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Asymptotic testing theory for generalized linear models. (English) Zbl 0618.62031

Consider a generalized linear model with the data consisting of a sequence \(\{(Y_ n,Z_ n)\}\), where \(Y_ n\) are q-dimensional stochastically independent responses whose distributions belong to a natural exponential family and \(Z_ n\), the regressors, are known (p\(\times q)\) matrices. The mean of \(Y_ n\) is related to a linear combination \(Z_ n'\beta\) of the regressors by a one-to-one mapping. Most of the test problems in this model are tests of linear hypotheses on the parameter \(\beta\).
The author considers three test statistics: the likelihood ratio \((\lambda_ n)\), the Wald \((w_ n)\) and the score statistic \((s_ n)\). It is established that under given conditions the test statistics \(\lambda_ n\), \(w_ n\) and \(s_ n\) are asymptotically equivalent and that they have certain limiting distributions which are shown.
Reviewer: K.Alam

MSC:

62F05 Asymptotic properties of parametric tests
62H15 Hypothesis testing in multivariate analysis
62J99 Linear inference, regression
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References:

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