More specifically, thispaper introduces a new class of hypothesis testing problems,referred to as almost-fixed-length hypothesis tests in whichthe number of samples is kept fixed ( ≤ n ) for almost all sample-paths except for an exponentially rare set for whichthe number of samples collected are allowed to be somewhatlarger (bounded by a polynomial function of n ). This suggests thatallowing some variability in the number of samples collectedis essential for achieving better reliability (error probabilities).The main contribution of this paper is to demonstrate thatthis flexibility need not be significant. In other words, by allowing the number ofsamples to be a random number, the sequential hypothesis testresolves the trade-off between error-types. ![]() It is well-known that while,in the fixed-length regime, the error exponents of the twotypes of errors can only be traded-off against each other,the sequential hypothesis tests can achieve both exponentssimultaneously. More specifically, the errorexponents in both variants of hypothesis testing is well-known and understood –. This paper considers the two well-known variants of simple binary hypothesis testing where adecision maker, after observing a sequence of i.i.d randomvariables, is tasked with identifying the most probable one.In the first version of the problem the number of samplesthat is provided to the decision maker is fixed ( ≤ n ), whilein the second variant of the problem (known as sequentialhypothesis testing due to Wald ), the decision maker isgiven the additional freedom to collect a random number ofsamples so long as the expected number of samples is keptconstant ( ≤ n ).There is a large body of literature on the asymptoticanalysis of type-I and type-II errors as the (expected) num-ber of samples n grows large. It is alsoshown to be at the core of many problems in informationtheory and statistics. Statistical hypothesis testing is an integral part of manyscientific discoveries and engineering systems. This class of hypothesis tests are shown to bridge the gapbetween the classical hypothesis testing with a fixed sample sizeand the sequential hypothesis testing, and improve the trade-offbetween type-I and type-II error exponents. ![]() In this class of tests, the decision-maker declares the true hypothesis almost always after collectinga fixed number of samples n however in very rare cases withexponentially small probability the decision maker is allowedto collect another set of samples (no more than polynomial in n ). RReliability of Sequential Hypothesis Testing CanBe Achieved by an Almost-Fixed-Length TestĮlectrical & Computer EngineeringUniversity of California, San DiegoEmail: Ībstract -The maximum type-I and type-II error exponentsassociated with the newly introduced almost-fixed-length hypoth-esis testing is characterized.
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