The effects of census undercount adjustment on congressional apportionment

"This article shows that undercount adjustment [of the 1990 U.S. census] will probably reallocate one [in] three House seats across the states. The adjustment's impact may depend on the method used and the assumptions underlying undercount estimates. Using regression analysis to reduce sampling error in undercount estimates from dual-systems analysis, however, eliminates sensitivity to all but the most extreme changes in assumptions. Generally, adjustment will more likely affect large states than small ones, and large states with proportionately many urban Black and Hispanic residents will likely gain seats at the expense of large states with few such residents."     

Using information from demographic analysis in post-enumeration survey estimation

"Population estimates from the 1990 Post-Enumeration Survey (PES), used to measure decennial census undercount, were obtained from dual system estimates (DSE's) that assumed independence within strata defined by age-race-sex-geography and other variables. We make this independence assumption for females, but develop methods to avoid the independence assumption for males within strata by using national level sex ratios from demographic analysis (DA).... We consider several...alternative DSE's, and use DA results for 1990 to apply them to data from the 1990 U.S. census and PES."     

Bayes/frequentist compromise decision rules for Gaussian sampling

Bayesian methods have the potential to confer substantial advantages over frequentist when the assumed prior is approximately correct, but otherwise can perform poorly. Therefore, estimators and other inferences that strike a compromise between Bayes and frequentist optimality are attractive. To evaluate potential trade-offs, we study Bayes vs. frequentist risk under Gaussian sampling for families of point estimators and interval estimators. Bayes/frequentist compromises for interval estimation are more challenging than for point estimation, since performance involves an interplay between coverage and length. Each family allows ‘purchasing’ improved frequentist performance by allowing a small increase in Bayes risk over the Bayes rule. Any degree of increase can be specified, thus enabling greater or lesser trade-offs between Bayes and frequentist risk.     

Multiple inverse sampling in post-stratification

In sample survey, post-stratification is often used when the identification of stratum cannot be achieved in advance of the survey. If the sample size is large, post-stratification is usually as effective as the ordinary stratification with proportional allocation. However, in the case of small samples, no general acceptable theory or technique has been well developed. One of the main difficulties is the possibility of obtaining zero sample sizes in some strata for small samples. In this paper, we overcome this difficulty by employing a sampling scheme referred to as the multiple inverse sampling such that each stratum is ensured to be sampled a specified number of observations. A Monte Carlo simulation is carried out to compare the estimator obtained from the multiple inverse sampling with some other existing estimators. The estimator under multiple inverse sampling is superior in the sense that it is unbiased and its variance does not depend on the values of stratum means in the population.     

Parameter estimation for a word frequency distribution based on occupancy theory

Compromise Estimators between the generalized Bayes and Bayes estimators with respect to conjugate gamma priors under entropy loss are proposed. The proposed compromise estimators are compared with some suitable generalized Bayes estimators in terms of their frequentist risk performance. Also the RSL approach will be employed to compare the proposed compromise estimators and some admissible generalized Bayes estimators in terms of their Bayes risk performance.     

Total error in PES estimates of population

"We describe a methodology for estimating the accuracy of dual systems estimates (DSE's) of population, census estimates of population, and estimates of undercount in the census. The DSE's are based on the census and a post-enumeration survey (PES). We apply the methodology to the 1988 dress rehearsal census of St. Louis and east-central Missouri and we discuss its applicability to the 1990 [U.S.] census and PES. The methodology is based on decompositions of the total (or net) error into components, such as sampling error, matching error, and other nonsampling errors. Limited information about the accuracy of certain components of error, notably failure of assumptions in the 'capture-recapture' model, but others as well, lead us to offer tentative estimates of the errors of the census, DSE, and undercount estimates for 1988. Improved estimates are anticipated for 1990." Comments are included by Eugene P. Ericksen and Joseph B. Kadane (pp. 855-7) and Kenneth W. Wachter and Terence P. Speed (pp. 858-61), as well as a rejoinder by Mulry and Spencer (pp. 861-3).     

Multivariate limited translation empirical Bayes estimators

The paper develops multivariate limited translation empirical Bayes estimators of the normal mean vector which serve as a compromise between the empirical Bayes and the maximum likelihood estimators. These compromise estimators perform better than the regular empirical Bayes estimators, in a frequentist sense, when there is wide departure of an individual observation from the grand average.     

Adjusting the 1980 census of population and housing

"In 1980, several cities and states sued the U.S. Census Bureau to correct census results. This correction would adjust for the differential undercounting of Blacks and Hispanics, especially in cities. In this article, the authors, each of whom testified for New York City and State in their joint lawsuit against the Census Bureau, describe the likely pattern of the undercount and present a method to adjust for it." The authors describe available methods for data adjustment and introduce a regression-based composite method of adjustment, which is used to estimate the undercounts for 66 areas. "As expected, we find that the highest undercount rates are in large cities, and the lowest are in states and state remainders with small percentages of Blacks and Hispanics. Next, we analyze how sensitive our estimates are to changes in data and modeling assumptions. We find that these changes do not affect the estimates very much. Our conclusion is that regardless of whether we use one of the simple methods or the composite method and regardless of how we vary the assumptions of the composite method, an adjustment reliably reduces population shares in states with few minorities and increases the shares of large cities."     

On robust empirical bayes analysis of means and variances from stratified samples

Ghosh and Lahiri (1987a,b) considered simultaneous estimation of several strata means and variances where each stratum contains a finite number of elements, under the assumption that the posterior expectation of any stratum mean is a linear function of the sample observations - the so called“posterior linearity” property. In this paper we extend their result by retaining the “posterior linearity“ property of each stratum mean but allowing the superpopulation model whose mean as well as the variance-covariance structure changes from stratum to stratum. The performance of the proposed empirical Bayes estimators are found to be satisfactory both in terms of “asymptotic optimality” (Robbins (1955)) and “relative savings loss” (Efron and Morris (1973)).     

Stratified sampling using double samples

Usual stratified sampling design assume that one is able to draw units directly from given strata. If this is not possible, one can use the following double sampling procedure: First take a large simple random sample out of the whole population and find out, to which stratum each sample unit belongs. Out of these chosen units take a second stratified sample. In this paper unbiased estimators for this procedure in the cases of known (part I) and unknown (part II) stratum weights are proposed for sampling with replacement and sampling without replacement and their variances are evaluated.     

Estimation for small areas in household surveys

In many socio-economic surveys the objective is estimation of total or proportion of persons with a particular attribute. Multi-stage area samples are drawn from geographic strata and population within areal units is used as an auxiliary variable in ratio estimation. For large administrative areas, the auxiliary variable totals are available as population projections based on the last census. However, for small areas population changes are significantly affected by non-demographic factors and hence projections with high enough reliability are not available for small areas. In such situations the efficiency of design-based estimators for small areas can be improved by a ratio adjustment based on the auxiliary variable total for a large area. An inequality on the efficiency of the ratio adjusted estimator is established and its bias and variance is investigated     

Theory & Methods: Incomplete detection in enumeration surveys: Whither distance sampling?

This paper considers the problem of undercount or incomplete detection in enumeration surveys that are intended to estimate population counts or population abundance. The problem is widespread in ecology but also occurs in other surveys: the census undercount is a well-known example of the problem. After framing the problem in a general context, the paper focuses on line transect sampling and the distance sampling method which has been widely applied in surveys of ecological populations. It describes distance sampling data and presents a graphical derivation of the distance sampling estimator. The graphical analysis leads to a new expression for the distance sampling estimator which gives useful insights into the nature of the estimator. The paper discusses the uniformity assumption on which distance sampling depends and describes the properties of the distance sampling estimator when uniformity does not hold. It then explores the relationship between this and other evaluations of distance sampling and mentions briefly some statistical ideas for treating the general incomplete detection problem. The paper concludes with some reflections on general insights arising from the research.     

The 1990 Post-Enumeration Survey: operations and results

The author assesses the 1990 Post-Enumeration Survey, which was "designed to produce Census tabulation of [U.S.] states and local areas corrected for the undercount or overcount of population....[He] discusses the process that produced the census adjustment estimates [as well as] the work aimed at improving the estimates.... The article then presents some of the principal results...."     

Variance estimation using auxiliary information under a one unit per stratum sample design

Several variance estimators using auxiliary information were compared for estimating the variance of the total or ratio under a one unit per stratum sample design. The auxiliary information used consisted of data from the 1977 and 1982 Economic Census. One thousand probability proportional to size one primary sampling unit per stratum samples were selected for the Monte Carlo study of characteristics of interest in a content evaluation survey as part of the 1982 Economic Census program. Six variance estimators were applied to each sample and their bias and mean square errors were evaluated. The results are strikingly different between the variance estimators of the estimated total and ratio.     

Bayesian Analysis of Stochastic Volatility Models

New techniques for the analysis of stochastic volatility models in which the logarithm of conditional variance follows an autoregressive model are developed. A cyclic Metropolis algorithm is used to construct a Markov-chain simulation tool. Simulations from this Markov chain converge in distribution to draws from the posterior distribution enabling exact finite-sample inference. The exact solution to the filtering/smoothing problem of inferring about the unobserved variance states is a by-product of our Markov-chain method. In addition, multistep-ahead predictive densities can be constructed that reflect both inherent model variability and parameter uncertainty. We illustrate our method by analyzing both daily and weekly data on stock returns and exchange rates. Sampling experiments are conducted to compare the performance of Bayes estimators to method of moments and quasi-maximum likelihood estimators proposed in the literature. In both parameter estimation and filtering, the Bayes estimators outperform these other approaches.     

Bayes methods in the ecological fallacy cntext:estimation of individual correlation from aggregate data

The ecological fallacy is related to Simpson's paradox (1951) where relationships among group means may be counterintuitive and substantially different from relationships within groups, where the groups are usually geographic entities such as census tracts. We consider the problem of estimating the correlation between two jointly normal random variables where only ecological data (group means) are available. Two empirical Bayes estimators and one fully Bayesian estimator are derived and compared with the usual ecological estimator, which is simply the Pearson correlation coefficient of the group sample means. We simulate the bias and mean squared error performance of these estimators, and also give an example employing a dataset where the individual level data are available for model checking. The results indicate superiority of the empirical Bayes estimators in a variety of practical situations where, though we lack individual level data, other relevant prior information is available.     

Empirical and Hierarchical Bayesian Estimation in Finite Population Sampling under Structural Measurement Error Models

Abstract.  This paper considers simultaneous estimation of means from several strata. A model-based approach is taken, where the covariates in the superpopulation model are subject to measurement errors. Empirical Bayes (EB) and Hierarchical Bayes estimators of the strata means are developed and asymptotic optimality of EB estimators is proved. Their performances are examined and compared with that of the sample mean in a simulation study as well as in data analysis.     

From the history of the 1937 census of population

This is a history of the 1937 census of the USSR, the results of which were suppressed because of an alleged massive undercount. The author attempts to determine whether there was in fact an undercount in this census, and if so how significant it was. He concludes that those responsible were not guilty of producing a significant undercount, and in fact employed the methods required to keep the undercount to a bare mimimum.     

A comparison of alternative variance estimation strategies for estimating the slope of a linear regression in sample surveys

The balanced half-sample, jackknife and linearization methods are used to estimate the variance of the slope of a linear regression under a variety of computer generated situations. The basic sampling design is one in which two PSU's are selected from each of a number of strata . The variance estimation techniques are compared with a Monte Carlo experiment. Results show that variance estimates may be highly biased and variable unless sizeable numbers of observations are available from each stratum. The jackknife and linearization estimates appear superior to the balanced half sample method - particularly when the number of strata or the number of available observations from each stratum is small.     

On Measuring Uncertainty of Benchmarked Predictors with Application to Disease Risk Estimate

Empirical Bayes (EB) estimates in general linear mixed models are useful for the small area estimation in the sense of increasing precision of estimation of small area means. However, one potential difficulty of EB is that the overall estimate for a larger geographical area based on a (weighted) sum of EB estimates is not necessarily identical to the corresponding direct estimate such as the overall sample mean. Another difficulty is that EB estimates yield over‐shrinking, which results in the sampling variance smaller than the posterior variance. One way to fix these problems is the benchmarking approach based on the constrained empirical Bayes (CEB) estimators, which satisfy the constraints that the aggregated mean and variance are identical to the requested values of mean and variance. In this paper, we treat the general mixed models, derive asymptotic approximations of the mean squared error (MSE) of CEB and provide second‐order unbiased estimators of MSE based on the parametric bootstrap method. These results are applied to natural exponential families with quadratic variance functions. As a specific example, the Poisson‐gamma model is dealt with, and it is illustrated that the CEB estimates and their MSE estimates work well through real mortality data.