Optimal Strategies in 2-Stage Sampling

In many survey situations simple random sampling of units and estimation of a total of interest by the expansion estimator are attractive methods, at least at first sight. Considering cost aspects suggests rather to use multiple stage sampling which, in general, is cheaper, but less effective. The design effect is an adequate criterion of the decrease of efficiency. We discuss this criterion for clusters (primary units) of equal size and derive exact conditions for a decrease of efficiency. The equality condition for cluster sizes seems not to be very restrictive, because in many cases one will be interested in clusters of approximately the same size, or, if sizes differ essentially, the clusters are partitioned into strata according to their sizes and the procedures for different strata are independent, each dealing with clusters of equal size or nearly so. In the context considered the use of the Horvitz–Thompson estimator is quite general. We examine a class of estimators with the Horvitz–Thompson estimator and a straight forward modification of it as special elements. As for the design effect all elements of the class are very similar, as for other aspects such as admissibility there are remarkable differences.     

On the efficiency of systematic sampling in ratio and product estimation

Summary In this paper the possibility of gain in efficiency in systematic sampling as compared to simple random sampling has been considered when a ratio or product estimator is used to improve upon the conventional unbiased estimator. The expression for the variance of the estimators are derived for multistage design where systematic selection is used at the ultimate-stage with any probability scheme at the previous stages. In particular the results for the uni-stage systematic sampling and for two-stage sampling with systematic selection at the second-stage have been obtained in section 3.     

Estimation in Two-Stage Models with Heteroscedasticity

A surprising number of important problems can be cast in the framework of estimating a mean and variance using data arising from a two-stage structure. The first stage is a random sampling of "units" with some quantity of interest associated with the unit. The second stage produces an estimate of that quantity and usually, but not always, an estimated standard error, which may change considerably across units. Heteroscedasticity in the estimates over different units can arise for a number of reasons, including variation associated with the unit and changing sampling effort over units. This paper presents a broad discussion of the problem of making inferences for the population mean and variance associated with the unobserved true values at the first stage of sampling. A careful discussion of the causes of heteroscedasticity is given, followed by an examination of ways in which inferences can be carried out in a manner that is robust to the nature of the within unit heteroscedasticity. Among the conclusions are that under any type of heteroscedasticity, an unbiased estimate of the mean and the variance of the estimated mean can be obtained by using the estimates as if they were true unobserved values from the first stage. The issue of using the mean versus a weighted average which tries to account for the heteroscedasticity is also discussed. An unbiased estimate of the population variance is given and the variance of this estimate and its covariance with the estimated mean is provided under various types of heteroscedasticity. The two-stage setting arises in many contexts including the one-way random effects models with replication, meta-analysis, multi-stage sampling from finite populations and random coefficients models. We will motivate and illustrate the problem with data arising from these various contexts with the goal of providing a unified framework for addressing such problems.     

Survey Estimates by Calibration on Complex Auxiliary Information

In the last decade, calibration estimation has developed into an important field of research in survey sampling. Calibration is now an important methodological instrument in the production of statistics. Several national statistical agencies have developed software designed to compute calibrated weights based on auxiliary information available in population registers and other sources. This paper reviews some recent progress and offers some new perspectives. Calibration estimation can be used to advantage in a range of different survey conditions. This paper examines several situations, including estimation for domains in one‐phase sampling, estimation for two‐phase sampling, and estimation for two‐stage sampling with integrated weighting. Typical of those situations is complex auxiliary information, a term that we use for information made up of several components. An example occurs when a two‐stage sample survey has information both for units and for clusters of units, or when estimation for domains relies on information from different parts of the population. Complex auxiliary information opens up more than one way of computing the final calibrated weights to be used in estimation. They may be computed in a single step or in two or more successive steps. Depending on the approach, the resulting estimates do differ to some degree. All significant parts of the total information should be reflected in the final weights. The effectiveness of the complex information is mirrored by the variance of the resulting calibration estimator. Its exact variance is not presentable in simple form. Close approximation is possible via the corresponding linearized statistic. We define and use automated linearization as a shortcut in finding the linearized statistic. Its variance is easy to state, to interpret and to estimate. The variance components are expressed in terms of residuals, similar to those of standard regression theory. Visual inspection of the residuals reveals how the different components of the complex auxiliary information interact and work together toward reducing the variance.     

Uncertainty Estimation for Pseudo‐Bayesian Inference Under Complex Sampling

Social and economic studies are often implemented as complex survey designs. For example, multistage, unequal probability sampling designs utilised by federal statistical agencies are typically constructed to maximise the efficiency of the target domain level estimator (e.g. indexed by geographic area) within cost constraints for survey administration. Such designs may induce dependence between the sampled units; for example, with employment of a sampling step that selects geographically indexed clusters of units. A sampling‐weighted pseudo‐posterior distribution may be used to estimate the population model on the observed sample. The dependence induced between coclustered units inflates the scale of the resulting pseudo‐posterior covariance matrix that has been shown to induce under coverage of the credibility sets. By bridging results across Bayesian model misspecification and survey sampling, we demonstrate that the scale and shape of the asymptotic distributions are different between each of the pseudo‐maximum likelihood estimate (MLE), the pseudo‐posterior and the MLE under simple random sampling. Through insights from survey‐sampling variance estimation and recent advances in computational methods, we devise a correction applied as a simple and fast postprocessing step to Markov chain Monte Carlo draws of the pseudo‐posterior distribution. This adjustment projects the pseudo‐posterior covariance matrix such that the nominal coverage is approximately achieved. We make an application to the National Survey on Drug Use and Health as a motivating example and we demonstrate the efficacy of our scale and shape projection procedure on synthetic data on several common archetypes of survey designs.     

Further developments in estimation of the largest mean ofK normal populations

We revisit the bounded maximal risk point estimation problem as well as the fixed-width confidence interval estimation problem for the largest mean amongk(≥2) independent normal populations having unknown means and unknown but equal variance. In the point estimation setup, we devise appropriate two-stage and modified two-stage methodologies so that the associatedmaximal risk can bebounded from aboveexactly by a preassigned positive number. Kuo and Mukhopadhyay (1990), however, emphasized only the asymptotics in this context. We have also introduced, in both point and interval estimation problems,accelerated sequential methodologies thereby saving sampling operations tremendously over the purely sequential schemes considered in Kuo and Mukhopadhyay (1990), but enjoying at the same time asymptotic second-order characteristics, fairly similar to those of the purely sequential ones.     

Using Remote Sensing for Agricultural Statistics

Remote sensing can be a valuable tool for agricultural statistics when area frames or multiple frames are used. At the design level, remote sensing typically helps in the definition of sampling units and the stratification, but can also be exploited to optimise the sample allocation and size of sampling units. At the estimator level, classified satellite images are generally used as auxiliary variables in a regression estimator or for estimators based on confusion matrixes. The most often used satellite images are LANDSAT-TM and SPOT-XS. In general, classified or photo-interpreted images should not be directly used to estimate crop areas because the proportion of pixels classified into the specific crop is often strongly biased. Vegetation indexes computed from satellite images can give in some cases a good indication of the potential crop yield.     

Sampling and Estimation Issues for Annual and Sub-annual Canadian Business Surveys

A typical Business Register (BR) is mainly based on administrative data files provided by organisations that produce them as a by-product of their function. Such files do not necessarily yield a perfect Business Register. A good BR should have the following characteristics: (1) It should reflect the complex structures of businesses with multiple activities, in multiple locations or with multiple legal entities; (2) It should be free of duplication, extraneous or missing units; (3) It should be properly classified in terms of key stratification variables, including size, geography and industry; (4) It should be easily updateable to represent the "newer" business picture, and not lag too much behind it. In reality, not all these desirable features are fully satisfied, resulting in a universe that has missing units, inaccurate structures, as well as improper contact information, to name a few defects.
These defects can be compensated by using sampling and estimation procedures. For example, coverage can be improved using multiple frame techniques, and the sample size can be increased to account for misclassification of units and deaths on the register. At the time of estimation, auxiliary information can be used in a variety of ways. It can be used to impute missing variables, to treat outliers, or to create synthetic variables obtained via modelling. Furthermore, time lags between the birth of units and the time that they are included on the register can be accounted for appropriately inflating the design-based estimates.     

HUMAN CAPITAL MEASUREMENT: A SURVEY

Abstract After a short history of the concept of human capital (henceforth HC) in economic thought ( Section 1 ), this study presents the two main methods for estimating the value of the stock of HC – the retrospective and prospective one – with a review of the models proposed ( Section 2 ). These methods are linked both to the theory of HC investment as a rational choice ( Section 3 ), the literature analysing the contribution of HC investment to economic growth and the HC estimating method through educational attainment ( Section 4 ). The more recent literature on HC as a latent variable is also assessed ( Section 5 ) and a new method of estimation where HC is seen both as an unknown function of formative indicators and as a ‘latent effect’ underlying earned income is proposed ( Section 6 ). Section 7 concludes.     

Estimation of variance in multi-stage sampling

In multi-stage sampling when selection is without replacement at the first stage, estimation of the variance of the estimate of the population total is often done assuming sampling with replacement. This estimate is biased and the degree of bias is not negligible. In this paper, a procedure which gives unbiased estimates of the variance making use of only estimated primary sampling unit totals is suggested for the case when sampling at the second and subsequent stages is simple random without replacement. This procedure is based on sub-samples drawn from the selected second and subsequent stage units.     

Estimating economic relations from incomplete cross-section/time-series data

Authors dealing with combined cross-section/time-series data usually assume that complete time-series exist for all units under observation. In the context of micro data, however, this may be a very restrictive assumption. The paper is concerned with problems of model specification and estimation when the data at hand are incomplete time-series from a sample of micro units. Particular attention is paid to a situation where the sample of micro units ‘rotates’ over time. The main results are compared with those derived by Nerlove and others for the standard specification with complete cross-section/time-series data. Some illustrative examples based on data from Norwegian household budget surveys are also given.     

Selecting Adaptive Survey Design Strata with Partial R‐indicators

Recent survey literature shows an increasing interest in survey designs that adapt data collection to characteristics of the survey target population. Given a specified quality objective function, the designs attempt to find an optimal balance between quality and costs. Finding the optimal balance may not be straightforward as corresponding optimisation problems are often highly non‐linear and non‐convex. In this paper, we discuss how to choose strata in such designs and how to allocate these strata in a sequential design with two phases. We use partial R‐indicators to build profiles of the data units where more or less attention is required in the data collection. In allocating cases, we look at two extremes: surveys that are run only once, or infrequent, and surveys that are run continuously. We demonstrate the impact of the sample size in a simulation study and provide an application to a real survey, the Dutch Crime Victimisation Survey.     

A Destructive Sampling Method Designed for Outsourcing Situations Involving High Quality Production Processes

In outsourcing situations involving manufacturing and assembly, the sampling of units produced is important since in many situations not all of the units can be tested. Destructive sampling, which commonly occurs in the assembly and manufacturing industry, is a form of sampling where all units produced cannot be tested since the parts are destructively tested. In this situation, sampling techniques are used to determine if an entire lot should be accepted or rejected based on the sampling results. The traditional sampling techniques include single or classical sampling, double sampling, multiple sampling, skip-lot sampling, chain sampling and MIL-STD-105E. The purpose of this paper is to develop a sophisticated technique that monitors quality at a lower cost than traditional methods yet has similar monitoring characteristics in situations where quality is high and tests are destructive. The proposed technique, Destructive Sampling Method for High Quality production processes (DSM-HQ), is based on a cost function, which balances the costs of sampling versus the costs of finding a defect on the field. DSM-HQ assumes to have a Poisson process defect pattern and uses an Empirical Bayesian analysis to allow the researcher to include prior knowledge.     

Procedures to determine optimum two-stage sampling plans by attributes

Summary In order to compare two sampling plans we use the minimax regret principle, i.e. the minimax principle applied to regret functions. It is shown that among all two-stage sampling plans there exists an optimum sampling plan which can be computed with the aid of a procedure presented in this paper; furthermore another procedure is described how to obtain an approximately optimum two-stage sampling plan in a more direct way. Finally only those two-stage sampling plans are regarded which satisfy an additional condition; among these sampling plans an optimum one exists and is to be determined, too.     

On the initialization of adaptive learning in macroeconomic models

We review and evaluate methods previously adopted in the applied literature of adaptive learning in order to initialize agents’ beliefs. Previous methods are classified into three broad classes: equilibrium-related, training sample-based, and estimation-based. We conduct several simulations comparing the accuracy of the initial estimates provided by these methods and how they affect the accuracy of other estimated model parameters. We find evidence against their joint estimation with standard moment conditions: as the accuracy of estimated initials tends to deteriorate with the sample size, spillover effects also deteriorate the accuracy of the estimates of the model’s structural parameters. We show how this problem can be attenuated by penalizing the variance of estimation errors. Even so, the joint estimation of learning initials with other model parameters is still subject to severe distortions in small samples. We find that equilibrium-related and training sample-based initials are less prone to these issues. We also demonstrate the empirical relevance of our results by estimating a New Keynesian Phillips curve with learning, where we find that our estimation approach provides robustness to the initialization of learning. That allows us to conclude that under adaptive learning the degree of price stickiness is lower compared to inferences under rational expectations.     

Efficient semiparametric estimation for endogenously stratified regression via smoothed likelihood

This paper presents efficient semiparametric estimators for endogenously stratified regression with two strata, in the case where the error distribution is unknown and the regressors are independent of the error term. The method is based on the use of a kernel-smoothed likelihood function which provides an explicit solution for the maximization problem for the unknown density function without losing information in the asymptotic limit. We consider both standard stratified sampling and variable probability sampling, and allow for the population shares of the strata to be either unknown or known a priori.     

Nonparametric function estimation subject to monotonicity, convexity and other shape constraints

This paper uses free-knot and fixed-knot regression splines in a Bayesian context to develop methods for the nonparametric estimation of functions subject to shape constraints in models with log-concave likelihood functions. The shape constraints we consider include monotonicity, convexity and functions with a single minimum. A computationally efficient MCMC sampling algorithm is developed that converges faster than previous methods for non-Gaussian models. Simulation results indicate the monotonically constrained function estimates have good small sample properties relative to (i) unconstrained function estimates, and (ii) function estimates obtained from other constrained estimation methods when such methods exist. Also, asymptotic results show the methodology provides consistent estimates for a large class of smooth functions. Two detailed illustrations exemplify the ideas.     

MAJORITY RULE AND ELECTION MODELS

Abstract. This paper surveys the theoretical work that has been done on majority rule and economic models of elections. Section I provides an overview of the topic. Section 2 reviews the most important results that have been obtained about majority rule as an abstract collective choice rule. Section 3 identifies some alternative inferences that can be made from those results. Section 4 covers some models that include additional features that are present in political institutions where majority rule is used. Section 5 concentrates on some alternative election models and equilibrium concepts. Section 6 focuses on election models with abstentions and/or candidate uncertainty about voter behaviour. Section 7 compares and contrasts models where candidates are certain about what the voters'choices will be (contingent upon about the choices made by the candidates) and models where they are uncertain about those choices. Section 8 closes the survey by identifying some emerging areas of research.     

A Simulation Study of Joint Uses of Data Envelopment Analysis and Statistical Regressions for Production Function Estimation and Efficiency Evaluation

A previous paper by Arnold, Bardhan, Cooper and Kumbhakar (1996) introduced a very simple method to estimate a production frontier by proceeding in two stages as follows: Data Envelopment Analysis (DEA) is used in the first stage to identify efficient and inefficient decision-making units (DMUs). In the second stage the thus identified DMUs are incorporated as dummy variables in OLS (ordinary least squares) regressions. This gave very satisfactory results for both the efficient and inefficient DMUs. Here a simulation study provides additional evidence. Using this same two-stage approach with Cobb-Douglas and CES (constant elasticity-of-substitution) production functions, the estimated values for the coefficients associated with efficient DMUs are found to be not significantly different from the true parameter values for the (known) production functions whereas the parameter estimates for the inefficient DMUs are significantly different. A separate section of the present paper is devoted to explanations of these results. Other sections describe methods for estimating input-specific inefficiencies from the first stage use of DEA in the two-stage approaches. A concluding section provides further directions for research and use.