On the number of observations necessary for the distinction between two proximate hypotheses

The problem of distinction between the following two proximate hypotheses $\mathbf H_0$: the population density is equal to $p_0(x)$ and $\mathbf H_\alpha$: the population density is equal to $p_\alpha(x)$, where $p_\alpha(x)\to p_0(x)$ as $\alpha\to0$, using the results of independent observations is considered.
In the case when $\alpha$ is a one dimensional parameter the Petrov–Aivazyan formula [1] for the number of observations nesessary for the distinction between hypotheses $\mathbf H_0$ and $\mathbf H_\alpha$ according to the Neumann–Pearson criterion with given probabilities of errors of the first $(\varepsilon)$ and second $(\omega)$ type is improved up to the members of order $O(1)$. A possibility of application of the results of this article to the problem of testing the hypotheses on the types of distributions given a large number of small simples is demonstrated by the example of the distinction between two gamma-types.