Non-Equivalent Comparisons of Experiments and Their Use for Experiments Involving Location Parameters

Abstract

Consider experiments of the following type. Observation is made of a univariate random variable $X$ whose absolutely continuous distribution function $F (x) \mid \theta)$ and probability density function $p(x \mid \theta)$ are functions of a real unknown parameter $\theta$. Different experiments of this type with random variables $X_1, X_2, \cdots$ will be denoted $\epsilon_1, \epsilon_2, \cdots$ In the following definitions, $\Theta$ represents a subset of $\theta$-values. (a) Following Blackwell [1], $\epsilon_1$ is sufficient for $\epsilon_2$ with respect to $\Theta$ or $\epsilon_1 \succ \epsilon_2(\Theta)$ when there exists a stochastic transformation of $X_1$ (given by a set of distribution functions $\{G(z | x_1) | - \infty < x_1 < \infty\})$ to a random variable $Z$ such that, for each $\theta \epsilon \Theta, Z(X_1)$ and $X_2$ have identical distributions. (b) Following Lindley [3], $\epsilon_1$ is not less Shannon informative than $\epsilon_2$ with respect to $\Theta$ or $\epsilon_1S \geqq \epsilon_2(\Theta)$ when $\mathscr{I}\lbrack\epsilon_1, F(\theta)\rbrack \geqq |mathscr{I}\lbrack\epsilon_2, F(\theta)\rbrack$ for all "prior" distribution functions $F(\theta)$ giving probability one to $\Theta,$ where $\mathscr{I}\lbrack\epsilon_i, F(\theta)\rbrack$ is the mean Shannon information given by $\epsilon_i$ about $\theta$ when $\theta$ has the prior distribution function $F(\theta).$ (c) When the Fisher imnformations $I_i(\theta) = \int^\infty_{-\infty} p(x_i | \theta) \bigg\lbrack\frac{\partial}{\partial\theta} \log p(x_i | \theta) \bigg\rbrack^2 dx_i, \quad i = 1, 2,$ are definable for $\theta \varepsilon |Theta, \epsilon_1$ will be said to be not less Fisher informative than $E_2$ with respect to $\Theta,$ or $\epsilon_1F \geqq \epsilon_2(\Theta),$ when $I_1(\theta) \geqq I_2(\theta)$ for $\theta \varepsilon \Theta.$ Lindley [3] has shown that $\epsilon_1 \succ \epsilon_2(\Theta) \longrightarrow \epsilon_1S \geqq \epsilon_2(\Theta).$ In Theorem 1, we show that under certain conditions $\epsilon_1S \geqq \epsilon_2(\Theta) \longrightarrow \epsilon_1F \geqq \epsilon_2(\Theta).$ If this implication always held, comparison by $F \geqq$ would be more widely applicable than comparison by $S \geqq$ (and a fortiori by $succ$). However the conditions of Theorem 1 suggest that cases exist where $\epsilon_1S \geqq |epsilon_2(\Theta)$ but where $I_1(\theta)$ and $I_2(\theta)$ are not even defined for $\theta \varepsilon \Theta.$ When $\theta$ is a location parameter, $p(x \mid \theta) = f\lbrack x - \theta\rbrack,$ say. For fixed $f\lbrack \cdot \rbrack$ consider the class of experiments $\{\epsilon(c) \mid c > 0\},$ where $\epsilon(c)$ is the experiment determined by the probability density function cf$\lbrack c(x - 0)\rbrack.$ The conditional distribution of $\epsilon(c_1)$ is a contraction of that of $\epsilon(c_2)$ when $c_1 > c_2.$ (Example: $\epsilon(c)$ consisting of $c^2$ observations from the normal distribution $N(\theta, 1)$ and $x$ their mean). In the theorems of Sections 3, 4 and 5, conditions for $\epsilon(c_1) \succ \epsilon(c_2), \epsilon(c_1)S \geqq \epsilon(c_1)F \geqq \epsilon(c_2)$ when $c_1 > C_2$ are given. Unless otherwise indicated integrals will be taken over $R^1.$