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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.

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

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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 |

[2] | R. R. Bahadur: An optimal property of the likelihood ratio statistic. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, pp. 13-26. University of California Press, Berkeley and Los Angeles 1967. · Zbl 0211.50901 |

[3] | R. R. Bahadur: Rates of convergence of estimates and test statistics. Ann. Math. Statist. 38 (1967), 303-324. · Zbl 0201.52106 · doi:10.1214/aoms/1177698949 |

[4] | R. R. Bahadur: Some Limit Theorems in Statistics. SIAM, Philadelphia 1971. · Zbl 0257.62015 |

[5] | R. R. Bahadur, M. Raghavachari: Some asymptotic properties of likelihood ratios on general sample spaces. Proceedings of the Sixth Berkeley Sympsium on Mathematical Statistics and Probability, pp. 129-152. University of California Press, Berkeley and Los Angeles 1972. · Zbl 0254.62021 |

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[8] | N. L. Johnson, S. Kotz: Continuous Univariate Distributions, Volume 1. J. Wiley and Sons, New York 1970. · Zbl 0213.21101 |

[9] | J. A. Koziol: Exact slopes of multivariate tests. Ann. Statist. 6 (1978), 546-557. · Zbl 0386.62035 |

[10] | M. Raghavachari: On a theorem of Bahadur on the rate of convergence of test statistics. Ann. Math. Statist. 41 (1970), 1695-1699. · Zbl 0223.62049 · doi:10.1214/aoms/1177696813 |

[11] | C. R. Rao: Linear Statistical Inference and Its Applications. J. Wiley and Sons, New York 1973. · Zbl 0256.62002 |

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