Large deviation theorems for extended random variables and some applications. (English) Zbl 0943.60014
The author studies the family \(Y_t\), \(t>0\), of extended real random variables which may also attain one of the values \(+\infty\) or \(-\infty\). Assuming a key condition on their corresponding (extended) moment generating functions \(m_t(\varepsilon)= E\exp (\varepsilon Y_t)\), \(\varepsilon\in (-\infty, \infty)\), the Chernoff type large deviation asymptotics of \(P(Y_t> \varphi_t x)\) and \(P(Y_t< \varphi_t x)\) are derived, where \(\varphi_t\to \infty\) as \(t\to\infty\). A related large deviation principle is also proved. With these tools in hand, the asymptotics of the error probabilities are studied e.g. for Neyman-Pearson tests, Bayes tests and minimax tests. The above results are complementary to earlier work of Sievers, Plachky, Steinebach, Gärtner, Ellis and other authors.
Reviewer: J.Steinebach (Marburg)
MSC:
| 60F05 | Central limit and other weak theorems |
| 62F05 | Asymptotic properties of parametric tests |
| 62F15 | Bayesian inference |
Keywords:
large deviations; large deviation principle; Chernoff theorem; likelihood ratio test; Bayes test; minimax testReferences:
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