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Pivot variables, tests and strong consistency. (English) Zbl 1167.62376

Summary: Let the random vector \(Z\) have a distribution depending on parameters \(\gamma\). A function \(U\) of both \(Z\) and \(\gamma\) is a pivot variable if its distribution does not depend on \(\gamma\). Assuming that \(\alpha_ n\) is an estimate of \(\alpha\) whose asymptotic distribution has an associated pivot variable and that \(\gamma_ n\) is a consistent estimate of \(\gamma\), it is shown how to derive tests for hypotheses on \(\alpha\). Conditions are obtained for these tests to be strongly consistent and to enjoy duality. The case in which \(\alpha_ n\) is asymptotically normal is considered and hence the Wald and Rao score tests are shown to be strongly consistent.

MSC:

62F05 Asymptotic properties of parametric tests
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