The Stein-Tikhomirov Method and Berry-Esseen Inequality for Sampling Sums from a Finite Population of Independent Random
We present a simplified version of the Stein-Tikhomirov method realized by defining a certain operator in class of twice differentiable characteristic functions. Using this method, we establish a criterion for the validity of a nonclassical central limit
- PDF / 165,032 Bytes
- 13 Pages / 439.36 x 666.15 pts Page_size
- 85 Downloads / 152 Views
Abstract We present a simplified version of the Stein-Tikhomirov method realized by defining a certain operator in class of twice differentiable characteristic functions. Using this method, we establish a criterion for the validity of a nonclassical central limit theorem in terms of characteristic functions, in obtaining of classical BerryEsseen inequality for sampling sums from finite population of independent random variables. Keywords Stein-Tikhomirov method • Distribution function • Characteristic function • Independent random variables • Berry-Esseen inequality • Sampling sums from finite population
Mathematics Subject Classification (2010): 60F05
1 The Stein-Tikhomirov Method and Nonclassical CLT Suppose that F .x/ is an arbitrary distribution function and 1 ˚.x/ D 2
Zx
e u
2 =2
du
1
S.K. Formanov () National University of Uzbekistan, Tashkent, Uzbekistan e-mail: [email protected] T.A. Formanova Tashkent Institute of Motor Car and Road Engineers, Tashkent, Uzbekistan e-mail: fortamara@yandex/ru A.N. Shiryaev et al. (eds.), Prokhorov and Contemporary Probability Theory, Springer Proceedings in Mathematics & Statistics 33, DOI 10.1007/978-3-642-33549-5 14, © Springer-Verlag Berlin Heidelberg 2013
261
262
S.K. Formanov and T.A. Formanova
is the standard distribution function for the normal law. In [9] Stein proposed a universal method for estimating the quantity ı D sup jF .x/ ˚ .x/ j; x
based on the following arguments. Suppose that h .u/ is a bounded measurable function on the line and 1 ˚h D p 2
Z1
2 h .u/e u =2 d u:
1
Consider the function g ./ which is a solution of the differential equation g 0 .u/ ug .u/ D h .u/ ˚h:
(1)
Suppose that is a random variable with distribution function P . < x/ D F .x/ : Setting h .u/ D hx .u/ D I.1;x/ .u/ in (1), we have
F .x/ ˚ .x/ D E g 0 ./ g ./ :
(2)
Thus, the problem of estimating ı can be reduced to that of estimating the difference of the expectations ˇ 0 ˇ ˇEg ./ Eg ./ˇ : Also note that for the case in which the random variable has normal distribution, the right-hand side of (2) vanishes. Using this method, Stein [9] obtained an estimate of the rate of convergence in the central limit theorem for stationary (in the narrow sense) sequences of random variables satisfying the strong mixing conditions (in the sense of Rosenblatt). Moreover, for the summands eighth-order moments must exits. In his paper, Stein stated his belief that his method is hardly related to that of characteristic functions. In [10,11] Tikhomirov refuted Stein’s suggestion. He showed that a combination of Stein’s ideas with the method of characteristic functions allows one to obtain the best possible estimates of the rate of convergence in the central limit theorem for sequences of weakly dependent random variables for less stringent conditions on the moments. He also used to best advantage the ideas [9] underling the proposed new method. The combination of methods our lined in [9, 10], later became known as the Stein-Tikhomirov method.
Data Loading...
