariates, whereas existing techniques can handle only one or the other, but not both. We prove that our methods yield asymptotically equivalent estimators that achieve the desirable asymptotic properties of unbiasedness, normality, and-consistency. Because we do not assume that the errors are identically distributed, our theoretical results are valid under heteroscedasticity, a particularly strong feature of our methods. Under the special case of identical error distributions, all of our proposed estimators achieve the semiparametric efficiency bound. To facilitate the practical implementation of these methods, we develop an iterative method based on the majorize/minimize algorithm for computing the quantile regression estimates, and a bootstrap method for computing their variances. Our simulation findings suggest that all three methods have good finite sample properties. We further illustrate these methods by a real data example. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

This article proposes a phase I/II clinical trial design for adaptively and dynamically optimizing each patient’s dose in each of two cycles of therapy based on the joint binary efficacy and toxicity outcomes in each cycle. A dose-outcome model is assumed that includes a Bayesian hierarchical latent variable structure to induce association among the outcomes and also facilitate posterior computation. Doses are chosen in each cycle based on posteriors of a model-based objective function, similar to a reinforcement learning or Q-learning function, defined in terms of numerical utilities of the joint outcomes in each cycle. For each patient, the procedure outputs a sequence of two actions, one for each cycle, with each action being the decision to either treat the patient at a chosen dose or not to treat. The cycle 2 action depends on the individual patient’s cycle 1 dose and outcomes. In addition, decisions are based on posterior inference using other patients’ data, and therefore, the proposed method is adaptive both within and between patients. A simulation study of the method is presented, including comparison to two-cycle extensions of the conventional 3 + 3 algorithm, continual reassessment method, and a Bayesian model-based design, and evaluation of robustness. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

Survival models such as the Weibull or log-normal lead to inference that is not robust to the presence of outliers. They also assume that all heterogeneity between individuals can be modeled through covariates. This article considers the use of infinite mixtures of lifetime distributions as a solution for these two issues. This can be interpreted as the introduction of a random effect in the survival distribution. We introduce the family of shape mixtures of log-normal distributions, which covers a wide range of density and hazard functions. Bayesian inference under nonsubjective priors based on the Jeffreys’ rule is examined and conditions for posterior propriety are established. The existence of the posterior distribution on the basis of a sample of point observations is not always guaranteed and a solution through set observations is implemented. In addition, we propose a method for outlier detection based on the mixture structure. A simulation study illustrates the performance of our methods under different scenarios and an application to a real dataset is provided. Supplementary materials for the article, which include R code, are available online. [ABSTRACT FROM AUTHOR]

Covariate measurement imprecision or errors arise frequently in many areas. It is well known that ignoring such errors can substantially degrade the quality of inference or even yield erroneous results. Although in practice both covariates subject to measurement error and covariates subject to misclassification can occur, research attention in the literature has mainly focused on addressing either one of these problems separately. To fill this gap, we develop estimation and inference methods that accommodate both characteristics simultaneously. Specifically, we consider measurement error and misclassification in generalized linear models under the scenario that an external validation study is available, and systematically develop a number of effective functional and structural methods. Our methods can be applied to different situations to meet various objectives. [ABSTRACT FROM AUTHOR]

Covariate-adaptive designs are often implemented to balance important covariates in clinical trials. However, the theoretical properties of conventional testing hypotheses are usually unknown under covariate-adaptive randomized clinical trials. In the literature, most studies are based on simulations. In this article, we provide theoretical foundation of hypothesis testing under covariate-adaptive designs based on linear models. We derive the asymptotic distributions of the test statistics of testing both treatment effects and the significance of covariates under null and alternative hypotheses. Under a large class of covariate-adaptive designs, (i) the hypothesis testing to compare treatment effects is usually conservative in terms of small Type I error; (ii) the hypothesis testing to compare treatment effects is usually more powerful than complete randomization; and (iii) the hypothesis testing for significance of covariates is still valid. The class includes most of the covariate-adaptive designs in the literature; for example, Pocock and Simon’s marginal procedure, stratified permuted block design, etc. Numerical studies are also performed to assess their corresponding finite sample properties. Supplementary material for this article is available online. [ABSTRACT FROM AUTHOR]

We propose a new nonparametric procedure (referred to as MuBreD) for the detection and estimation of multiple structural breaks in the autocovariance function of a multivariate (second-order) piecewise stationary process, which also identifies the components of the series where the breaks occur. MuBreD is based on a comparison of the estimated spectral distribution on different segments of the observed time series and consists of three steps: it starts with a consistent test, which allows us to prove the existence of structural breaks at a controlled Type I error. Second, it estimates sets containing possible break points and finally these sets are reduced to identify the relevant structural breaks and corresponding components which are responsible for the changes in the autocovariance structure. In contrast to all other methods proposed in the literature, our approach does not make any parametric assumptions, is not especially designed for detecting one single change point, and addresses the problem of multiple structural breaks in the autocovariance function directly with no use of the binary segmentation algorithm. We prove that the new procedure detects all components and the corresponding locations where structural breaks occur with probability converging to one as the sample size increases and provide data-driven rules for the selection of all regularization parameters. The results are illustrated by analyzing financial asset returns, and in a simulation study it is demonstrated that MuBreD outperforms the currently available nonparametric methods for detecting breaks in the dependency structure of multivariate time series. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

K-sample testing problems arise in many scientific applications and have attracted statisticians’ attention for many years. We propose an omnibus nonparametric method based on an optimal discretization (aka “slicing”) of continuous random variables in the test. The novelty of our approach lies in the inclusion of a term penalizing the number of slices (i.e., the resolution of the discretization) so as to regularize the corresponding likelihood-ratio test statistic. An efficient dynamic programming algorithm is developed to determine the optimal slicing scheme. Asymptotic and finite-sample properties such as power and null distribution of the resulting test statistic are studied. We compare the proposed testing method with some existing well-known methods and demonstrate its statistical power through extensive simulation studies as well as a real data example. A dynamic slicing method for the one-sample testing problem is further developed and studied under the same framework. Supplementary materials including technical derivations and proofs are available online. [ABSTRACT FROM AUTHOR]

This work is concerned with marginal sure independence feature screening for ultrahigh dimensional discriminant analysis. The response variable is categorical in discriminant analysis. This enables us to use the conditional distribution function to construct a new index for feature screening. In this article, we propose a marginal feature screening procedure based on empirical conditional distribution function. We establish the sure screening and ranking consistency properties for the proposed procedure without assuming any moment condition on the predictors. The proposed procedure enjoys several appealing merits. First, it is model-free in that its implementation does not require specification of a regression model. Second, it is robust to heavy-tailed distributions of predictors and the presence of potential outliers. Third, it allows the categorical response having a diverging number of classes in the order ofO(nκ) with some κ ⩾ 0. We assess the finite sample property of the proposed procedure by Monte Carlo simulation studies and numerical comparison. We further illustrate the proposed methodology by empirical analyses of two real-life datasets. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

Clustered failure times often arise from studies with stratified sampling designs where it is desired to reduce both cost and sampling error. Semiparametric accelerated failure time (AFT) models have not been used as frequently as Cox relative risk models in such settings due to lack of efficient and reliable computing routines for inferences. The challenge roots in the nonsmoothness of the rank-based estimating functions, and for clustered data, the asymptotic properties of the estimator from the weighted version have not been available. The recently proposed induced smoothing approach, which provides fast and accurate rank-based inferences for AFT models, is generalized to incorporate weights to accommodate stratified sampling designs. The estimator from the induced smoothing weighted estimating equations are shown to be consistent and have the same asymptotic distribution as that from the nonsmooth version, which has not been developed before. The variance of the estimator is estimated by computationally efficient sandwich estimators aided by a multiplier bootstrap. The proposed method is assessed in extensive simulation studies where the estimators appear to provide valid and efficient inferences. A stratified case-cohort design with clustered times to tooth extraction in a dental study illustrates the usefulness of the method. [ABSTRACT FROM AUTHOR]

In this article, we consider bootstrapping the Lasso estimator of the regression parameter in a multiple linear regression model. It is known that the standard bootstrap method fails to be consistent. Here, we propose a modified bootstrap method, and show that it provides valid approximation to the distribution of the Lasso estimator, for all possible values of the unknown regression parameter vector, including the case where some of the components are zero. Further, we establish consistency of the modified bootstrap method for estimating the asymptotic bias and variance of the Lasso estimator. We also show that the residual bootstrap can be used to consistently estimate the distribution and variance of the adaptive Lasso estimator. Using the former result, we formulate a novel data-based method for choosing the optimal penalizing parameter for the Lasso using the modified bootstrap. A numerical study is performed to investigate the finite sample performance of the modified bootstrap. The methodology proposed in the article is illustrated with a real data example. [ABSTRACT FROM AUTHOR]

Principal components analysis (PCA) is a classic method for the reduction of dimensionality of data in the form of n observations (or cases) of a vector with p variables. Contemporary datasets often have p comparable with or even much larger than n. Our main assertions, in such settings, are (a) that some initial reduction in dimensionality is desirable before applying any PCA-type search for principal modes, and (b) the initial reduction in dimensionality is best achieved by working in a basis in which the signals have a sparse representation. We describe a simple asymptotic model in which the estimate of the leading principal component vector via standard PCA is consistent if and only if p(n)/n → 0. We provide a simple algorithm for selecting a subset of coordinates with largest sample variances, and show that if PCA is done on the selected subset, then consistency is recovered, even if p(n) ⪢ n. [ABSTRACT FROM AUTHOR]

Microarray technology allows a scientist to study genomewide patterns of gene expression. Thousands of individual genes are measured with a relatively small number of replications, which poses challenges to traditional statistical methods. In particular, the gone-specific estimators of variances are not reliable and gene-by-gene tests have low powers. In this article we propose a family of shrinkage estimators for variances raised to a fixed power. We derive optimal shrinkage parameters under both Stein and squared loss functions. Our results show that the standard sample variance is inadmissible under either loss function. We propose several estimators for the optimal shrinkage parameters and investigate their asymptotic properties under two scenarios: large number of replications and large number of genes. We conduct simulations to evaluate the finite sample performance of the data-driven optimal shrinkage estimators and compare them with some existing methods. We construct F-like statistics using these shrinkage variance estimators and apply them to detect differentially expressed genes in a microarray experiment. We also conduct simulations to evaluate performance of these F-like statistics and compare them with some existing methods. [ABSTRACT FROM AUTHOR]

We present two improvements on the technique of importance sampling. First, we show that importance sampling from a mixture of densities, using those densities as control variates, results in a useful upper bound on the asymptotic variance. That bound is a small multiple of the asymptotic variance of importance sampling from the best single component density. This allows one to benefit from the great variance reductions obtainable by importance sampling, while protecting against the equally great variance increases that might take the practitioner by surprise. The second improvement is to show how importance sampling from two or more densities can be used to approach a zero sampling variance even for integrands that take both positive and negative values. [ABSTRACT FROM AUTHOR]

This article reports on the proposal for a Bayesian sequential two-phase sampling design to accommodate applications where conventional two-phase sampling cannot be used. Researchers first derive the optimal, the optimal myopic, and several approximate optimal myopic sequential decision procedures for subsampling a first-phase sample. Then they compared the procedure to that of the optimal (nonsequential) conventional two-phase sample design. In our application, the objective is inference about the age distribution of a population of fish using data on covariates like lengths and weight obtained from all members of the first-phase sample and data on age obtained from the fish in the second-phase subsample. The routine monitoring of commercial fish stocks often requires estimates of the age distribution for modeling and forecasting. Typically, an initial random sample of fish is selected from either a commercial catch or research vessel catch, and these fish are assigned to strata, defined by length or weight measurements.

Establishment surveys based on list frames often use stratified random sampling with a small number of strata, H, and relatively large sample sizes, nhr within strata. For such surveys, a grouped balanced half-sample (GBHS) method is often used for variance estimation and for construction of confidence intervals on population parameters of interest. In this method the sample in each stratum is first randomly divided into two groups, and then the balanced half-sample (BHS) method is applied to the groups. We show that the OBHS method leads to asymptotically incorrect inferences as the strata sample sizes nh → &inifiin; with H fixed. To overcome this difficulty, we propose a repeatedly grouped balanced half-sample (RGBHS) method, which essentially involves independently repeating the grouping T times and then taking the average of the resulting T GBHS variance estimators. This method retains the simplicity of the CBHS method. We establish its asymptotic validity as min nh → &inifin; and T → ∞. We also study an alternative method by forming substrata within each stratum, consisting of a pair of sampling units, and then applying the BHS method on the total set of substrata, treating them as strata. We establish its asymptotic validity as min nh → ∞. We provide simulation results on the finite-sample properties of the GBHS, RGBHS, the jackknife, and the alternative BHS method. Our results indicate that the proposed RGBHS method performs well for T as small as 15, thus providing flexibility in terms of the number of half-samples used. The alternative BHS method has also performed well in the simulation study. [ABSTRACT FROM AUTHOR]

Fitting local polynomials in nonparametric regression has a number of advantages. The attractive theoretical features are in a partial contradiction to variance properties for random design and to practical experience over a broad range of situations. No upper bound can be given for the conditional variance. The unconditional variance is infinite when using optimal weights with compact support. Properties are better for Gaussian weights. We analyze local polynomials for finite sample size, both theoretically and numerically. It turns out that difficulties arise in sparse regions in the realization, of the design, when the realization has locally a small variance and/or a skew empirical distribution. Two small-sample modifications of local polynomials are presented: local increase of bandwidth in sparse regions of the design, and local polynomial ridge regression. Both modifications combine a good finite-sample behavior with the asymptotic advantages of local polynomials. [ABSTRACT FROM AUTHOR]

The present article examines the robustness of sampling strategies, that is, estimators and sample designs, from the point of view of attaining the minimum value of the Godambe-Joshi lower bound to the expected variance of a strategy. It is shown that asymptotic design unbiasedness is necessary but insufficient for robustness defined in the above sense. The concepts of weak and strong robustness of estimators, for given sampling strategies, are introduced. The article establishes that a sufficient condition for a sampling strategy to be robust is that the estimator is model unbiased and weakly robust. Sufficient conditions are also given for an estimator to be weakly robust. These conditions are then used to examine the robustness of strategies when the design matrix or the covariance matrix of the superpopulation model is misspecified. The robustness of model-based predictors is also examined. [ABSTRACT FROM AUTHOR]

This article examines a methodology to improve the efficiency of surveys to locate and estimate characteristics of rare populations. Two multiplicity counting rules are compared with a conventional rule in terms of reporting errors, sample variances, and survey yields. The analysis is based on data from two survey field experiments with diagnosed cancer patients and a subsample of their siblings or children who reside outside their households. The article concludes with a brief discussion of the circumstances in which a multiplicity counting rule is the preferred methodology and those in which other approaches should be pursued. [ABSTRACT FROM AUTHOR]

In standard Bayes and empirical Bayes component decision problems, estimating inidividual parameters is the primary goal. In multiple comparison problems and in comparisons of histograms of estimates, however, the primary goal is to produce parameter estimates that can be considered as an ensemble. For example, the histogram of estimates should be a good estimate of the histogram of parameters. Standard methods of estimating by the posterior expectation do minimize symmetric, componentwise losses such as squared error, but they produce ensembles of estimates with a sample variance smaller than the posterior expected sample variance for parameters. In this article we propose new Bayes and empirical Bayes estimates that minimize a distance function between the empirical cdf of the estimates and the true parameters. These estimators are weighted averages of the prior mean and the data, with weight on the data being approximately the square root of that for the posterior expectation. We give theoretical and applied examples, including subgroup analysis in a clinical trial. [ABSTRACT FROM AUTHOR]

A "t"-like statistic, replacing the classical mean by a biweight location estimator in the numerator and the sample variance by a corresponding variance term in the denominator, is proposed as a modification to that used by Gross (1976) and is evaluated for its efficiency in constructing confidence intervals in symmetric, stretched-tailed situations. The one-sample biweight "t" is shown, via Monte Carlo simulations, to be efficient for samples of moderate sizes (in terms of expected length of the confidence intervals). For smaller samples (size five), the sum of the biweight weights is useful in rescaling biweight "t". For several samples of common population width, a root mean square of the variances affords greater stability when the underlying distribution is not extremely stretched-tailed. [ABSTRACT FROM AUTHOR]

The article presents the author's comment on an article on moment generating function method (MGF) and method of moments (MM). The author feels that authors of the article are to be congratulated not only for proposing a new method for estimating the unknown parameters for a mixture of two normal distributions, but also for proposing one which seems to work with small sample sizes. There are, however, a number of important questions regarding the performance of the method that are not addressed in the article. The author is confused as to why authors chose to compare their MGF to MM. Simulation studies have shown that the sampling variances and mean squared errors of the maximum likelihood estimates are, for almost all parameter values, considerably smaller than those of the moment estimates. The reason for the authors' choice may be that they were unaware of the work by various researchers on maximum likelihood estimation with mixtures of two normal distributions and the work extending the maximum likelihood procedure to the switching regressions model.

Consider a system of two seemingly unrelated regression equations: y[sub i] = X[sub i] beta[sub i]-u[sub i](i = 1, 2), where x[sub 2] is a proper subset of x[sub 1], i.e., X[sub 1] Is equivalent to (X[sub 2]:L[sub 2]), and SIGMA = (sigma[sub ij]) is the disturbance covariance matrix. Let beta[sub i] and beta[sub i] (i = 1, 2) be the beta-estimators based, respectively, on the restricted estimate S of SIGMA and on the unrestricted estimate S of SIGMA. This paper derives finite sample variances of the beta[sub i], and examines their efficiency with respect to the beta[sub i], beta[sub 2] and beta[sub 2] are identical to the OLS estimator obtained directly from y[sub 2] = X[sub 2]beta[sub 2] + u[sub 2], and are efficient. On the other hand, beta[sub 1](beta[sub 1]) is more (less) efficient at the lower (upper) end of the scale of |rho| = |sigma[sub 12]/(sigma[sub 11] x sigma[sub 22])[sup 1/2]|. In practical applications, it may be preferable to use beta[sub 1] when |rho| less than or equal to .5 (approximately)and df less than or equal to 5, and beta[sub 1] otherwise. [ABSTRACT FROM AUTHOR]

For three symmetric distributions and six sample sizes, this article presents estimates of the small-sample variance of Pitman's location-invariant and location-scale-invariant estimators of location. It then compares these two estimators and investigates the closeness of the Cramer-Rao bound when estimators are required to be invariant. Expressions of the form c/(n -- d) prove quite effective in fitting variances of Pitman estimators, and d can be interpreted as the amount of Fisher information "lost." In terms of relative efficiency, maximum-likelihood estimators and linear combinations of order statistics offer computationally attractive alternatives to the Pitman estimators. [ABSTRACT FROM AUTHOR]

Four methods for estimating heteroscedastic variances are discussed in this article: the MINQUE introduced by C.R. Rao [7], the AUE introduced by S.D. Horn, R.A. Horn, and D.B. Duncan [4], the average of the squared residuals, and the sample variance. Properties of these estimators, including translation invariance, existence, bias, consistency, existence of negative estimates, and mean square error (MSE) are compared. In particular, it is shown that the AUE has smaller MSE than either the MINQUE or the sample variance in a wide range of situations. [ABSTRACT FROM AUTHOR]

In the context of the usual regression model, Y = X beta + epsilon, this article shows that the coefficient obtained when Y is regressed on the first principal component of the independent variables may be interpreted as an estimator of that c' beta, c'c = 1, which can be estimated with least variance. Analogous results hold for succeeding components, and the property can be expressed in terms of volumes of confidence ellipsoids. To estimate beta using r principal components, choice of those components associated with the r largest characteristic roots minimizes the trace of a variance matrix. The results are illustrated with data on French imports. [ABSTRACT FROM AUTHOR]

Four multiple range tests for specifying homogeneous subsets of treatment variances are proposed and investigated. The procedures are based on Hartley's F-max statistic, Cochran's statistic, Bartlett and Kendall's normalizing log transformation of the sample variances, and Box's approximate test. The philosophy underlying these tests, with respect to Typed errors, is that advocated by Newman and later by Keuls. Empirical significance levels are obtained by simulating independent random samples from normal, uniform, double exponential and Cauchy distributions. The Box test was the only procedure not seriously affected by nonnormality. [ABSTRACT FROM AUTHOR]

A procedure for allocation of sample sizes to different strata consists of drawing a preliminary sample of fixed size from each stratum to estimate the strata variances and test their homogeneity. If the strata variances are found homogeneous, the sample sizes to be drawn from different strata are allocated according to proportional allocation; otherwise, they are allocated according to Neyman allocation using estimated variances. The efficiency of the proposed allocation, based on preliminary test of significance with respect to proportional allocation and modified Neyman allocation, is investigated. [ABSTRACT FROM AUTHOR]

Given k normal populations with unknown variances and known or unknown means, suppose we wish to select N individuals in two stages as follows: in the first stage, n individuals are selected from each of the k populations; in the second stage, the remaining (N-kn) individuals are selected from the population which gave rise to the smallest variance in the first stage. This article considers a minimax method for determining how large n should be. [ABSTRACT FROM AUTHOR]

This article presents a subset selection procedure for correlated variances with emphasis on the asymptotic case. An application to selecting the best set of independent variables in a regression problem is given. [ABSTRACT FROM AUTHOR]

The variance of the maximum likelihood estimator in each of the one-and two-sample alternate question randomized response models is compared to that of the original Warner model. In both cases, the former variance is shown to be uniformly less than that of the latter for all values of the parameter being estimated and appropriate ranges for the design parameters. [ABSTRACT FROM AUTHOR]

Several one-pass and two-pass algorithms for the computation of sample means and variances are compared by their performance on sets of randomly generated data and systematically generated data with random noise. The relation between the performance of each algorithm and the coefficient of variation of the population from which random data sets are generated is explored. [ABSTRACT FROM AUTHOR]

It has been established in Part I [3] that for estimating a finite population total, the estimators which are optimal under simple superpopulation models can be made insensitive to certain departures from the models by the choice of balanced samples. Here stratification on a size variable is considered as another technique for protecting against model failure. Together the two techniques, stratification and balanced sampling, are shown to provide more efficient protection than does balanced sampling alone. Questions of definition of strata and of allocation of sampling units among strata are also considered. [ABSTRACT FROM AUTHOR]

A lower bound is derived concerning a special form of the entropy function of information theory. It is applied to Hoeffding's bound for the probability of the deviation of the sample mean from its expected value and for the corresponding problem concerning the sample variance. Furthermore, applying the bound one can obtain a discard function for a binary symmetric channel, a lower bound for the total error probability in testing two simple hypotheses, given the number of observations, and a lower bound for the deviation of the empirical distribution from the true distribution. [ABSTRACT FROM AUTHOR]

Let a variate (x, say) have a power-series distribution with a given series function. A constructive method is given for finding (apart from an arbitrary sign and an additive constant) the function y[sub x] of x for which any one cumulant of y[sub x] is a given function of the parameter of the distribution. The application of this result to finding transformations with approximately constant variance is discussed, though a fully satisfactory solution has not been obtained. A non-monotone transformation has been found which transforms a Poisson, binomial or negative binomial variate into one for which the rate at which the variance approaches constancy is of a higher order than for the usual transformations, but it is normally very inefficient as an estimator. [ABSTRACT FROM AUTHOR]

This article deals with BLUS residuals in regression analysis, which have the property of being uncorrelated and having constant variance under the null hypothesis that the "true" disturbances have the same property. It is a continuation of the author's earlier article in this journal [7] and the results reported here are the following: (1) the BLUS residuals can be expressed conveniently in terms of the least-squares residuals by means of matrix operations of the order K (K being the number of unknown coefficients in the regression), (2) the BLUS residuals satisfy a stronger optimality condition than the one stated in [7], and (3) a simple expression is obtained for the coefficient vector which is implied by the BLUS residuals. [ABSTRACT FROM AUTHOR]