On Berry-Esseen type bounds and asymptotic expansions for slightly trimmed means

The paper is concerned with the second-order asymptotics of distributions of trimmed means $T_n = (\sum\nolimits_{i = k_n + 1}^{n - m_n } {X_{i:n} } )/n$ , where k _n , m _n are sequences of integers, 0 ≤ k _n < n ? m _n ≤ n, and r _n := min(k _n , m _n ) → ∞ as n → ∞, X _i:n are order statistics that correspond to the sample X ₁, …, X _n of independent identically distributed random variables with distribution function F. We focus on the case of slightly trimmed means, when max(k _n , m _n )/n → 0 as n → ∞. In 11], Berry-Esseen-type bounds were obtained for the normal approximation of T _n ; under certain regularity conditions, these bounds are of the order O(r _n ^?1/2 ). In 11], it is also shown that this order cannot be improved if E X ₁ ² = ∞. Moreover, asymptotic Edgeworth-type expansions were found in 11] for slightly trimmed means and their Studentized versions. In the present paper, we supplement the results of 11] by Berry-Esseen-type bounds and asymptotic approximations for the case E X ₁ ² < ∞.