The Annals of Mathematical Statistics

On Continuous Sufficient Statistics

J. L. Denny

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In [1], [2] we construct for each integer $n \geqq 2$, a real-valued, bounded, uniformly continuous statistic defined on $R^n$, nondecreasing in each real argument, which is a minimal sufficient statistic for the family of all probability distributions defined on the Borel field $\beta^n$ in $R^n$ and dominated by Lebesgue measure $\lambda_n$. In this paper let $\{P_\theta\}$ be a family of probability distributions dominated by Lebesgue measure and defined on the restriction of $\beta^n$ to a Borel set $A \subset R^n$. Let $f = (f_1, \cdots, f_k)$ and $g = (g_{11}, \cdots, g_{1n_1}, \cdots, g_{k1}, \cdots g_{kn_k})$ be continuous sufficient statistics for $\{P_\theta\}$ defined on $A$, with $f_i$ and $g_{ij}$ real-valued. If there are $k$ functions $h_i:R^1 \rightarrow R^{n_i}, i = 1, \cdots, k$ so that $(g_{i1}, \cdots, g_{in_i}) = h_i \circ f_i$ a.e. $(\lambda_n)$, then is $g$ everywhere a continuous function of $f$, i.e., $g = h \circ f$ for continuous $h:f\lbrack A \rbrack \rightarrow g\lbrack A \rbrack$? If in addition $n_i = 1, i = 1, \cdots, k$ and each $h_i$ is a 1-1 function, are $f$ and $g$ identical, i.e., $g = h \circ f$ for bicontinuous $h:f\lbrack A \rbrack \rightarrow g\lbrack A \rbrack$? Now if (1) $A$ is connected, (2) $A$ has a dense interior, and (3) almost every linear section of each $f_i$ (and $g_i$ in the second case) satisfy Lusin's condition $(N)$, the answer to the above questions is affirmative (see Section 2 for definitions). But if at least one of (1), (2), or (3) is not satisfied, an affirmative answer is not in general possible (see Examples, Section 3). In Section 5 we show that this implies it is not possible to find a real-valued continuous minimal sufficient statistic $f$ defined on $R^n$ such that almost every linear section of $f$ satisfies Lusin's condition $(N)$, for some familiar probability distributions.

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Ann. Math. Statist., Volume 35, Number 3 (1964), 1229-1233.

First available in Project Euclid: 27 April 2007

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Denny, J. L. On Continuous Sufficient Statistics. Ann. Math. Statist. 35 (1964), no. 3, 1229--1233. doi:10.1214/aoms/1177703280.

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