Interval Estimation of Non-Linear Parametric Functions

Suppose x 1 and x 2 are observations on a bivariate normal distribution with known variance matrix Σ and unknown means vector (μ1, μ2). Suppose an interval estimate of f(μ 1, μ2), is desired with 1 – α confidence. Let an interval estimate be obtained as [min f(μ1, μ2), max f(μ1, μ2)] subject to [(x 1 – μ1) (x 2 – μ2)] Σ−1[(x 1 – μ1)(x 2 – μ2)]′ ≤ R 2. It is shown that under certain conditions on f(μ1, μ2) and Σ, R need be taken only slightly larger than x 1(α), the value of a chi-variate on one degree of freedom exceeded with probability α. For example under the given conditions at a nominal .95 confidence one may use R = 2.06 rather than 1.96. Similar results are obtained when variances must be estimated for some special assumptions about Σ.