Lynne Stokes

PARAMETRIC RANKED SET SAMPLING

Ranked set sampling was introduced by McIntyre (1952,Australian Journal of Agricultural Research,3, 385–390) as a cost-effective method of selecting data if observations are much more cheaply ranked than measured. He proposed its use for estimating the population mean when the distribution of the data was unknown. In this paper, we examine the advantage, if any, that this method of sampling has if the distribution is known, for a specific family of distributions. Specifically, we consider estimation of μ and σ for the family of random variables with cdfs of the formF(x−μ/σ). We find that the ranked set sample does provide more information about both μ and σ than a random sample of the same number of observations. We examine both maximum likelihood and best linear unbiased estimation of μ and σ, as well as methods for modifying the ranked set sampling procedure to provide even better estimation.

Estimating the Number of Classes in a Finite Population

Abstract We use an extension of the generalized jackknife approach of Gray and Schucany to obtain new nonparametric estimators for the number of classes in a finite population of known size. We also show that generalized jackknife estimators are closely related to certain Horvitz–Thompson estimators, to an estimator of Shlosser, and to estimators based on sample coverage. In particular, the generalized jackknife approach leads to a modification of Shlossers estimator that does not suffer from the erratic behavior of the original estimator. The performance of both new and previous estimators is investigated by means of an asymptotic variance analysis and a Monte Carlo simulation study.

Considerations of cost trade-offs in insurance solvency surveillance policy

Abstract We propose a process for identifying potentially insolvent insurers on a cost-effective basis. A loss cost function is developed such that the effectiveness of monitoring is maximized relative to a cost constraint. The loss cost function is supported by a model that provides a rank ordering of financial institutions according to their probability of insolvency. When tested against a full sample of property-liability insurance companies , the procedure provides information critical to maximizing the effectiveness of regulatory resources available for solvency surveillance and performs well as a predictor of insolvency. Likewise, the rank ordering of insurers overcomes an estimation problem critical to establishing risk-adjusted guaranty assessments.

Estimation of the CDF of a finite population in the presence of a calibration sample

We compare the performance of a number of estimators of the cumulative distribution function (CDF) for the following scenario: imperfect measurements are taken on an initial sample from afinite population and perfect measurements are obtained on a small calibration subset of the initial sample. The estimators we considered include two naive estimators using perfect and imperfect measurements; the ratio, difference and regression estimators for a two-phasesample; a minimum MSE estimator; Stefanski and Bays SIMEX estimator (1996); and two proposed estimators. The proposed estimators take the form of a weighted average of perfect and imperfect measurements. They are constructed by minimizing variance among the class of weighted averages subject to an unbiasedness constraint. They differ in the manner of estimating the weight parameters. The first one uses direct sample estimates. The second one tunes the unknown parameters to an underlying normal distribution. We compare the root mean square error (RMSE) of the proposed estimator against other potential competitors through computer simulations. Our simulations show that our second estimator has the smallest RMSE among thenine compared and that the reduction in RMSE is substantial when the calibration sample is small and the error is medium or large.

Special Issue on Statistical Design and Analysis with Ranked Set Samples

Ranked set sampling (rss) is an old concept, as statistical ideas go. It was introduced nearly fifty years ago as an efficient method to gather data for estimating pasture yield (McIntyre 1952). The statistical theory that confirmed the improved efficiency was developed nearly two decades later (Takahasi and Wakimoto 1968, Dell and Clutter 1972). Ranked set sampling provides a way of selecting a sample of units from a population that will allow more efficient estimation of certain parameters than would be possible from a simple random sample. The methods of survey sampling, or sampling from a finite population, have been doing this for years. However, most improved designs from survey sampling require knowledge about each member of the population prior to sampling, such as stratum membership or a proxy size measurement. Exceptions to this are post-stratification or double sampling methods, which do not require any prior knowledge about individual population units. Both of these techniques have some conceptual similarities to rss. In the last few years there has been an explosion of interest in and a tremendous amount of methodological development of ranked set sampling procedures. (Figure 1 illustrates this increasing interest by showing the number of papers on the topic published in refereed journals over time.) One reason for this increase is the recognition by statisticians of the need for more cost-effective sampling procedures, such as those that use a priori knowledge or can otherwise provide the needed information with a significant reduction in cost over the more traditional simple random sampling (srs) approaches. Nowhere is this need more evident than in the field of environmental monitoring and assessment. In the past, the assessment of most environmental problems was relatively straight forward. Many of the major environmental problems could be detected using the human senses (the Cuyahoga river in Ohio was burning because of chemical wastes being discharged directly into the river; the air in LA and other large cities could be seen and smelled). In the last thirty years, our knowledge of anthropogenic pollution and our ability to measure minute quantities (parts per billion or trillion) of toxic chemicals in our environment has dramatically improved. Now we have identified hundreds of man made toxic chemicals in our environment. The cost of measuring and monitoring these chemicals and assessing their environmental impact is extremely high. It requires sophisticated measurement and careful sampling of large potentially Environmental and Ecological Statistics 6, 5±9 (1999)

Estimation of Interviewer Effects for Categorical Items in a Random Digit Dial Telephone Survey

Abstract The loss of precision in estimates of means due to variability among interviewers can be substantial for some questionnaire items and survey designs. The most commonly used methods for estimating the magnitude of this loss are inappropriate for binary items in complex surveys. This article shows how parameters from a model for variance components in binary variables (Anderson and Aitkin 1985) are related to the increased variance of population estimates. It is suggested that one of these parameters, a measure of correlation between interviewer observations on a latent variable, may be more appropriate than the intrainterviewer correlation p for measuring the magnitude of interviewer effects. This is because it is unaffected by the level of the attribute in the population, which is not true of p. Interviewer effects for a household respondents recorded labor-force status in a random digit dial telephone survey are examined using the new model and estimation process. Small but positive effects are...

Estimating the Size of a Subdomain: An Application in Auditing

This article suggests an alternative to the ratio estimator for estimating the total size of a subdomain of a population. The application that served as the genesis for this work is from auditing. The problem is to estimate the total of sales transactions that are not tax exempt from an audit sample of the population of nontaxed sales transactions. A superpopulation approach, which models the units probability of belonging to the subdomain as a function of its size, leads to a family of estimators. The simplest member of this famiiy is one in which that function is specified to be a constant. The optimal estimator for this model performs markedly better than the ratio estimator when the assumption is true and often performs better when it is not, though in that case it is biased. Stratification is shown to reduce this bias and at the same time make the ratio estimator more similar to the optimal estimator. A simulation experiment shows that the theoretical advantages hold in a real audit population.

Introduction to Variance Estimation

This book covers stochastic integration with respect to square-integrable martingales. Although a general construction is given for right-continuous martingales with left limits, and the compensated Poisson process is treated as an example, the book’s focus is very much on continuous martingales, which places it in the tradition of introductory texts like those by Arnold (1974), Øksendal (2000), and Mikosch (1998). Because the material is somewhat classical, the value of this book very much depends on how the topic is presented. And here the author succeeds indeed by giving a well-motivated and detailed pedagogical account. His approach can be best explained by the way in which he introduces the stochastic integral. In the introductory chapter, he reviews the Riemann–Stieltjes integral in a way not often found in calculus textbooks. In the special case where the integrand coincides with the integrator function, two different approximations by sums are considered, depending whether the integrand is sampled at the left or right endpoint of the partition interval. The limit of the difference of these two sums, for shrinking grid size, is called (if it exists) the quadratic variation of the function. In this way, the author provides a link to the established knowledge of the reader and at the same time gives one of the main motivations for considering more general integrals: In the event that the integrator has nonvanishing quadratic variation, as is the case with Brownian sample paths, we cannot expect the Riemann–Stieltjes theory to be carried forward. Next, the Wiener integral of a square-integrable deterministic function (possibly of unbounded variation) toward a Brownian motion is discussed. This is done by the usual isometric extension, which can be carried out in this context in a rather painless way. After introducing the basic concepts like conditional expectation, Brownian motion, and martingales, the text next defines the stochastic integral with respect to a Brownian motion followed by simple examples that are evaluated by using the very definition of the integral. Then the author moves forward to extend the theory to more general integrators. The next part of the book deals with stochastic calculus. Again, the author follows his pattern by motivating theorems like Itô’s formula with illustrative examples before giving the actual proof. He then proceeds to cover many of the classical applications of Itô’s formula. One whole chapter is devoted to multiple Wiener–Itô integrals, which are used as a vehicle to prove the martingale representation theorem for a Brownian filtration. Stochastic differential equations are treated in detail and linked to Markovian diffusion processes. Historically, the construction of such processes in a single step was one of the great successes of Itô’s theory, because the previous constructions from infinitesimal generators required several steps. In the final chapter, several applications to finance, filtering, and electric circuits, among others, are given. I am sure that this book will be very welcomed by students and lecturers of this subject alike, who will find many illustrative exercises provided. Reader also should not miss out on the Preface, which includes some anecdotes about K. Itô.