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## The Method of Random Groups

### Introduction to Variance Estimation (2007-01-01): 21-106 , January 01, 2007

###
*Abstract*

The random group method of variance estimation amounts to selecting two or more samples from the population, usually using the same sampling design for each sample; constructing a separate estimate of the population parameter of interest from each sample and an estimate from the combination of all samples; and computing the sample variance among the several estimates. Historically, this was one of the first techniques developed to simplify variance estimation for complex sample surveys. It was introduced in jute acreage surveys in Bengal by Mahalanobis (1939, 1946), who called the various samples *interpenetrating samples*. Deming (1956), the United Nations Subcommission on Statistical Sampling (1949), and others proposed the alternative term *replicated samples*. Hansen, Hurwitz, and Madow (1953) referred to the *ultimate cluster* technique in multistage surveys and to the *random group* method in general survey applications. Beginning in the 1990s, various writers have referred to the *resampling* technique. All of these terms have been used in the literature by various authors, and all refer to the same basic method. We will employ the term *random group* when referring to this general method of variance estimation.

## The Bootstrap Method

### Introduction to Variance Estimation (2007-01-01): 194-225 , January 01, 2007

###
*Abstract*

In the foregoing chapters,wediscussed three replication-based methods of variance estimation. Here we close our coverage of replication methods with a presentation of Efron’s (1979) bootstrap method, which has sparked a massive amount and variety of research in the past quarter century. For example, see Bickel and Freedman (1984), Booth, Butler, and Hall (1994), Chao and Lo (1985, 1994), Chernick (1999), Davison and Hinkley (1997), Davison, Hinkley, and Young (2003), Efron (1979, 1994), Efron andTibshirani (1986, 1993, 1997), Gross (1980), Hinkley (1988), Kaufman (1998), Langlet, Faucher, and Lesage (2003), Li, Lynch, Shimizu, and Kaufman (2004), McCarthy and Snowden (1984), Rao,Wu, and Yue (1992), Roy and Safiquzzaman (2003), Saigo, Shao, and Sitter (2001), Shao and Sitter (1996), Shao and Tu (1995), Sitter (1992a, 1992b), and the references cited by these authors.

## Introduction to Variance Estimation

### Introduction to Variance Estimation (2007-01-01) , January 01, 2007

## Variance Estimation for Systematic Sampling

### Introduction to Variance Estimation (2007-01-01): 298-353 , January 01, 2007

###
*Abstract*

The method of systematic sampling, first studied by the Madows (1944), is used widely in surveys of finite populations. When properly applied, the method picks up any obvious or hidden stratification in the population and thus can be more precise than random sampling. In addition, systematic sampling is implemented easily, thus reducing costs.

## Hadamard Matrices

### Introduction to Variance Estimation (2007-01-01): 367-368 , January 01, 2007

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*Abstract*

The orthogonal matrices used in defining half-sample replicates in Chapter 3 are known in mathematics as Hadamard matrices. A Hadamard matrix *H* is a *k* × *k* matrix all of whose elements are +1 or −1 that satisfies *H*′*H* = *k**I*, where *I* is the identity matrix of order *k*. The order *k* is necessarily 1, 2, or 4*t*, with *t* a positive integer.

## Introduction

### Introduction to Variance Estimation (2007-01-01): 1-20 , January 01, 2007

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*Abstract*

The theory and applications of survey sampling have grown dramatically in the last 60 years. Hundreds of surveys are now carried out each year in the private sector, the academic community, and various governmental agencies, both in the United States and abroad. Examples include market research and public opinion surveys; surveys associated with academic research studies; and large nationwide surveys about labor force participation, health care, energy usage, and economic activity. Survey samples now impinge upon almost every field of scientific study, including agriculture, demography, education, energy, transportation, health care, economics, politics, sociology, geology, forestry, and so on. Indeed, it is not an overstatement to say that much of the data undergoing any form of statistical analysis are collected in surveys.

## Asymptotic Theory of Variance Estimators

### Introduction to Variance Estimation (2007-01-01): 369-383 , January 01, 2007

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*Abstract*

Inferences from large, complex sample surveys usually derive from the pivotal quantity

## The Jackknife Method

### Introduction to Variance Estimation (2007-01-01): 151-193 , January 01, 2007

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*Abstract*

In Chapters 2 and 3, we discussed variance estimating techniques based on random groups and balanced half-samples. Both of these methods are members of the class of variance estimators that employ the ideas of subsample replication. Another subsample replication technique, called the jackknife, has also been suggested as a broadly useful method of variance estimation. As in the case of the two previous methods, the jackknife derives estimates of the parameter of interest from each of several subsamples of the parent sample and then estimates the variance of the parent sample estimator from the variability between the subsample estimates.

## Generalized Variance Functions

### Introduction to Variance Estimation (2007-01-01): 272-297 , January 01, 2007

###
*Abstract*

In this chapter, we discuss the possibility of a simple mathematical relationship connecting the variance or relative variance of a survey estimator to the expectation of the estimator. If the parameters of the model can be estimated from past data or from a small subset of the survey items, then variance estimates can be produced for all survey items simply by evaluating the model at the survey estimates rather than by direct computation. We shall call this method of variance estimation the method of *generalized variance functions* (GVF).

## Transformations

### Introduction to Variance Estimation (2007-01-01): 384-397 , January 01, 2007

###
*Abstract*

Transformations find wide areas of application in the statistical sciences. It often seems advantageous to conduct an analysis on a transformed data set rather than on the original data set. Transformations are most often motivated by the need or desire to