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For a given Graph HT ht , we would like to predict what Graph HT ht would be. Later we relax this requirement and only assume that Graph HT ht relates to Graph HT ht the same way as Graph HT ht relates to Graph HT ht , but Graph HT ht follows a marginal distribution that is different from that of Graph HT ht . [Extracted from the article]

Model averaging generally provides better predictions than model selection, but the existing model averaging methods cannot lead to parsimonious models. Parsimony is an especially important property when the number of parameters is large. To achieve a parsimonious model averaging coefficient estimator, we suggest a novel criterion for choosing weights. Asymptotic properties are derived in two practical scenarios: (i) one or more correct models exist in the candidate model set and (ii) all candidate models are misspecified. Under the former scenario, it is proved that our method can put the weight one to the smallest correct model and the resulting model averaging estimators of coefficients have many zeros and thus lead to a parsimonious model. The asymptotic distribution of the estimators is also provided. Under the latter scenario, prediction is mainly focused on and we prove that the proposed procedure is asymptotically optimal in the sense that its squared prediction loss and risk are asymptotically identical to those of the best—but infeasible—model averaging estimator. Numerical analysis shows the promise of the proposed procedure over existing model averaging and selection methods. [ABSTRACT FROM AUTHOR]

In Equation (1), assume that Graph HT ht and Graph HT ht . In Lemma 1, the assumption that Graph HT ht may be relaxed. One issue with such an approach is that the effective sample data for estimation of Graph HT ht reduces for the sake of estimation of the optimal I i . However, one may use the same training data for estimation of Graph HT ht and tuning I i . [Extracted from the article]

The inferential models (IM) framework provides prior-free, frequency-calibrated, and posterior probabilistic inference. The key is the use of random sets to predict unobservable auxiliary variables connected to the observable data and unknown parameters. When nuisance parameters are present, a marginalization step can reduce the dimension of the auxiliary variable which, in turn, leads to more efficient inference. For regular problems, exact marginalization can be achieved, and we give conditions for marginal IM validity. We show that our approach provides exact and efficient marginal inference in several challenging problems, including a many-normal-means problem. In nonregular problems, we propose a generalized marginalization technique and prove its validity. Details are given for two benchmark examples, namely, the Behrens–Fisher and gamma mean problems. [ABSTRACT FROM AUTHOR]

In epidemiologic studies, partial questionnaire design (PQD) can reduce cost, time, and other practical burdens associated with lengthy questionnaires by assigning different subsets of the questionnaire to different, but overlapping, subsets of the study participants. In this article, we describe methods for semiparametric inference for regression model under PQD and other study settings that can generate nonmonotone missing data in covariates. In particular, motivated from methods for multiphase designs, we develop three estimators, namely mean score, pseudo-likelihood, and semiparametric maximum likelihood, each of which has some unique advantages. We develop the asymptotic theory and a sandwich variance estimator for each of the estimators under the underlying semiparametric model that allows the distribution of the covariates to remain nonparametric. We study the finite sample performances and relative efficiencies of the methods using simulation studies. We illustrate the methods using data from a case-control study of non-Hodgkin’s lymphoma where the data on the main chemical exposures of interest are collected using two different instruments on two different, but overlapping, subsets of the participants. This article has supplementary material online. [ABSTRACT FROM AUTHOR]

We describe a method for evaluating an ensemble of predictive models given a sample of observations comprising the model predictions and the outcome event measured with error. Our formulation allows us to simultaneously estimate measurement error parameters, true outcome--the "gold standard"--and a relative weighting of the predictive scores. We describe conditions necessary to estimate the gold standard and to calibrate these estimates and detail how our approach is related to, but distinct from, standard model combination techniques. We apply our approach to data from a study to evaluate a collection of BRCA1/BRCA2 gene mutation prediction scores. In this example, genotype is measured with error by one or more genetic assays. We estimate true genotype for each individual in the data set, operating characteristics of the commonly used genotyping procedures, and a relative weighting of the scores. Finally, we compare the scores against the gold standard genotype and find that Mendelian scores are, on average, the more refined and better calibrated of those considered and that the comparison is sensitive to measurement error in the gold standard. [ABSTRACT FROM AUTHOR]

The PB scheme is based on assuming a parametric linear model with iid normal errors: Graph HT ht To complete the framework, the authors also need to estimate the marginal distribution of I x i , and this is also assumed to be normal. We can demonstrate that this is not the case for "high-dimensional" cases, Graph HT ht by explicitly analyzing least squares regression with similar concepts. Consider the simplest case Graph HT ht with Graph HT ht normally distributed. [Extracted from the article]

This article discusses aspects of focused information criteria (FIC) and frequentist model averaging (FMA) in the model selection process. The local neighborhood framework allows one to extend familiar standard iid and regression models in several parametric directions. This may be utilized for robustness purposes and sensitivity analyses and leads to a theory for model averaging and focused model selection criteria. Regression analyses have different aims on different occasions, and even the same dataset may be analyzed with different goals in mind. The three main strands of references in Bayesian model averaging (BMA) literature are: General Bayes theory, simulations and cross-validation-type predictive performance. The development of the FMA methodology includes understanding the behavior of BMA strategies. Comments from Cook and Li as well as Raftery and Zheng point to the usefulness of developing the FIC and FMA apparatus to assess prediction quality when averaged in suitable ways, rather than for one focus parameter at a time.

The traditional use of model selection methods in practice is to proceed as if the final selected model had been chosen in advance, without acknowledging the additional uncertainty introduced by model selection. This often means underreporting of variability and too optimistic confidence intervals. We build a general large-sample likelihood apparatus in which limiting distributions and risk properties of estimators post-selection as well as of model average estimators are precisely described, also explicitly taking modeling bias into account. This allows a drastic reduction in complexity, as competing model averaging schemes may be developed, discussed, and compared inside a statistical prototype experiment where only a few crucial quantities matter. In particular, we offer a frequentist view on Bayesian model averaging methods and give a link to generalized ridge estimators. Our work also leads to new model selection criteria. The methods are illustrated with real data applications. [ABSTRACT FROM AUTHOR]

This article discusses the performance of Bayesian model averaging (BMA) relative to other methods for real datasets. The performance of Bayesian model selection and BMA has been studied. There are three main strands of results: (1) General theoretical results from Jeffreys in 1939, (2) Simulation studies and (3) Results on out-of-sample predictive performance. There are several realistic simulation studies of the performance of BMA relative to other methods in a variety of situations, including linear regression (George and McCulloch 1993 and Raftery, Madigan and Hoeting 1997), log-linear models (Clyde 1999), logistic regression (Viallefont, Raftery and Richardson 2001) and wavelets (Clyde and George 2000). The BMA distribution of a quantity of interest can be viewed as the posterior distribution from the full model, with a mixed discrete-continuous prior distribution that assigns weight to the events that the individual components are zero. A simple normal example has been presented to assess numerical differences in performance and also of the extent to which results hold event if the prior distribution is not the same as the practical distribution. Posterior model probabilities can be computed in four ways.

This article presents information on a study which analyzed inference and interpretability considerations in frequentist model averaging (FMA) and selection. Model averaging and model selection as well as their associated inferential aspects are important in both statistical theory and practice. One key ingredient of statistical data analysis is model development. Others believe model development should be separated from the data and based on the theory of the process involved or derived from a completely algorithmic or black box approach. To highlight the differences in possible goals, the study considered the model for mixed enzyme inhibition from applied biochemistry, which describes a wide class of reaction kinetics. If predictive accuracy of the model is the stated goal, then some parameter of interest is better than another in achieving that goal via FMA when using focused information criterion. In many practical situations the number of models involved in averaging is so large that the computation of model averaging is prohibitive. A fundamental tenet of frequentist analysis is that unplanned contrasts should be avoided. FMA yields a biased estimator having smaller variance than the single model estimator that is unbiased.

Payphones, parking meters, and vending machines illustrate the modem business practice of substituting machinery for manpower. They do not eliminate manual labor completely, because full coin-boxes still must be replaced and vending machines still must be stocked. Deciding when to replace a coin-box is important, with unequal losses resulting from underestimation and overestimation. This article derives optimal methodology for this problem by incorporating collection history and specifying common prior distributions over average daily fill rate and standard deviation at each box. The approach is implemented and analyzed on collection records from 11.308 pay phones over a large geographical region. When the loss from overestimation is 19 times that from underestimation, our methods outperform the one in current use at least 69.9% of the time, translating into average potential collection-cost reductions exceeding 21%. [ABSTRACT FROM AUTHOR]

This article highlights the historical events that took place within 50 years since the American Statistical Association was founded in 1839 and the Journal of the American Statistical Association (JASA) was first published in 1888. For the first years, JASA contained almost exclusively nonmathematical papers. Many were mere repositories of extensive data sets, including many compilations from census counts with interpretations of what these data purportedly revealed. Others were from investigations undertaken by sociologists, economists, political scientists and historians. One area that attracted theoretical attention dealt with the concepts of averages, variation and distributions. The second area that received theoretical attention during these years was correlation and related concepts.

The sequential imputation procedure is applied to adaptively and sequentially reconstruct discrete input signals that are blurred by an unknown linear moving average channel and contaminated by additive Gaussian noises, a problem known as blind deconvolution in digital communication. A rejuvenation procedure for improving the efficiency of sequential imputation is introduced and theoretically justified. The proposed method does not require the channel to be nonminimum phase and can be used in real time signal restoration. Two simulated systems are studied to illustrate the proposed method. Our result shows that the ideas of multiple imputations and flexible simulation techniques are as powerful in engineering as in survey sampling. [ABSTRACT FROM AUTHOR]

In this article we introduce a new type of positive-breakdown regression method, called a generalized S-estimator (or GS-estimator), based on the minimization of a generalized M-estimator of residual scale. We compare the class of GS-estimators with the usual S-estimators, including least median of squares. It turns out that GS-estimators attain a much higher efficiency than S-estimators, at the cost of a slightly increased worst-case bias. We investigate the breakdown point, the maxbias curve, and the influence function of GS-estimators. We also give an algorithm for computing GS-estimators and apply it to real and simulated data. [ABSTRACT FROM AUTHOR]

We show that the familiar bootstrap plug-in rule of Efron has a natural analog in finite population settings In our method a characteristic of the population is estimated by the average value of the characteristic over a class of empirical populations constructed from the sample Our method extends that of Gross to situations in which the stratum sizes are not integer multiples of their respective sample sizes Moreover, we show that our method can be used to generate second-order correct confidence intervals for smooth functions of population means, a property that has not been established for other resampling methods suggested in the literature A second resampling method is proposed that also leads to second-order correct confidence intervals and is less computationally intensive than our bootstrap But a simulation study reveals that the second method can be quite unstable in some situations, whereas our bootstrap performs very well. [ABSTRACT FROM AUTHOR]

We investigate the behavior of M estimators of the location parameter for stochastic processes with long-range dependence. The processes considered are Gaussian or one-dimensional transformations of Gaussian processes. It turns out that, up to a constant, all M estimators are asymptotically equivalent to the arithmetic mean. For Gaussian processes this constant is always equal to one, independently of the ψ function. In view of the case of ud observations, the results are surprising. They are related to earlier work by Gastwirth and Rubin. Some simulations illustrate the results. [ABSTRACT FROM AUTHOR]

This article presents a kernel estimate of a curve that uses an adaptive procedure for local selection of the bandwidth. A two-step procedure is proposed for estimating the local bandwidth that minimizes the mean squared error (MSE) of a kernel estimator for nonparametric regression. First, a consistent estimate of the exact MSE is constructed. Then the bandwidth that minimizes the estimate of the MSE is calculated. Sufficient conditions under which this bandwidth is asymptotically optimal and normally distributed are given. The local bandwidth selection procedure was implemented on some simulated data and compared to a global bandwidth selection procedure. A 68%-91% reduction in the average MSE of a kernel estimator was realized with the local bandwidth selection procedure. Such a scheme was also studied by another statistician and he termed a direct pilot estimator approach. The result is slightly more general in the specification of the interval that is searched for the optimizing bandwidth.

Optimal rank-based procedures were derived in Hallin, Ingenbleek, and Puri (1985, 1987) and Hallin and Puri (1988) for some fundamental testing problems arising in time series analysis. The optimality properties of these procedures are of an asymptotic nature, however, whereas much of the attractiveness of rank-based methods lies in their small-sample applicability and robustness features. Accordingly, the objective of this article is twofold: (a) a study of the finite-sample behavior of the asymptotically optimal tests for randomness against first-order autoregressive moving average dependence proposed in Hallin et al. (1985), both under the null hypothesis (tables of critical values) and under alternatives of serial dependence (evaluation of the power function), and (b) an (heuristic) investigation of the robustness properties of the proposed procedures (with emphasis on the identification problem in the presence of "outliers"). We begin (Sec. 2) with a brief description of the rank-based measures of serial dependence to be considered throughout: (a) Van der Waerden, (b) Wilcoxon, (c) Laplace, and (d) Spearman-Wald-Wolfowitz autocorrelations. The article is mainly concerned with first-order (lag 1) coefficients of these types. Tables of the critical values required for performing tests of randomness are provided (Sec. 3), and the finite-sample power of the resulting tests is compared with that of their parametric competitors (Sec. 4). Although the exact level of classical parametric procedures is only approximately correct (whereas the distribution-free rank tests are of the correct size), the proposed rank-based tests compare quite favorably with the classical ones, and appear to perform at least as well as (often strictly better than) their classical counterparts. The examples of Section 5 emphasize the... [ABSTRACT FROM AUTHOR]

Integrated processes as models for time series data have proved to be an important component of the highly flexible class of ARIMA(p, d, q) models. Determining the amount of differencing, d, has been a difficult task: too little and the process is not yet second-order stationary; too much and the process is more variable than it need be. It is shown that by introducing the notion of generalized covariances, developed by Matheron (1973) for spatial processes, the amount of differencing needed can be read easily from a sequence of graphs showing averages of squares of primary data increments. Formal inference to determine if the last difference really is necessary can then be carried out. Time series data are analyzed in this way and compared with the hypothesis-testing approach illustrated by Dickey, Bell, and Miller (1986). Once the order of differencing has been diagnosed, either the differenced time series can be analyzed or the generalized covariance of the undifferenced series can be estimated. [ABSTRACT FROM AUTHOR]

This article examines the phenomenon of negative regret in sequential point estimation, in which a stopping rule used in ignorance of the variance of the distribution sometimes performs better than the best fixed sample size when the variance is known. An explanation in terms of a certain robustifying effect of the stopping rule is given. Results of simulation studies illustrating both negative regret and its explanation are also given. Suppose X[sub 1], X[sub 2], . . . are iid with mean mu and variance sigma[sup 2]/X. One may stop observing the sequence of X,'s after any number of observations n and estimate mu by the sample mean X[sub n], subject to the loss L[sub n] = A(X[sub n] - mu)[sup 2] + n (A > 0). If sigma is known, then the best fixed (i.e., nonrandom) sample size, in the sense of minimum risk, can be used. If a is unknown, the best fixed sample size is unknown, necessitating a sequential procedure. For a sequential procedure of the type first proposed by Robbins (1959), the "regret'--expected additional loss from not knowing a and using the sequential procedure instead of the best fixed sample size--can take arbitrarily large negative values. This is explained by noting that the sequential procedure acts to control extreme values and can be viewed as a sequential analog of trimming and Winsorizing in the nonsequential setting. I illustrate analytically by showing that the sequential procedure reduces the mean squared error of the sample mean when the distribution of the X,'s is symmetric about mu and has large kurtosis. Simulation results show that both negative regret and its explanation can occur when the expected sample size is as small as 25 or 30. [ABSTRACT FROM AUTHOR]

Single replicate factorial designs for incomplete block experiments are obtained by first constructing a single replicate preliminary design in incomplete blocks for the same number of factors but an excessive number of levels of the first factor, then deleting the excess treatment combinations to obtain a deletion design. Any single replicate preliminary design yields a single replicate deletion design. Furthermore, if the preliminary design is orthogonal, then the resulting deletion design is shown to be nearly orthogonal and, under certain reduced models, to provide efficient estimation of lower-order effects and in some cases an orthogonal analysis. For example, a 2 x 3[sup 2] deletion design is constructed in three blocks of size 6, the 2 df confounded being the sum of interaction effects of F[sub 2] and F[sub 3] and second-order interaction effects. If second-order interactions are assumed negligible, then the deletion design provides efficient estimation of interactions between F[sub 2] and F[sub 3] and an orthogonal analysis. In another example, a 2 x 3[sup 2] deletion design is constructed in nine blocks of size 2 with main effects of F[sub 1] unconfounded. In a main effects model, main effects of F[sub 2] and F[sub 3] are estimable with optimal average efficiency. Experimental settings involving more factor levels are also considered, tables are given showing the efficiency of the resulting deletion designs to compare favorably with optimal upper bounds, and the deletion designs are shown by comparison to be more efficient on lower-order effects than competing orthogonal designs. [ABSTRACT FROM AUTHOR]

The problem of simultaneously estimating the pairwise differences of means of three independent normal populations with equal variances is considered. A computational method involving a bivariate t density is used to form confidence intervals with simultaneous coverage probability equal to 1 - a. For equal sample sizes, the method is the Tukey studentized range procedure. With unequal sample sizes, the method is superior to the various generalized Tukey methods. A table of probability points is presented for small sample sizes. A large-sample approximation based on the bivariate normal and studentized range distributions is given. [ABSTRACT FROM AUTHOR]

In multiple comparison procedures often the ordering of populations is of more interest than the testing of hypotheses. In such a situation it is important to control the probability of a Type III error (introduced by Hatter 1957 to indicate that one population is concluded to be better than another when actually it is worse). The studentized differences between means may be used to give ordering conclusions on all pairs of populations under consideration. Although this may be done by setting up confidence intervals using Tukey's honest significant difference, a more efficient method (when the confidence intervals themselves are not of interest) is suggested here, and tables are provided for its implementation. [ABSTRACT FROM AUTHOR]

When two or more information sources ("experts") provide a decision maker with information on two or more random variables, the decision maker using Bayes's rule has an opportunity to (a) update a prior about the random variables and (b) calibrate the experts. (Calibration is the process of adjusting the decision maker's likelihood about the experts' assessments.) This article presents a model for this two-way process and specializes to the case in which the experts' assessment errors have a multivariate normal density. In general, we find that variables which the decision maker and the experts regard as independent a priori will be dependent a posteriori because of dependence in the assessment errors. Formulas for posterior densities are given for the normal model. In this model the posterior density of the random variables depends on only a weighted average of the expert's means, with weights that depend on the experts' assessments of previously known quantities. I also present a special case of the model for which the mean of the posterior density is correctly given by a simple (unweighted) average of assessments. [ABSTRACT FROM AUTHOR]

In this article a modified definition of the G-spectral estimator is given. It is shown that the resulting estimator is a method of moments autoregressive moving average (ARMA) spectral estimator that does not require an estimate of the moving average parameters. As a result, a new formula for the power spectrum of an ARMA process is given that does not explicitly involve the moving average (MA) parameters. This formula then leads to a closed-form expression for the MA parameters and their corresponding moment estimators. [ABSTRACT FROM AUTHOR]

Expressions derived by Mann and Grubbs [9] for obtaining confidence bounds on series-system reliability from exponential subsystem Type-II censored life-test data are simplified. The simplification allows calculation of confidence bounds based on weighted averages of only the first and second powers of the inverses of "total time on test" for the various independent subsystems tested. These approximate well (within about a unit in the second significant figure) optimum (uniformly most accurate unbiased) confidence bounds obtained iteratively by computer techniques from results of Lentner and Buehler [5] and El Mawaziny [1]. [ABSTRACT FROM AUTHOR]

The article shows that if the original variable follows a pth order autoregressive system then the non-overlapping moving sum follows a pth order autoregression with at most a pth order moving-average of an independent sequence regardless of the length of the summation. From such an aggregate model we derive the optimal predictor of the aggregate variable and show that it performs remarkably well compared to the optimal disaggregate predictor. The article contains both theoretical and numerical analysis. [ABSTRACT FROM AUTHOR]

Following Duncan [3], the per hour cost of the process under the surveillance of a mean chart for controlling the mean of non-normal variables whose density function is represented by the first four terms of an Edgeworth series is obtained. The sample size and sampling interval which minimize the cost for detecting a particular shift in the process average are derived when the width of the control limits is specified. They are numerically determined for different non-normal situations by assuming various cost values when a particular shift in the process average is to be detected. They are presented in a table which compares the changes in optimum sample size and sampling interval due to changes in (a) the non-normal situation, (b) the shifts to be detected in the process average and (c) width of the control limits specified. [ABSTRACT FROM AUTHOR]

This article considers the problem of adjusting monthly or quarterly time series to make them accord with independent annual totals or averages without introducing artificial discontinuities. A general approach and some specific procedures involving constrained minimization of a quadratic form in the differences between revised and unrevised series are proposed. Some computational advantages are noted. Attention is given to the relationships between the adjustment problem and earlier work by other authors on the creation of monthly or quarterly series when only annual figures are available. An example is provided to illustrate the application of the proposed adjustment procedures. [ABSTRACT FROM AUTHOR]

The early decision version of the uniformly most powerful (UMP) fixed-sample-size test for binomial probabilities, as described for instance by Alling [1], is modified by changing all continuation points which lie on or below a line with specified slope into acceptance points. The modification reduces the average sample number (ASN) for small values of the binomial probability with only a slight loss of power. Approximate formulas for the ASN and power function of the resulting test are presented and a small sample comparison made with other sequential tests. [ABSTRACT FROM AUTHOR]

The concepts of the "median significance level" (MSL) and the 'significance level of the average" (SLA) are introduced and some relationships among these measures and the recently introduced "expected significance level" (ESL), "average critical value" (ACV), and "median critical value" (MCV) are considered. The median significance level is defined as the median of the distribution of the observed significance level for a given alternative and is shown to be equivalent to the significance level attained by the median of the test statistic for one-sided tests. The 'significance level of the average" is analogously defined to be the significance level attained by the average (expectation) of the test statistic and the MSL and SLA are shown to be inverse functions of Geary's MCV and ACV. Some relations between these small sample measures of test efficacy, and Pitmans' and Bahadur's asymptotic measures are described. The MCV is shown to be formally related to Hamaker's "indifference quality" method of classifying acceptance sampling plans. Several simple examples are given illustrating some relationships among the several criteria. [ABSTRACT FROM AUTHOR]

Suppose there are a finite number of possible states of a system, the state is observable at the beginning of each period, and it can be changed at that time to any other state with a cost for changing the state or resetting. In addition, an immediate loss is incurred depending on the state at the beginning of the period, but after resetting. The problem is, then, for which states to reset and where to reset to. For the infinite horizon problem with discount factor 0 is less than or equal to beta < 1 and for average cost per period, a computational procedure is given which amounts essentially to inverting a matrix the size of the number of states for which it is optimal to not reset. In terms of the corresponding linear program, once certain columns enter the basis, they do not subsequently drop from the basis. Some results on structure of optimal policies are also given. [ABSTRACT FROM AUTHOR]

In a recent paper in this Journal, Rothenberg, Fisher and Tilanus [1] discuss a class of estimators of tile location parameter of the Cauchy distribution, taking the form of the arithmetic average of a central subset, of the sample order statistics. They show that the average of roughly the middle quarter of the ordered sample has minimum asymptotic variance within this class, and that asymptotically it eliminates about. 36 per cent of the efficiency loss of the median (the most commonly used estimate,r) in comparison to the maximum likelihood estimator (m.l.e.). Of course both the m.l.e, and the best linear unbiased estimator based on the order statistics (BLUE) achieve full asymptotic efficiency in the Cramer-Rao sense and there can be no dispute about the relative merits of the three estimators asymptotically, or about the inferiority of the median (with asymptotic efficiency 8/pi[sup 2] * This character cannot be converted in ASCII text) 0.8 compared with about 0.88 for the estimator of Rothenberg et al.). In any practical situation however, we will be concerned with estimation from samples of finite size and asymptotic properties will not necessarily give any guidance here. We are essentially concerned with two points in assessing the relative merits of estimators in small samples, their case of application and "small-sample efficiency" which is conveniently measured as the ratio of the Cramer-Rao lower bound to the variance of the estimator. In this paper various estimators of the location of the Cauchy distribution arc compared in these two respects for samples of up to 20 observations. The small-sample properties of the m.l.e. have been extensively discussed elsewhere (Barnett [2]) and relevant results are summarized where necessary. The main purpose of the paper is to discuss general linear estimators based on the order statistics, and to assess their utility in the present context. Since this paper was prepared a further interesting 'quick estimator', b [ABSTRACT FROM AUTHOR]