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Power losses of some asymptotically efficient tests. (English) Zbl 1104.62303

From the text: Let \(\{\mathbf P_\theta, \theta\in\Theta\subset\mathbb R^1\}\) be a family of probability measures on a measurable space \(({\mathcal X},{\mathcal A})\) having densities \(p(x,\theta)\) with respect to a \(\sigma\)-finite measure \(\nu\). Assume without loss of generality that \(\Theta\subset\mathbb R^1\) contains an interval \([0,\epsilon],\;\epsilon>0\). Suppose we have independent and identically distributed \(\mathcal X\)-valued observations \((X_1,\dots,X_n)\) distributed according to \(\{P_\theta, \theta\in\Theta\subset\mathbb R^1\}\). Our problem is to test the hypothesis \[ H_0\colon \theta=0 \text{ against } H_1: \theta>0. \]
Note that the alternative hypothesis is composite. This is the most general one-sided hypothesis that we need to consider. The test \( H_0\colon \theta=\theta_0,\;\theta_0\) specified, against \( H_1\colon \theta>\theta_0\), can be reduced to the above case by considering the family \(\{P_{\theta_0+\theta},\;\theta\in\Theta\subset\mathbb R^1\}\).
For any \(0<t\leq C\), \(C>0\), we consider testing \(H_0: \theta=0\) against \(H_{n1}: \theta=\tau t\), \(0<t\leq C.\) Throughout the paper, we use the abbreviation \(\tau=n^{-1/2}\).
There are many (first-order) asymptotically efficient tests, i.e., tests whose power function \(\beta_n(t)\) converges to the same limit as \(\beta^*_n(t)\), that of the Neyman-Pearson test, i.e., the most powerful size-\(\alpha\) test for \(\mathbf H_0\) against \(\mathbf H_{n,t}: \theta=\tau t\). They can be compared with each other by higher-order terms of their power. This is done in Section 2. In Section 3 the power loss \(\beta^*_n(t)-\beta_n(t),\) in Section 4 tests based on \(L\)-, \(R\)-, and \(U\)-statistics, and, in Section 5, combined \(L\)-tests are studied.

MSC:

62F05 Asymptotic properties of parametric tests
62G10 Nonparametric hypothesis testing
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