Bayesian Methods for Finite Population Sampling

Malay Ghosh, Glen Meeden

• Publisher: Routledge
• Year: 2021
• eBook ISBN: 9781351464413
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

Assuming a basic knowledge of the frequentist approach to finite population sampling, Bayesian Methods for Finite Population Sampling describes Bayesian and predictive approaches to inferential problems with an emphasis on the likelihood principle. The authors demonstrate that a variety of levels of prior information can be used in survey sampling in a Bayesian manner. Situations considered range from a noninformative Bayesian justification of standard frequentist methods when the only prior information available is the belief in the exchangeability of the units to a full-fledged Bayesian model. Intended primarily for graduate students and researchers in finite population sampling, this book will also be of interest to statisticians who use sampling and lecturers and researchers in general statistics and biostatistics.

CHAPTER 1

Bayesian foundations

In this chapter we present the underlying foundations of finite population sampling and describe the Bayesian and predictive approach to inferential problems in this area. In Section 1.1 we introduce the notation we will be using throughout the book. In Section 1.2 we restate the work of Basu and Ghosh (1967) which identifies the minimal sufficient statistic for a finite population sampling problem. In Section 1.3 we summarize Basu’s work (1969) which describes how the likelihood principle should be applied in finite population sampling. In Section 1.4 we outline the usual Bayesian approach to finite population sampling with particular emphasis on the work of Ericson (1969b). In Section 1.5 we review an approach to finite population sampling which only assumes that the posterior expectation is linear. Finally in Section 1.6 we give a brief outline of the rest of the book.

1.1 Notation

In this section we will introduce the notation which we shall use throughout the book. For now, we will concentrate on the simplest situation in finite population sampling. Let $\mathcal{U}$ denote a finite population which consists of N units labelled 1, 2,…, N. We will assume that these labels are known and that they often can contain some information about the units. Attached to unit i let y_i be the unknown value of some characteristic of interest. Typically, y_i will be a real number. For this problem y = (y₁,…, y_N)^T is the unknown state of nature or parameter. y is assumed to belong to $\mathcal{Y}$, a subset of N-dimensional Euclidean space, ${\mathcal{R}}^N$. The statistician usually has some prior information about y and this could influence the choice of $\mathcal{Y}$. However, in most cases convenience and tradition seem to dictate the choice of the parameter space and usually $\mathcal{Y}$ is taken to be ${\mathcal{R}}^N$. In what follows we will sometimes assume that the parameter space is a finite subset of ${\mathcal{R}}^N$ of a particular special form. If b = (b₁,…, b_k)^T is a k-dimensional vector of distinct real numbers then we let

$\mathcal{Y}(b)=\left\{y:\text{such that for}i=1,\dots ,N,y_i=b_j\text{for}\text{some}j=1,\dots ,k\right\}.$

(1.1)

In many problems in finite population sampling there are additional characteristics or variables associated with each unit which are known to the statistician. For unit i let x_i denote a possible vector of other characteristics, all of which are assumed to be known. We let x denote the collection of these vectors for the entire population. Hence in the usual frequentist theory the x_i’s and their possible relationship to the y_i’s summarize the statistician’s prior information about y.

A subset s of {1, 2,…, N } is called a sample. Let n(s) denote the number of elements belonging to s. Let S denote the set of all possible samples. A (nonsequential) sampling design is a function p defined on S such that p(s) ∈ [0, 1] for every nonempty s ∈ S and Σ_s∈S p(s) = 1. Given $y\in \mathcal{Y}$ and s = {i₁, …, i_n(s)}, where 1 ≤ i₁ < · · · < i_n(s) < N let y(s) = (y_i1...

Table of contents

• Cover
• Dedication
• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 Bayesian foundations
• 2 A noninformative Bayesian approach
• 3 Extensions of the Polya posterior
• 4 Empirical Bayes estimation
• 5 Hierarchical Bayes estimation
• References
• Author index
• Subject index