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

Consider experiments of the following type. Observation is made of a univariate random variable $X$ whose absolutely continuous distribution function $F (x) mid heta)$ and probability density function $p(x mid heta)$ are functions of a real unknown parameter $ heta$. 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 $ heta$-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 $ heta 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 $mathscr{I}lbrackepsilon_1, F( heta) brack geqq |mathscr{I}lbrackepsilon_2, F( heta) brack$ for all "prior" distribution functions $F( heta)$ giving probability one to $Theta,$ where $mathscr{I}lbrackepsilon_i, F( heta) brack$ is the mean Shannon information given by $epsilon_i$ about $ heta$ when $ heta$ has the prior distribution function $F( heta).$ (c) When the Fisher imnformations $I_i( heta) = int^infty_{-infty} p(x_i | heta) igglbrackfrac{partial}{partial heta} log p(x_i | heta) igg brack^2 dx_i, quad i = 1, 2,$ are definable for $ heta 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 $I_1( heta) geqq I_2( heta)$ for $ heta 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 $I_1( heta)$ and $I_2( heta)$ are not even defined for $ heta varepsilon Theta.$ When $ heta$ is a location parameter, $p(x mid heta) = flbrack x - heta brack,$ say. For fixed $flbrack cdot brack$ consider the class of experiments ${epsilon(c) mid c > 0},$ where $epsilon(c)$ is the experiment determined by the probability density function cf$lbrack c(x - 0) brack.$ 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 $N( heta, 1)$ 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 $R^1.$