CsiszÁr's Cutoff Rates for the General Hypothesis Testing Problem

Abstract

In , Csisza/spl acute/r established the concept of forward /spl beta/-cutoff rate for the error exponent hypothesis testing problem based on independent and identically distributed (i.i.d.) observations. Given /spl beta/<0, he defined the forward /spl beta/-cutoff rate as the number R/sub 0//spl ges/0 that provides the best possible lower bound in the form /spl beta/(E-R/sub 0/) to the type 1 error exponent function for hypothesis testing where 0<E<R/sub 0/ is the rate of exponential convergence to 0 of the type 2 error probability. He then demonstrated that the forward /spl beta/-cutoff rate is given by D/sub 1/(1-/spl beta/)/(X/spl par//spl circ/X), where D/sub /spl alpha//(X/spl par//spl circ/X) denotes the Re/spl acute/nyi /spl alpha/-divergence [19], /spl alpha/>0, /spl alpha//spl ne/1. Similarly, for 0</spl beta/<1, Csisza/spl acute/r also established the concept of reverse /spl beta/-cutoff rate for the correct exponent hypothesis testing problem. In this work, we extend Csisza/spl acute/r's results by investigating the forward and reverse /spl beta/-cutoff rates for the hypothesis testing between two arbitrary sources with memory. We demonstrate that the lim inf Re/spl acute/nyi /spl alpha/-divergence rate provides the expression for the forward /spl beta/-cutoff rate. We also show that if the log-likelihood large deviation spectrum admits a limit, then the reverse /spl beta/-cutoff rate equals the liminf /spl alpha/-divergence rate, where /spl alpha/=(1/1-/spl beta/) and 0</spl beta/</spl beta//sub max/, where /spl beta//sub max/ is the largest /spl beta/<1 for which the lim inf (1/1-/spl beta/)-divergence rate is finite. For /spl beta//sub max//spl les//spl beta/<1, we show that the reverse cutoff rate is in general only upper-bounded by the lim inf Re/spl acute/nyi divergence rate. Unlike in , where the alphabet for the source coding cutoff rate problem was assumed to be finite, we assume arbitrary (countable or continuous) source alphabet. We also provide several examples to illustrate our forward and reverse /spl beta/-cutoff rates results and the techniques employed to establish them.