Moderate Deviations for a Class of L-Statistics
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Moderate Deviations for a Class of L-Statistics Hui Jiang
Received: 28 October 2008 / Accepted: 7 November 2008 / Published online: 15 November 2008 © Springer Science+Business Media B.V. 2008
Abstract We study a class of L-statistics based on linear combinations of order statistics divided by the sample mean. The moderate deviation and functional moderate deviation are obtained by the method of Rényi representation. Moreover, we also apply our result to Jackson, Gini and Fortiana-Grané tests and obtain their asymptotic properties. Keywords Asympotic properties · L-statistics · Large deviations · Moderate deviations · Order statistics · Rényi representation Mathematics Subject Classification (2000) 62F12 · 62N02 · 60F15
1 Introduction Let {Xn , n ≥ 1} be a sequence of independent identically distributed random variables and have the density f (s) = e−x ,
x ≥ 0.
The order statistics of X1 , X2 , . . . , Xn are denoted as X1∗ ≤ X2∗ ≤ · · · ≤ Xn∗ . Shorack and Wellner [9] proposed a class of L-statistics of the form n ∗ i=1 wi,n Xi Tn = , (1.1) n i=1 Xi where {wi,n }, i = 1, 2, . . . , n, is an array of coefficients. Motivated by their simple, wellstudied asymptotic behavior and nice properties of order statistics under exponentiality, the tests Tn is widely used. Moreover, many statistics belong to this class (cf. [3, 4, 9]).
Research supported by the National Natural Science Foundation of China (10571139). H. Jiang () School of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China e-mail: [email protected]
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H. Jiang
The asymptotic behavior and applications of Tn have been studied widely. Tchirina [11] obtained the consistency and large deviations of Tn . Mason and Shorack [6] proved the necessary and sufficient conditions for the central limit theorem. For further references, on can see [5, 10]. Set [nt] ∗ i=1 wi,n Xi , 0 ≤ t ≤ 1. Tn (t) = n i=1 Xi Since the law of large numbers, central limit theorem and large deviation have been well investigated, as a complement, the purpose of this paper is to study further estimations about this estimator, i.e. moderate deviation. We study the moderate deviation for Tn (t) from two points of view: the parametrical statistical one when t ∈ [0, 1] is fixed and the nonparametrical statistical one when t varies in [0, 1]. Moreover, we also apply our result to the statistics in Jackson, Gini and Fortiana-Grané tests (cf. [3, 4, 9]). And their asymptotic properties can be obtained. More precisely, we are interested in the estimations of t n W (u)du ∈ A , Tn (t) − P bn 0 where A is a given domain of deviation, (b(n), n > 0) is some sequence denoting the √ scale of deviation. When b(n) = n, this is exactly the estimation of central limit √ theorem. When b(n) = n, it becomes the large deviations. Furthermore, when b(n)/ n → ∞ and b(n) = o(n), this is the so called moderate deviations. In other words, the moderate deviations investigate the convergence speed between the large deviations and central limit theorem. Let {bn , n ≥
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