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,Yi=μ(xi)+εi(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.
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Research partially supported by the NSF Grant DMS-8502346.
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Tong, Y.L. Interval estimation of the critical value in a general linear model. Ann Inst Stat Math 39, 289–297 (1987). https://doi.org/10.1007/BF02491468
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DOI: https://doi.org/10.1007/BF02491468