Csiszár's cutoff rates for the general hypothesis testing problem

Fady Alajaji*, Po-Ning Chen, Ziad Rached

*Corresponding author for this work

Research output: Contribution to journalLetterpeer-review

5 Scopus citations


In [6], Csiszár established the concept of forward β-cutoff rate for the error exponent hypothesis testing problem based on independent and identically distributed (i.i.d.) observations. Given β < 0, he defined the forward β-cutoff rate as the number R0 ≥ 0 that provides the best possible lower bound in the form β(E - Ro) to the type 1 error exponent function for hypothesis testing where 0 < E < R0 is the rate of exponential convergence to 0 of the type 2 error probability. He then demonstrated that the forward β-cutoff rate is given by D1/(1-β) (X∥X̄), where Dα(X∥X̄) denotes the Rényi α-divergence [19], α > 0, α ≠ 1. Similarly, for 0 < β < 1, Csiszár also established the concept of reverse β-cutoff rate for the correct exponent hypothesis testing problem. In this work, we extend Csiszár's results by investigating the forward and reverse β-cutoff rates for the hypothesis testing between two arbitrary sources with memory. We demonstrate that the lim inf Rényi α-divergence rate provides the expression for the forward β-cutoff rate. We also show that if the log-likelihood large deviation spectrum admits a limit, then the reverse β-cutoff rate equals the liminf α-divergence rate, where α = 1/1-β and 0 < β < βmax, where βmax is the largest β < 1 for which the lim inf 1/1-β-divergence rate is finite. For βmax ≤ β ≤ 1, we show that the reverse cutoff rate is in general only upper-bounded by the lim inf Rényi divergence rate. Unlike in [4], 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 β-cutoff rates results and the techniques employed to establish them.

Original languageEnglish
Pages (from-to)663-678
Number of pages16
JournalIEEE Transactions on Information Theory
Issue number4
StatePublished - 1 Apr 2004


  • α-divergence rate
  • Arbitrary sources with memory
  • Forward and reverse cutoff rates
  • Hypothesis testing error and correct exponents
  • Information spectrum
  • Large deviation theory


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