Hypothesis testing with communication constraints

A new class of statistical problems is introduced, involving the presence of communication constraints on remotely collected data. Bivariate hypothesis testing, H_{0}: P_{XY} against H_{1}: P_{\={XY}} , is considered when the statistician has direct access to Y data but can be informed about X data only at a preseribed finite rate R . For any fixed R the smallest achievable probability of an error of type 2 with the probability of an error of type 1 being at most \epsilon is shown to go to zero with an exponential rate not depending on \epsilon as the sample size goes to infinity. A single-letter formula for the exponent is given when P_{\={XY}} = P_{X} \times P_{Y} (test against independence), and partial results are obtained for general P_{\={XY}} . An application to a search problem of Chernoff is also given.