Interactionism,Social Organization,and Social Processes: Looking Back and Moving Ahead

With its roots in American pragmatism, symbolic interactionism has created a distinctive perspective and produced numerous important contributions and now offers significant prospects for the future. In this article, I review my intellectual journey with this perspective over forty years. This journey was initiated within the American society, sociology, and symbolic interaction of circa 1960. I note many of the contributions made by interactionists since that time, with particular focus on those who have contributed to the study of social organization and social process. I offer an agenda for the future based on currently underdeveloped areas that have potential. These are inequality orders, institutional analysis, collective action across space and time, and the integration of temporal and spatial orders. The article concludes with calls for further efforts at cross‐perspective dialogues, more attention to feminist scholars, and an elaborated critical pragmatism.     

Biased Bootstrap Methods for Reducing the Effects of Contamination

Contamination of a sampled distribution, for example by a heavy-tailed distribution, can degrade the performance of a statistical estimator. We suggest a general approach to alleviating this problem, using a version of the weighted bootstrap. The idea is to 'tilt' away from the contaminated distribution by a given (but arbitrary) amount, in a direction that minimizes a measure of the new distribution's dispersion. This theoretical proposal has a simple empirical version, which results in each data value being assigned a weight according to an assessment of its influence on dispersion. Importantly, distance can be measured directly in terms of the likely level of contamination, without reference to an empirical measure of scale. This makes the procedure particularly attractive for use in multivariate problems. It has several forms, depending on the definitions taken for dispersion and for distance between distributions. Examples of dispersion measures include variance and generalizations based on high order moments. Practicable measures of the distance between distributions may be based on power divergence, which includes Hellinger and Kullback–Leibler distances. The resulting location estimator has a smooth, redescending influence curve and appears to avoid computational difficulties that are typically associated with redescending estimators. Its breakdown point can be located at any desired value ε∈ (0, ½) simply by 'trimming' to a known distance (depending only on ε and the choice of distance measure) from the empirical distribution. The estimator has an affine equivariant multivariate form. Further, the general method is applicable to a range of statistical problems, including regression.     

Improving coverage accuracy of nonparametric prediction intervals

Methods are suggested for improving the coverage accuracy of intervals for predicting future values of a random variable drawn from a sampled distribution. It is shown that properties of solutions to such problems may be quite unexpected. For example, the bootstrap and the jackknife perform very poorly when used to calibrate coverage, although the jackknife estimator of the true coverage is virtually unbiased. A version of the smoothed bootstrap can be employed for successful calibration, however. Interpolation among adjacent order statistics can also be an effective way of calibrating, although even there the results are unexpected. In particular, whereas the coverage error can be reduced from O ( n ^-1) to orders O ( n ^-2) and O ( n ^-3) (where n denotes the sample size) by interpolating among two and three order statistics respectively, the next two orders of reduction require interpolation among five and eight order statistics respectively.     

STEIN'S METHOD AND THE BERRY-ESSÉEN THEOREM

Stein's method is used to prove the Lindeberg-Feller theorem and a generalization of the Berry-Esséen theorem. The arguments involve only manipulation of probability inequalities, and form an attractive alternative to the less direct Fourier-analytic methods which are traditionally employed.     

Deconvolution methods for non-parametric inference in two-level mixed models

Summary.  We develop a general non-parametric approach to the analysis of clustered data via random effects. Assuming only that the link function is known, the regression functions and the distributions of both cluster means and observation errors are treated non-parametrically. Our argument proceeds by viewing the observation error at the cluster mean level as though it were a measurement error in an errors-in-variables problem, and using a deconvolution argument to access the distribution of the cluster mean. A Fourier deconvolution approach could be used if the distribution of the error-in-variables were known. In practice it is unknown, of course, but it can be estimated from repeated measurements, and in this way deconvolution can be achieved in an approximate sense. This argument might be interpreted as implying that large numbers of replicates are necessary for each cluster mean distribution, but that is not so; we avoid this requirement by incorporating statistical smoothing over values of nearby explanatory variables. Empirical rules are developed for the choice of smoothing parameter. Numerical simulations, and an application to real data, demonstrate small sample performance for this package of methodology. We also develop theory establishing statistical consistency.     

ESTIMATING A PARAMETER WHEN IT IS KNOWN THAT THE PARAMETER EXCEEDS A GIVEN VALUE

In some statistical problems a degree of explicit, prior information is available about the value taken by the parameter of interest, θ say, although the information is much less than would be needed to place a prior density on the parameter's distribution. Often the prior information takes the form of a simple bound, ‘θ > θ₁ ’ or ‘θ < θ₁ ’, where θ₁ is determined by physical considerations or mathematical theory, such as positivity of a variance. A conventional approach to accommodating the requirement that θ > θ₁ is to replace an estimator, , of θ by the maximum of and θ₁. However, this technique is generally inadequate. For one thing, it does not respect the strictness of the inequality θ > θ₁ , which can be critical in interpreting results. For another, it produces an estimator that does not respond in a natural way to perturbations of the data. In this paper we suggest an alternative approach, in which bootstrap aggregation, or bagging, is used to overcome these difficulties. Bagging gives estimators that, when subjected to the constraint θ > θ₁ , strictly exceed θ₁ except in extreme settings in which the empirical evidence strongly contradicts the constraint. Bagging also reduces estimator variability in the important case for which is close to θ₁, and more generally produces estimators that respect the constraint in a smooth, realistic fashion.     

Bootstrap confidence regions computed from autoregressions of arbitrary order

Given a linear time series, e.g. an autoregression of infinite order, we may construct a finite order approximation and use that as the basis for confidence regions. The sieve or autoregressive bootstrap, as this method is often called, is generally seen as a competitor with the better-understood block bootstrap approach. However, in the present paper we argue that, for linear time series, the sieve bootstrap has significantly better performance than blocking methods and offers a wider range of opportunities. In particular, since it does not corrupt second-order properties then it may be used in a double-bootstrap form, with the second bootstrap application being employed to calibrate a basic percentile method confidence interval. This approach confers second-order accuracy without the need to estimate variance. That offers substantial benefits, since variances of statistics based on time series can be difficult to estimate reliably, and—partly because of the relatively small amount of information contained in a dependent process—are notorious for causing problems when used to Studentize. Other advantages of the sieve bootstrap include considerably greater robustness against variations in the choice of the tuning parameter, here equal to the autoregressive order, and the fact that, in contradistinction to the case of the block bootstrap, the percentile t version of the sieve bootstrap may be based on the 'raw' estimator of standard error. In the process of establishing these properties we show that the sieve bootstrap is second order correct.     

Importance of interpolation when constructing double-bootstrap confidence intervals

We show that, in the context of double-bootstrap confidence intervals, linear interpolation at the second level of the double bootstrap can reduce the simulation error component of coverage error by an order of magnitude. Intervals that are indistinguishable in terms of coverage error with theoretical, infinite simulation, double-bootstrap confidence intervals may be obtained at substantially less computational expense than by using the standard Monte Carlo approximation method. The intervals retain the simplicity of uniform bootstrap sampling and require no special analysis or computational techniques. Interpolation at the first level of the double bootstrap is shown to have a relatively minor effect on the simulation error.     

Confidence intervals for secondary parameters following a sequential test

In sequential studies, formal interim analyses are usually restricted to a consideration of a single null hypothesis concerning a single parameter of interest. Valid frequentist methods of hypothesis testing and of point and interval estimation for the primary parameter have already been devised for use at the end of such a study. However, the completed data set may warrant a more detailed analysis, involving the estimation of parameters corresponding to effects that were not used to determine when to stop, and yet correlated with those that were. This paper describes methods for setting confidence intervals for secondary parameters in a way which provides the correct coverage probability in repeated frequentist realizations of the sequential design used. The method assumes that information accumulates on the primary and secondary parameters at proportional rates. This requirement will be valid in many potential applications, but only in limited situations in survival analysis.