Interval Estimation of Non-Linear Parametric Functions, III

原文摘要

Let f(μ1, μ2) = constant be a function which is monotone and has no finite termini in the (μ1, μ2) plane. Let x 1 and x 2 be either independent normal with unknown means μ1, μ2 and unknown variances σ2 1, σ2 2 or correlated normal with unknown mean vector (μ1, μ2) and unknown variance matrix σ. Suppose, in the first instance, that estimates, s 2 1 and s 2 2, of σ2 1 and σ2 2 with v 1 and v 2 degrees of freedom are available and, in the second instance, an estimate, S, of σ with v degrees of freedom is available. Let vis 2 i /σ2 i , i = 1, 2, be distributed as independent chi-squares and let vS have the Wishart distribution. Under the above assumptions we study the characteristics of two procedures for obtaining interval estimates of f(μ1, μ2). One procedure bases interval estimation of f(μ1, μ2) on separate 1 - α confidence intervals on μ1 and μ2. The other procedure bases estimation on an elliptical confidence region for (μ1, μ2). For the first procedure it is shown that the interval estimate of f(μ1, μ2) is at least at the 1 - α confidence level, providing v 1 = v 2. (which is automatically satisfied in the correlated case); if v 1 v 2 (case of independence) we give arguments which strongly suggest, but do not prove, that the same result holds. Further, for this case, if x 1 and x 2 are sample means from samples of size (v 1 + 1) and (v 2 + 1) respectively, a well known randomization argument due to Scheffe can be used to reduce the problem to the equal degree of freedom case, with no loss of precision in estimation of mean values. For the second procedure no definite conclusions are drawn.

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