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This article revisits the problem of sequential point estimation of the autogressive parameter in an autoregressive model of order 1, where the errors are independent and identically distributed with mean 0 and unknown variance. This problem was originally considered in Sriram (1988), where first-order efficiency properties and a second-order expansion for the expected value of a stopping rule were established. Here, we obtain an asymptotic expression for the so-called regret due to not knowing σ, as the cost of estimation error tends to infinity. Under suitable assumptions, our extensive analysis shows that all but one term in the regret are asymptotically bounded. If the errors have a bounded support, however, then the regret remains asymptotically bounded. Finally, we illustrate the performance of our sequential procedure and the associated regret for well-known blowfly data (Nicholson, 1950) and Internet traffic data using the residual bootstrap method for autoregressive models. [ABSTRACT FROM AUTHOR]

For count data having an unknown mass functiong0, we use the minimum Bhattacharyya-Hellinger distance (MBHD) estimator and a stopping rule to construct a sequential fixed-width confidence interval for a functionalT(g0) = θ0, wherefθ0is the best-fitting parametric model achieving the MBHD betweeng0and any member of a parametric class of mass functions. We establish the asymptoticconsistencyandefficiencyproperties of the sequential confidence interval and the expected sample size, respectively, as the half-widthd → 0. When the count data come from a gross-error contamination modelgα, L = (1 − α)fθ + αδ{L}for α ∈ (0, 1), where a parametric modelfθis mixed with a point mass δ{L}located at a valueL, we reparameterizeL = Ldsuch thatLd → ∞ asd → 0 and theoretically show that the expected sample size is affected by α, whereas the coverage probability of the sequential confidence interval depends on the rate of [T(gα, Ld) − θ]/d, asd → 0. Our reparameterization fully exploits the MBHD estimator's inherent ability to progressively ignore increasing values ofL, providing an asymptotically consistent sequential fixed-width interval estimator of θ. Whenfθis Poisson (θ), simulations are conducted to corroborate our theoretical results and to contrast the performance of the MBHD with that of the maximum likelihood estimator (MLE) of θ. A real data onDeath Notice, modeled as a negative binomial, are analyzed to compare and contrast the performance of our sequential MBHD procedure with that of the MLE. [ABSTRACT FROM PUBLISHER]

This article contains our responses to the eight discussion pieces written by experts, commenting on the results presented in Sriram and Iaci (2014). The majority of the discussants described complementary sequential problems arising in time series and other models involving dependent data, while two of the discussants provided thoughtful views on the phenomenon ofnegative regretinvestigated in our article. Prompted by an excellent observation made in one of the discussion pieces, we have attempted to shed further light on the phenomenon ofnegative regretin the context of sequential estimation of the autoregressive parameter. [ABSTRACT FROM AUTHOR]

This article revisits sequential estimation of the autoregressive parameter β in a first-order autoregressive (AR(1)) model and construction of a sequential confidence region for a parameter vector θ in a first-order threshold autoregressive (TAR(1)) model. To resolve a theoretical conjecture raised in Sriram (1986), we provide a comprehensive numerical study that strongly suggests that theregretin using a sequential estimator of β can be significantlynegativefor many heavy-tailed error distributions and even for normal errors. Secondly, to investigate yet another conjecture about the limiting distribution of a sequential pivotal quantity for θ in a TAR(1) model, we conduct an extensive numerical study that strongly suggests that the sequential confidence region has much better coverage probability than that of a fixed sample counterpart, regardless of whether the θ values are inside or on or near the boundary of the ergodic region of the series. These highlight the usefulness of sequential sampling methods in fitting linear and nonlinear time series models. [ABSTRACT FROM AUTHOR]

We propose a notion of central mean dimension reduction subspace for time series {x1} which does not require specification of a model but seeks to find a p x d matrix Φd, d ≤ p, so that the d x 1 vector where ΦdTXt-1 where Xt-1= (xt-1,…Xt-p)T for some p ≥ 1, includes all the information about xt that is available from E(xt∣Xt-1). For known p and d, we estimate the mean central subspace through the Nadaraya-Watson kernel smoother and establish the strong consistency of our estimator. In addition, we propose estimation of d and p using a modified Schwarz Bayesian criterion, if either of d and p is unknown. Finally, we examine the performance of all the estimators extensively through a variety of simulations and provide a new analysis of the well-known Canadian lynx data. Supplemental materials for this article are available online. [ABSTRACT FROM AUTHOR]

In this article we propose a new general index that measures relationships between multiple sets of random vectors. This index is based on Kullback-Leibler (KL) information, which measures linear or nonlinear dependence between multiple sets using joint and marginal densities of affine transformations of the random vectors. Estimates of the matrixes are obtained by maximizing the KL information and are shown to be consistent. The motivation for introducing such an index comes from morphological integration studies, a topic in biological science. As a special case of this index, we define an overall measure of association and two other measures for dimension reduction. The use of these measures is illustrated through real data analysis in morphometric studies and extensive simulations, and their performance is compared with that of approaches based on canonical correlation analysis. Extensions of the aforementioned measures to multiple groups are also discussed. In contrast to canonical correlation analysis, our general index not only provides an overall measure of association, but also determines nonlinear relationships, thereby making it useful in many other applications. [ABSTRACT FROM AUTHOR]

In many applications, it is important to find the mixture with fewest number of components, known as the mixture complexity, that provides a satisfactory fit to the data. This article focuses on developing an estimator of mixture complexity that is consistent when the form of component densities are unknown but are postulated to be members of some parametric family and is simultaneously robust against model misspecification. We treat the estimation of mixture complexity as a model selection problem and construct an estimator of mixture complexity as a byproduct of minimizing a Hellinger information criterion. This estimator is shown to be consistent for any parametric family of mixtures. When the model is correctly specified, Monte Carlo simulations for a wide variety of normal mixtures show that our estimator is very competitive with several others in the literature in correctly identifying the true mixture complexity. The basic construction, being firmly rooted in the minimum Hellinger distance approach, enables our estimator to naturally inherit the property of robustness, which is examined, through simulations, under symmetric departures from postulated component normality. In terms of correctly identifying the mixture complexity under model misspecification, our estimator performs much better than an estimator based on the Kullback--Leibler distance due to James, Priebe, and Marchette. An example concerning hypertension is revisited to further illustrate the performance of our estimator. [ABSTRACT FROM AUTHOR]

When it is necessary to manually count a large number N s of items, significant time and energy can be saved by weighing the items. When the weights are random with unknown mean and variance, one procedure is to take a small sample of n items, weigh them, and add more items until the total weight reaches N s times the average weight of the n items. This procedure yields a batch of items. An estimation criterion used in Yu (1989) yields an optimal value n * for the original sample, but it is not useful when the coefficient of variation is unknown. To overcome this, Yu (1989) proposed a fully, sequential scheme. Here, we propose a relatively new sequential sampling scheme due to Liu (1997a). Our scheme and Yu's (1989) scheme are shown to be second-order efficient, but our scheme requires substantially fewer sampling operations. These show that a little bit of sampling using substantially fewer sampling operations can significantly reduce the effort and cost of complete counting. [ABSTRACT FROM AUTHOR]

Considerable time and energy involved in complete counting of large, but pre-specified, batch of Ns items could be saved by using weights of items. This article considers the case when the underlying distribution of weights, and its mean and variance are unknown. The problem is then reduced to that of finding an estimator of the mean and an optimal (small) sample size based on which a number [Ncirc]n of items in the batch can be determined. Using a fixed-width interval criterion, Nickerson [Nickerson, D.M. Another look at counting by weighing. Commun. Statist. Simula. 1993, 22 (2), 323–343] derived an optimal sample size, but it depends on the unknown coefficient of variation. For this case, we propose a batch-type sequential sampling scheme which requires substantially fewer sampling operations and no prior knowledge of the coefficient of variation, but performs as well as Nickerson's and other available procedures in the literature. This shows that a little bit of sampling using substantially fewer sampling operations can significantly reduce the effort of complete counting. [ABSTRACT FROM AUTHOR]