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On optimality of the LR tests in the sense of exact slopes. (English) Zbl 0692.62015

The first part contains the treatment of the following problem: Let (X,\({\mathcal F},P_{\vartheta_ j})\), \(1\leq j\leq q\), be a set of q probability spaces (populations) with \((\vartheta_ 1,...,\vartheta_ q)\in \Theta =\Sigma^ q\) and \(\emptyset \leq \Omega_ 0\subset \Omega_ 1\subset \Theta\). For testing the hypothesis \[ H_ 0: \vartheta \in \Omega_ 0\quad against\quad H_ 1: \vartheta \in \Omega_ 1\setminus \Omega_ 0 \] a sequence \(\{T_ u\}_{u\in {\mathbb{N}}}\) of statistics is wanted that maximizes the exact slope \[ C(\vartheta):=-\lim_{u\to \infty}(2 \ln L_ u(s))/n_ u\quad where\quad L_ u(s):=1-\inf_{\vartheta \in \Omega_ 0}P_{\vartheta}(T_ u\leq T_ u(s)) \] and \(n_ u^{(j)}\) is the sample size of the u-th sample in the j-th population, \(n_ u:=\sum^{q}_{j=1}n_ u^{(j)}\). Optimality in this sense was first shown by R. R. Bahadur [Proc. 5th Berkeley Sympos. math. Statist. Probab., Univ. Calif. 1965/1966, 1, 13-26 (1967; Zbl 0211.509)] for the likelihood ratio (LR) test statistic when the sample space is finite. Under various conditions the optimality was later proved e.g. by Bahadur, Koziol and Hsieh.
Here some other regularity conditions are assumed in order to prove the optimality of the LR statistic in the sense of the exact slope. The advantage of these conditions compared with those assumed in former investigations is the fact that they are relatively easy to verify for some important probability distributions, e.g. the k-dimensional normal distribution, the exponential, the Laplace and the Poisson distribution. This is worked out in Part II, see the following entry, Zbl 0692.62016, together with some applications mainly concerning the hypothesis of monotonicity of means.
Reviewer: B.Rauhut

MSC:

62F03 Parametric hypothesis testing
62L10 Sequential statistical analysis
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References:

[1] R. R. Bahadur: Stochastic comparison of tests. Ann. Math. Statist. 31 (1960), 276-295. · Zbl 0201.52203 · doi:10.1214/aoms/1177705894
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