Statistics Sequential statistics Theorem

Wald–Wolfowitz optimality of the sequential probability ratio test

Builds on Sequential probability ratio test Sequential analysis Independent and identically distributed random variables Stopping time Type I and type II errors Expected value Neyman–Pearson lemma

Statement

Let $X_1, X_2, \dots$ be independent and identically distributed observations with density $f$ , and test $H_0 \colon f = f_0$ against $H_1 \colon f = f_1$ , where $f_0$ and $f_1$ are distinct densities. Let $S$ be the sequential probability ratio test with boundaries $0 < A < 1 < B$ . Let $N$ be its sample size, and let $\alpha = P_0(S \text{ rejects } H_0)$ and $\beta = P_1(S \text{ accepts } H_0)$ be its error probabilities. Let $S'$ be any sequential procedure for the same hypotheses with sample size $N'$ and error probabilities $\alpha' \le \alpha$ and $\beta' \le \beta$ . Then $E_0[N] \le E_0[N'] \quad \text{and} \quad E_1[N] \le E_1[N'],$ where $E_i$ is the expectation when $f = f_i$ .

Remarks

The proof is Bayesian. Fix a prior probability of $H_1$ , a cost per observation and losses for wrong decisions. The Bayes sequential procedure is then an optimal stopping rule, and it has the form of an SPRT on the posterior odds. One shows that every SPRT with $A < 1 < B$ is Bayes for suitable priors and costs. Comparing the Bayes risks of $S$ and $S'$ over a range of priors and costs for which $S$ is Bayes then gives both inequalities. The theorem was proved by Wald and Wolfowitz in 1948. The optimality holds only at the two simple hypotheses. For a composite family of distributions, the expected sample size of an SPRT can be large at parameter values between $f_0$ and $f_1$ .

Going deeper

For a student

Fixed-sample tests, such as those from the Neyman–Pearson lemma, optimize power for a given sample size. The SPRT optimizes the opposite quantity: the expected stopping time for given error probabilities. In practice the SPRT often needs far fewer observations on average than the best fixed-sample test with the same error rates. The theorem is the starting point of optimal stopping methods in sequential statistics.

Reading path

To understand this page from the ground up, read these 66 pages in order. The foundations come first.

Construction of the reals from Cauchy sequences concept
Element definition
Expected value concept
Independent and identically distributed random variables concept
Integers object
Ordered field definition
Proposition concept
Random experiment concept
Rational numbers object
Real numbers object
Sequential analysis concept
Sequential probability ratio test concept
Stopping time concept
Type I and type II errors concept
Set definition
Ring of integers statement
Disjunction definition
Implication definition
Predicate definition
Ordered pair definition
Union definition
Sample space definition
Existential quantifier definition
Universal quantifier definition
The rationals form a field statement
Cartesian product definition
Subset definition
Uniqueness quantifier definition
The rationals form an ordered field statement
Binary relation definition
Universal set definition
Power set definition
Order on the reals definition
Complement definition
Function definition
Interval definition
Sigma-algebra definition
Injective function definition
Preimage of a set definition
Countable set definition
Event definition
Borel sigma-algebra definition
Probability axioms concept
Probability measure definition
Probability space definition
Probability distribution definition
Conditional probability definition
Random variable definition
Independence of two events definition
Discrete random variable definition
Cumulative distribution function definition
Law of a random variable definition
Random variable with a density definition
Mutual independence of events definition
Probability mass function definition
Identically distributed random variables definition
Probability density function definition
Independent sigma-algebras definition
Independence of random variables definition
Independent and identically distributed random variables definition
Statistic definition
Likelihood function definition
Statistical hypothesis test concept
Type I and type II errors definition
Neyman–Pearson lemma statement
Wald–Wolfowitz optimality of the sequential probability ratio test statement

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About	Sequential probability ratio test
Uses	Sequential probability ratio test, Sequential analysis, Independent and identically distributed random variables, Stopping time, Type I and type II errors, Expected value, Neyman–Pearson lemma