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By Van Der Merwe A. J., Du Plessis J. L.

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Extra info for A Bayesian Approach to Selection and Ranking Procedures: The Unequal Variance Case

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This idea 44 V. Bentkus, F. Gotze, V. Paulauskas and A. RaCkauskas has been exploited in several papers using in addition either iteration or induction arguments. It yields bounds of at most order O(n- 1 / 6 ) by assuming a finite third moment. 6 below). Iteration arguments appeared in Paulauskas (1976b) and under certain assumptions, he gave bounds On balls of order O(n- 1 / 6 ). Refinements and generalizations were given by Butzer et al. (1979), Ul'yanov (1981), Bernotas (1980) and Bentkus and Raekauskas (1982), (1983).

Let fJ. £. v(t) = 1. exp(itx)v(dx). This is the so-called Berry~Esseen lemma (Berry (1941), Esseen (1942)). It is used to prove convergence rates and higher order approximations to a measure fJ. by a measure v with smooth density by means of their characteristic functions. Here T is chosen so that T~l has a desired error size. he proof of the classical Berry~Esseen estimate sup IP{Sn xEIR where Sn that = n~1/2(Xl < x} - P{Y < x}1 = + ... d. v. such that EY = 0 and f32 := EX 2 = Ey2 > O. 1) fJ. 2) to the estimate II.

A similar remark applies to Trotter's proof given in 1959. Actually Trotter's proof differs from Lindeberg's only in terminology, but the method is presented in an intuitively more understandable manner. Besides the original papers of Lindeberg (1920, 1922) and Trotter (1959), we mention the books of Billingsley (1968), Thomasian (1969) and Feller (1971). In this section we shall concentrate our attention on some extensions of Lindeberg's method that have been used to investigate the rate of convergence in the CLT in Banach spaces.

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