Asymptotic properties of the Neyman-Pearson test in the case of completely asymptotically distinguishable hypotheses. (English. Russian original) Zbl 0632.62022

Lin’kov, Yu. N.

Theory Probab. Math. Stat. 35, 65-73 (1987); translation from Teor. Veroyatn. Mat. Stat. 35, 60-69 (1986).

The author investigates the asymptotic behavior of the probability of type I and II errors and the parameters of the Neyman-Pearson test in a general scheme of observations where the hypotheses are completely asymptotically distinguishable. He obtains dependence of the behavior of the probability of type II error on the behavior of the probability of type I error in two cases:
1) Under the null hypothesis the logarithm of the likelihood ratio satisfies the law of large numbers; 2) the logarithm of the likelihood ratio, after normalization by an increasing function, converges weakly under the null hypothesis to a negative random variable.

MSC:

62F05	Asymptotic properties of parametric tests
62F03	Parametric hypothesis testing
60G42	Martingales with discrete parameter
60J60	Diffusion processes
60J65	Brownian motion

Keywords:

Neyman-Pearson test; general scheme of observations; probability of type II error; probability of type I error; logarithm of the likelihood ratio; law of large numbers