We study the distribution of the ratio $\,Z_n\,$ of sums of random variables. Berry--Esseen-type estimates of the rate of convergence in the central limit theorem for $\,Z_n\,$ are given. In addition, second-order asymptotic expansions for $\,\bE Z_n\,$ and $\,\bE Z^2_n\,$ are presented. The results are applied to the problems of nonparametric regression curve estimation, nonparametric tail index estimation, and nonparametric estimation of a hazard function.