The distribution and quantiles of a function of parameter estimates

Summary

Let$\hat \omega _n$ be an estimate of a parameter ω inR ^p,n a known real parameter, andt(·) a real function onR ^p. Suppose that the variance of$n^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} (t(\hat \omega _n ) - t(\omega ))$ tends to σ²>0 asn→∞, and that$\hat \sigma _n$ is an estimate of σ. We give asymptotic expansions for the distributions and quantiles of

$$Y_{n1} = n^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \sigma ^{ - 1} (t(\hat \omega _n ) - t(\omega ))and Y_{n2} = n^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \hat \sigma _n^{ - 1} (t(\hat \omega _n ) - t(\omega ))$$

to withinO(n^−5/2). It is assumed that (i) E$\hat \omega _n \to \omega$ asn→∞; (ii)t(·) is suitably differentiable at ω; (iii) forr≧1 therth order cross-cumulants of$\hat \omega _n$ have magnituden ^1−r asn→∞ and can be expanded as a power series inn−1; (iv) that$\hat \omega _n$ has a valid Edgeworth expansion. (Bhattacharya and Ghosh [1] have given easily verifiable sufficient conditions for commonly used statistics like functions of sample moments and the m.l.e.)

As an application we investigate for what parameter ranges common confidence intervals for a linear combination of the means of normal samples are adequate.