Multivariate sequential procedures for testing means (1959)

We consider a multivariate situation with means µ₁,...,µ<sub>p</sub> and covariance matrix Σ. We wish to derive sequential procedures for testing the hypothesis: H₀: (µ̲ - µ̲ₒ)Σ⁻¹(µ̲ - μ̲₀)’= λ₀²( usually zero) against the alternative: μ̲₀ H₁: (µ̲ - µ̲ₒ)Σ⁻¹µ̲ - μ̲₀)’=λ₁² both for the case where Σ is known (the sequential X²-test) and where Σ is unknown and must be estimated from the sample (the sequential T²-test). These sequential procedures should guarantee that the probability of accepting H₁ when H₀ is true is equal to a and the probability of accepting H₀ when H₁ is true is equal to β. For the case where Σ is known, λ₀² = 0 and λ₁² = λ², the test procedure is as follows: for a sample of n observations form the probability ratio: P<sub>1n</sub>/P<sub>0n</sub> = e<sup>-nλ²</sup><sub>0</sub>F₁(p/2;nλ²X<sub>n²</sub>/4) where p denotes the number of variables, <sub>n</sub>x[with horizontal bars above and below the x] denotes the vector of the sample means based on n observations, X²<sub>n</sub> = n(<sub>n</sub>x[with horizontal bars above and below the x] - μ̲₀) Σ⁻¹(<sub>n</sub>x[with horizontal bars above and below the x] - μ̲₀)’ and ₀F₁ (c;x) is a type of generalized hypergeometric function. a. If P<sub>1n</sub>/P<sub>0n</sub> ≤ β/(1-α), accept H₀; b. If P<sub>1n</sub>/P<sub>0n</sub> ≥ (1- β)/α, accept H₁; c. If β/(1-α) < P<sub>1n</sub>/P<sub>0n</sub> < (1-β)/α, continue sampling. For the case where Σ is unknown, the procedure is exactly the same except that the probability ratio is now: P<sub>1n</sub>/P<sub>0n</sub> = e⁻<sup>-nλ²/2</sup> ₁F₁[n/2,p/2;nλ²T<sub>n</sub>²/2(n-1+T<sub>n</sub>²)] where T<sub>n</sub>² = n(<sub>n</sub>x[with horizontal bars above and below the x] - μ̲₀)S<sub>n</n>⁻¹(<sub>n</sub>x[with horizontal bars above and below the x] - μ̲₀)' , S<sub>n</sub>denotes the sample covariance matrix based on n observations and ₁F₁(a,c;x) is a confluent hypergeometric function. Procedures are also given for the case λ²₀ ≠ 0. Similar procedures are given to test the hypothesis: H₀ = (μ̲₁ - μ̲₂- δ ̲)Σ⁻¹(μ̲₁ - μ̲₂ - δ̲)’ = λ₀² (usually zero) against the alternative: H₁ = (μ̲₁ - μ̲₂- δ ̲)Σ⁻¹(μ̲₁ - μ̲₂ - δ̲)’ = λ₁² It is shown that these sequential procedures all exist in the sense that the risks of accepting H₀ when H₁ is true and of accepting H₁ when H₀ is true are approximately α and β respectively and that these sequential procedures terminate with probability unity. Some of these situation have been generalized to give simultaneous tests and the means and covariance matrix of a sample. No expressions yet exist for the OC or ASN functions although some conjectured values have been determined for the latter and suggest, in comparison with their corresponding fixed-sample tests, substantial reductions in the sample sizes required when either H₀ or H₁ is true. The general problem of tolerances is discussed and then some of these procedures are demonstrated with a numerical example drawn from the field of ballistic missiles. The determination of P<sub>1n</sub>/P<sub>0n</sub> is quite laborious for both the sequential X² - and T²-tests since it requires the evaluation of a hypergeometric function each time an observation is made. It would be better for each value of n, given p, α, β and λ² under H₁, to compute the values of X²<sub>n</sub> or T²<sub>n</sub> which would correspond to the boundaries of the tests indicated by β/(1-α) and (1-β)/α. Tables to facilitate both the sequential X²- and T²-tests are given for p = 2,3,...,9; λ² = 0.5, l.0, 2.0; α = β = 0.05 for n ranging from the minimum value necessary to reach a decision to 30, 45 and 60 for λ² = 0.5, 1.0, 2.0 respectively. These tables were prepared on the IBM 650 computer using the Newton-Raphson iterative procedure. Finally, a discussion is given for the hypergeometric function ₀F₁ (c;x) and a table given of this function for c = .5(.5)5.0 and x = .1(.1)1(1) 10(10)100(50)1000.