Interval estimation of the critical value in a general linear model

Summary

This paper concerns interval estimation of the critical value θ which satisfies$\mu (\theta ) = \mathop {\sup }\limits_{x \in \mathfrak{X}} \mu (x)$ under the general linear model,Y_i=μ(x_i)+ε_i(i=1,2,···), where$\mu (x) = \sum\limits_{j = 1}^p {\beta _j f_j (x)}$ for$x \in \mathfrak{X}$ and the functional forms off ^′_j s are known. From an asymptotic expansion it is shown that, under reasonable conditions, the limiting distribution of$\sqrt n (\hat \theta _n - \theta )$ is normal. Thus in the large-sample case a confidence interval for θ can be obtained. Such a result is useful when one is interested in carrying out a retrospective analysis rather than designing the experiment (as in the Kiefer-Wolfowitz procedure). In Section 3 a sequential procedure is considered for confidence intervals with fixed width 2d. It is shown that, for a given stopping variableN,$\sqrt n (\hat \theta _n - \theta )$ is also asymptotically normal asd→0. Thus the coverage probability converges to 1−α (preassigned) asd→0. An example of application in estimating the phase parameter in circadian rhythms is given for the purpose of illustration.