Admissibility and Bayes Estimation in Sampling Finite Populations--V

Abstract

Joshi (1965), (1966) studied in great detail admissible estimation, in relation to survey-sampling. He (1966) also established a property more demanding than admissibility namely uniform admissibility (previously called global admissibility by Godambe (1966)) for the conventional sample mean while estimating the population total. In this paper we establish uniform admissibility of a class of Bayes estimators. Using the notation similar to that of Godambe and Joshi (1965) we denote the population units by integers $1, 2, \cdots, N$. Any subset $s$ of the integers $1,\cdots, N$ is called a sample. If $S$ denotes the set of all possible samples, $(s\varepsilon S)$, any real function $p$ on $S$ such that $\sum_s p(s) = 1$ and $1 \geqq p(s) \geqq 0$, for all $s \varepsilon S$ is called a sampling design. Next we denote by $x_i$ the real value associated with the unit $i (i = 1, \cdots, N)$ of the population. $\mathbf{x} = (x_1, \cdots, x_i, \cdots, x_N)$ is a vector in the $N$-dimensional Euclidean space $R_N$. Any real function $e(\mathbf{x}, s)$ on the product space $R_N \mathbf{\times} S$, such that $e$ depends on $\mathbf{x}$ only through those $x_i$ for which $i \varepsilon s$, is called an estimator. Since in this paper we would be concerned with estimation of the population total $T(\mathbf{x}) = \sum^N_1 x_i$, the terms such as estimator, admissibility, uniform admissibility etc. used subsequently are to be understood in relation to estimation of $T$. Now to distinguish `admissibility' from `uniform admissibility' we introduce the following four definitions. DEFINITION 1.1. For a given sampling design $p$, an estimator $e'$ is said to be superior to the estimator $e$ if for all $\mathbf{x} \varepsilon R_N$, $\sum_s p(s)\lbrack e'(s, \mathbf{x}) - T(\mathbf{x})\rbrack^2 \leqq \sum_s p(s)\lbrack e(s, \mathbf{x}) - T(\mathbf{x})\rbrack^2$ strict inequality being true for at least one $\mathbf{x}$. DEFINITION 1.2. For a given sampling design $p$, an estimator $e$ is said to be admissible if no estimator $e'$ is superior (Definition 1.1) to $e$. DEFINITION 1.3. A pair $(e', p')$ of an estimator $e'$ and a sampling design $p'$ is said to be uniformly superior to another pair $(e, p)$ if for all $\mathbf{x} \varepsilon R_N$, $\sum_s p'(s)\lbrack e'(s, \mathbf{x}) - T(\mathbf{x})\rbrack^2 \leqq \sum_s p(s)\lbrack e(s, \mathbf{x}) - T(\mathbf{x})\rbrack^2$ strict inequality holding for at least one $\mathbf{x}$. DEFINITION 1.4. With respect to a class $C$ of sampling designs, a pair $(e, p)$ of an estimator $e$ and a sampling design $p$ is said to be uniformly admissible if no other pair $(e', p')$ such that $p' \varepsilon C$, is uniformly superior to $(e, p)$, (Definition 1.3). For the discussion of the practical significance of the notion of uniform admissibility, especially if in Definition 1.4, the class $C = C_n$, where \begin{equation*}\tag{1.1}C_n = \{p:\sum_s p(s) \cdot n(s) = \operatorname{const.} = n\}\end{equation*} $n(s)$ being the number of units $i$ such that $i\varepsilon s$, we refer to Joshi ((1966), Section 7). Obviously $C_n$ above is the class of all sampling designs having a fixed `average sample size.' The main result of this paper is the following THEOREM 1.1 With respect to the class $C_n$ in (1.1) of sampling designs the pair $(e^\ast, p^\ast)$, where $e^\ast$ is the estimator given by, \begin{equation*}\tag{1.2}e^\ast(s, \mathbf{x}) = \sum_{i\varepsilon s} x_i + \sum_{i\not\varepsilon s} \lambda_i,\end{equation*} $\lambda_1, \cdots, \lambda_i, \cdots, \lambda_N$ being any arbitrarily fixed numbers and $p^\ast$ is any sampling design belonging to the class $C_n$ in (1.1), is uniformly admissible. (Definition 1.4).