An adaptive omnibus test for exponentiality
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Let be independent and identically distributed observations on a non-negative random variable X. If X has the exponential density , its Laplace transform satisfies the differential equation . We show that a strong omnibus test for expo-nentiality may be based on , where is the Maximum-Likelihood- estimate of λ and Ψn is the empirical Laplace transform, each based on .Throughout this paper we shall be concerned with a sequence of mutually independent and identically distributed random variables ξ 1 ξ 2 , · · ·, ξ n , · · · taking on real values. We shall use the notation ζ n = ξ 1 + · · · + ξ n for n = 1, 2, · · · and ζ 0 = 0.
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Pairwise independence
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In a recent paper R. Dudley gave a characterization of those sequences of independent and identically distributed random variables which are l p {l_p} -compatible for p ≧ 1 p \geqq 1 . In the present note we extend his result into p ∈ ( 0 , 1 ] p \in (0,1] and provide some conditions (necessary or sufficient) for l φ {l_\varphi } -compatibility of a sequence of independent random variables not necessarily identically distributed.
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A sequence of random variables is said to be extended negatively dependent (END) if the tails of its finite-dimensional distributions in the lower-left and upper-right corners are dominated by a multiple of the tails of the corresponding finite-dimensional distributions of a sequence of independent random variables with the same marginal distributions. The goal of this paper is to establish the strong law of large numbers for a sequence of END and identically distributed random variables. In doing so we derive some new inequalities of large deviation type for the sums of END and identically distributed random variables being suitably truncated. We also show applications of our main result to risk theory and renewal theory.
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Expected value
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The problem is considered of obtaining bounds for the (cumulative) distribution function of the sum of $n$ independent, identically distributed random variables with $k$ prescribed moments and given ranger. For $n = 2$ it is shown that the best bounds are attained or arbitrarily closely approach with discrete random varibles which take on at most $2k + 2$ values. For nonnegative random variables with given mean, explicit bounds are obtained when $n = 2$; for arbitrary values of $n$, bounds are given which are asymptotically best in the "tail" of the distribution. Some of the results contribute to the more general problem of obtaining bounds for the expected values of a given function of independent, identically distributed random variables when the expected values of certain functions of the individual variables are given. Although the results are modest in scope, the authors hope that the paper will draw attention to a problem of both mathematical and statistical interest.
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Sequence (biology)
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1. Introduction. Let {X k : k ≥ 1} be a sequence of independent, but not necessarily identically distributed, random variables. Suppose that the random variables X n are uniformly bounded by a random variable X in the sense that (1) P(|X n | ≥x) ≤ P(|X| ≥ x)for all x > 0. Write q n (x) = P (|X n | ≥ x) and q (x) = P (|X| ≥ x).
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Let $X,X_1,\ldots,X_n$ be independent identically distributed random variables. The paper deals with the question about the behavior of the concentration function of the random variable $\sum\limits_{k=1}^{n}X_k a_k$ according to the arithmetic structure of vectors $a_k$. Recently, the interest to this question has increased significantly due to the study of distributions of eigenvalues of random matrices. In this paper we formulate and prove multidimensional generalizations of the results Eliseeva and Zaitsev (2012). They are also the refinements of the results of Friedland and Sodin (2007) and Rudelson and Vershynin (2009).
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In this paper,the similarity and difference of identically distributed random variables and exchangeable random variables sequences in certain relevant conditions are researched.This paper uses reverse martingale approach to solve the approximate behavior problems of finite exchangeable random variables sequences.As the De Finetti′s theorem states that infinite exchangeable random variables sequences is independent and identically distributed with the condition of the tailσ-algebra.So some results about independent identically distributed random variables is similar to exchangeable random variables.
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