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Empirical Bayes estimation in finite population sampling. (English) Zbl 0616.62012
Consider a finite population \({\mathfrak U}\) with units labelled 1,2,...,N. Let \(y_ i\) be the value of a single characteristics attached to the unit i. The vector \(y=(y_ 1,...,y_ n)'\) is the unknown state of nature, and it is assumed to belong to \(\Theta ={\mathbb{R}}^ N\). A subset s of \(\{\) 1,...,N\(\}\) is called a sample. Let S be the set of all possible samples of size n. In an empirical Bayes model, it is assumed that we are at the mth stage of the sampling procedure and that sampling has been repeated (m-1) times. The population size at the jth stage of the experiment is denoted by \(N_ j\), and at that stage a certain characteristic, say, \(y_ i^{(j)}\) \((i=1,...N_ j\); \(j=1,...,m)\) is associated with the ith unit. A fixed sample of size \(n_ j\) is taken at the jth stage, and a typical sample is denoted by \(s_ j\) \((j=1,...,m)\). Let the model \(y_ i^{(j)}=\theta^{(j)}+\epsilon_ i^{(j)}\) \((i=1,...N_ j\); \(j=1,...,m)\), where \(\theta^{(j)}\)’s and \(\epsilon_ i^{(j)}\)’s are all independently distributed with \(\theta^{(j)}\)’s iid \(N(\mu,\sigma^ 2)\) and \(\epsilon_ i^{(j)}\)’s iid \(N(0,\tau^ 2)\). Writing \(B_ j=M/(M+n_ j)\) \((j=1,...,m)\), it follows that at the mth stage of the experiment, the Bayes estimator of \(\gamma (y^{(m)})=N_ m^{-1}\sum^{N_ m}_{i=1}y_ i^{(m)}\) is \[ E[\gamma (y^{(m)})| s_ m,y_{s_ m}]=N_ m^{-1}[n_ m\bar y_{s_ m}+(N_ m-n_ m)(B_ m\mu +(1-B_ m)\bar y_{s_ m})] \] where \(\bar y{}_{s_ j}=n_ j^{-1}\sum_{i\in s_ j}y_ i^{(j)}\) \((j=1,...,m)\). Therefore, one or both of the parameters M and \(\mu\) are unknown and need to be estimated from the data.
In this paper the authors propose an estimator of M as a function of the usual F ratio of between and within mean squares and an estimator of \(\mu\) based on the principle of maximum likelihood. A variety of properties of these empirical Bayes estimators are established both in the general case and in the special case when \(n_ 1=...=n_ m\).
Reviewer: J.P.Vilaplana

MSC:
62D05 Sampling theory, sample surveys
62F15 Bayesian inference
62C12 Empirical decision procedures; empirical Bayes procedures
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