Hypothesis testing with multiterminal data compression

The multiterminal hypothesis testing <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H: XY</tex> against <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H̄: X̄Ȳ</tex> is considered where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X^{n} (X̄^{n})</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y^{n} (Ȳ^{n})</tex> are separately encoded at rates <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_{1}</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_{2}</tex> , respectively. The problem is to determine the minimum <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\beta_{n}</tex> of the second kind of error probability, under the condition that the first kind of error probability <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha_{n} \leq \epsilon</tex> for a prescribed <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0 &lt; \epsilon &lt; 1</tex> . A good lower bound <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\theta_{L}(R_{1}, R_{2})</tex> on the power exponent <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\theta (R_{1}, R_{2},\epsilon)= \lim \inf_{n \rightarrow \infty}(-1/n \log \beta_{n})</tex> is given and several interesting properties are revealed. The lower bound is tighter than that of Ahlswede and Csiszár. Furthermore, in the special case of testing against independence, this bound turns out to coincide with that given by them. The main arguments are devoted to the special case with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_{2} = \infty</tex> corresponding to full side information for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y^{n}(Ȳ^{n})</tex> . In particular, the compact solution is established to the complete data compression cases, which are useful in statistics from the practical point of view.