An Approximation to the Sample Size in Selection Problems

Let f(x P₁) be the pdf of a $(k - 1)$-dimensional normal distribution with zero means, unit variances, and correlation matrix P₁. Consider the integral, for δ > 0, {equation*}{1}∫^∞-δ ⋯ ∫^∞-δ f(x P_1)dx ⋯ dxₖ₋₁ = α(δ), say.{equation*} Assume that no element of P₁ is a function of δ. Note that α(δ) is an increasing function of δ and α(δ) → 1 as δ → ∞. The problem is to obtain an approximation to δ, for a large specified value, α, of α(δ). This is given by the theorem of Section 1. This result is used to obtain approximations to the sample size in a selection procedure of Bechhofer and in a problem of selection from a multivariate normal population. The closeness of the approximation is illustrated for the procedure of Bechhofer (Table 1).