On the Edgeworth Expansion and the Bootstrap Approximation for a Studentized $U$-Statistic

On the Edgeworth expansion and the M out of N bootstrap accuracy for a Studentized trimmed mean

“…The consistency of the standard nonparametric, or naive, bootstrap was proved for many interesting statistics, at least for the asymptotically normal ones (see [1,4,6,7,12,13,15,25] and references therein). One of the main reasons of interest in the bootstrap and its application in statistics is the second order accuracy property: under proper conditions the bootstrap approximation to the distribution function (df ) of a pivotal statistic is more accurate than the normal one.…”

“…So, the application of a relevant version of the Law of Large Numbers implies the second order accuracy of the bootstrap (cf. [12,15,25]). However, the case of the trimmed mean is a special one.…”

“…This beneficial property of the bootstrap was proved for the sample mean [25], for the class of Hoeffding's U -statistics (cf. [15]) and for some other statistics. The usual way to prove this fact is based on the congruence of the one-term EE for the pivotal statistic (in the 'real world') and the EE for its bootstrap counterpart (in the 'bootstrap world').…”

On a crossroad of resampling plans: bootstrapping elementary symmetric polynomials

“…It can be proved by a slight adaptation of the proof given in MAESONO (1995) (cf. also HELMERS (1991) and VAN Es and HELMERS (1988)) that G~k)(x) = <P(x) + ~n-112 </J(x)…”

Аппроксимация Второго Порядка Для Слабо Усеченных Средних

“…− 1 суммой независи-мых случайных величин Это позволит нам получить неравенства типа Берри-Эссеена и разложения типа Эджворта для статистики T n и ее стьюдентизованной версии путем применения известных результатов этого типа для U -статистик степени 2 (см., например, статьи [1]- [4], [12], [19], [25], [27], в которых получены соответствующие результаты по аппроксимации второго порядка для U -статистик). Этот метод до-казательства хорошо известен в литературе, имеется много работ, в ко-торых используется данный подход (см., например [3]).…”