On a Minimal Essentially Complete Class of Experiments

Abstract

The purpose of this paper is to show that a certain class of experiments is minimal essentially complete asymptotically and to demonstrate that this is not generally the case for finite sample sizes. The model to be discussed involves linear experiments consisting of uncorrelated observations. That is, the experimenter may choose $n$ uncorrelated random variables $Y(x_1), Y(x_2), \cdots, Y(x_n)$ with expectation, \begin{equation*}\tag{1.1}E\lbrack Y(x)\rbrack = x'\theta\end{equation*} where $\theta$ is a parameter vector in $k$ dimensional space $E^{(k)}$ and $x$ is a vector also in $E^{(k)}$, to some extent to be chosen by the experimenter, and the variance $V\lbrack Y(x)\rbrack = \sigma^2$. An experiment, $e$, of size $n$ is fully specified, using the terminology of Elfving [6], by a spectrum $(x_1, x_2, \cdots, x_r)$ consisting of the different $x$'s and an allocation $(n_1, n_2, \cdots, n_r)$ where $n_1 + n_2 + \cdots + n_r = n$. Thus, $e$ can be represented by $e = (n_1, n_2, \cdots, n_r; x_1, x_2, \cdots, x_r)$. We are concerned with the case where $x$ is restricted to lie in a set $A$. The questions to be discussed relate to the choice of the $x$'s in $A$. Some results in this direction have previously been obtained and are discussed by Elfving [5], [6], [7], Ehrenfeld [2], [3] and Kiefer [9], [10]. The information matrix, $F(e)$, associated with experiment $e$, is given by \begin{equation*}\tag{1.2}F(e) = n_1x_1x_1' + \cdots + n_rx_rx_r' = n(p_1x_1x_r' + \cdots + p_rx_rx_r')\end{equation*} where $p_j = n_j/n$ is called the relative allocation at $x_j$. Also, $p_j \geqq 0$ and $p_1 + p_2 + \cdots + p_r = 1$. Strictly speaking, in the exact theory, the $p_j$'s can only range over multiples of $1/n$. However, in the approximate or asymptotic theory, the $p_j$'s have been allowed to range continuously from 0 to 1, see Elfving [5], [6] and Kiefer [9], [10]. We will show later that the exact and asymptotic theory differ in essential ways. We will assume throughout that the set $A$ is symmetric about the origin. This can be done, without essential restriction, since $-Y(x)$ is automatically available with $Y(x)$, see Elfving [6]. Before stating some results we have to introduce some notation for certain classes of experiments. In the approximate theory we denote by $\mathfrak{E}\lbrack A\rbrack$ the set of experiments $e$ with the $x$'s restricted to be in $A$. We assume that any $e \varepsilon \mathfrak{E}\lbrack A\rbrack$ is described by a spectrum $(x_1, x_2, \cdots, x_r)$ consisting of a finite number of $x$'s in $A$ and a relative allocation $(p_1, p_2, \cdots, p_r)$. The sample size plays no role here since we are dealing with the asymptotic case. Also, the value of $r$ is part of the choice of the experiment. In the exact theory, we denote by $\mathfrak{E}_N\lbrack A\rbrack$ the set of experiments $e$ where the $x$'s are restricted to be in set $A$ and the sample size $n \leqq N$. To compare experiments, we introduce a partial ordering $e_1 \geqq e_2$ which will mean that \begin{equation*}\tag{1.3}V_{e_1}\lbrack t'\hat\theta\rbrack \leqq V_{e_2}\lbrack t'\hat\theta\rbrack\quad \text{for all} t\end{equation*} where $V_e\lbrack t'\hat\theta\rbrack$ denotes the variance of the least square estimate of $t'\theta$ with experiment $e$. When $t'\theta$ is not estimable with respect to $e, V_e\lbrack t'\hat\theta\rbrack$ is set equal to infinity. It was shown, Ehrenfeld [3], that relation (1.3) is equivalent to $F(e_1) - F(e_2)$ being a non-negative definite matrix. Finally, we consider comparing two classes of experiments $\mathfrak{E}_1$ and $\mathfrak{E}_2$. We say that $\mathfrak{E}_1$ is essentially complete with respect to $\mathfrak{E}_2$ when for any $e_2 \varepsilon \mathfrak{E}_2$ there exists an $e_1 \varepsilon \mathfrak{E}_1$ such that $e_1 \geqq e_2$. Furthermore, $\mathfrak{E}_1$ is said to be a minimal essentially complete class when no proper subset of $\mathfrak{E}_1$ is complete with respect to $\mathfrak{E}_2$. We also define a weaker kind of minimality, designated by minimal essential completeness ($W$), in the exact or asymptotic case. Suppose $\mathfrak{E}$, depends on a set $B$ of experimental points. The essentially complete class $\mathfrak{E}_1$, is minimal essentially complete $(W)$ with respect to $\mathfrak{E}_2$. That is, one cannot remove any elements of $B$ without losing essential completeness. Various results concerning essentially complete classes were proven and discussed by Ehrenfeld [3], Elfving [6] and Kiefer [9], [10]. Let $E(A)$ denote the extreme points of the set $A$. A point $x$ is an extreme point of $A$ if it is not a convex combination of two or more points of the set $A$. We will show when $A$ is compact that, in the asymptotic case, $\mathfrak{E}\lbrack E(A)\rbrack$ is minimal essentially complete $(W)$ with respect to $\varepsilon \lbrack A\rbrack$. Furthermore, we will demonstrate, with an example and some general theorems, that in the exact case $\mathfrak{E}_N\lbrack E(A)\rbrack$ is not necessarily even essentially complete with respect to $\mathfrak{E}_N\lbrack A\rbrack$.