On the Sampling Interpretation of Confidence Intervals and Hypothesis Tests in the Context of Conditional Maximum Likelihood Estimation

In the context of conditional maximum likelihood (CML) estimation, confidence intervals can be interpreted in three different ways, depending on the sampling distribution under which these confidence intervals contain the true parameter value with a certain probability. These sampling distributions are ( a) the distribution of the data given the incidental parameters, (b) the marginal distribution of the data (i.e., with the incidental
parameters integrated out), and (c) the conditional distribution of the data given the sufficient statistics for the incidental parameters. Results on the asymptotic distribution of CML estimates under sampling scheme (c) can be used to construct asymptotic confidence intervals using only the CML estimates. This is not possible for the· results on the asymptotic distribution under sampling schemes (a) and (b). However, it is shown that the conditional asymptotic confidence intervals are also valid under the other two sampling schemes.