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2008, Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen
The Simes inequality has received considerable attention recently because of its close connection to some important multiple hypothesis testing procedures. We revisit in this article an old result on this inequality to clarify and strengthen it and a recently proposed generalization of it to offer an alternative simpler proof.
Statistics & Probability Letters, 2013
showed that certain positively dependent (MTP 2 ) random variables satisfy the Simes Inequality. The multivariate-t distribution does not satisfy this property, so other means are necessary to show that it also satisfies the Simes inequality. A corollary was given in Sarkar (1998) to handle this distribution, but there is an error. In this paper a direct proof is given to show the multivariate-t does satisfy the Simes inequality.
The Annals of Statistics, 1998
Statistics & Probability Letters, 2008
Considering independent test statistics, Simes' critical values are modified and newer sets of critical values, each providing an exact control of the type I error rate, are obtained. These modifications, as simulations show, quite often yield more powerful tests than the original Simes' test.
Journal of Modern Optics, 2004
Journal of Mathematical Analysis and Applications, 1997
The Annals of Statistics, 2008
Given a collection of null hypotheses and the corresponding p-values in multiple testing, one encounters two types of problem: (i) global testing of the intersection null hypothesis and (ii) simultaneous testing of the individual null hypotheses. Traditionally, one seeks to control the probability of falsely rejecting the intersection null hypothesis, the global Type I error rate, in global testing and the probability of rejecting at least one true null hypothesis, the familywise (Type I) error rate (FWER), in simultaneous testing. As a global test, Simes' [19] test has received considerable attention. With the p-values marginally having
arXiv: Statistics Theory, 2017
The aim of this paper is to discuss various concentration inequali- ties for U-statistics and most recent results. A special focus will be on providing short proofs for bounds on the U-statistics using classical concentration inequalities, which, although well known, are not easy to locate in the literature.
arXiv: Statistics Theory, 2017
The aim of this paper is to discuss various concentration inequalities for U-statistics and most recent results. A special focus will be on providing proofs for bounds on the U-statistics using classical concentration inequalities, which, although the results well known, the proofs are not found in the literature.
Applied Mathematics Letters, 2002
The authors use their recently proved integral inequality to obtain bounds for the covariance of two random variables (a) in a general setup and (b) for a class of special joint distributions. The same inequality is also used to estimate the difference of the expectations of two random variables. Finally, the authors study the attainability of a related inequality.
We show that P ′ Q L p (I) ≤ c 1+1/p (N + M) log(min(N, M + 1) + 1) P Q L p (I) for all real trigonometric polynomials P and Q of degree N and M , respectively, where 0 < p ≤ ∞, I := [−π, π], and c > 0 is a suitable absolute constant. We also show that f ′ g L p (J) ≤ c 1+1/p (N + M) 2 f g L p (J) for all algebraic polynomials f and g of degree N and M , respectively, where 0 < p ≤ ∞, J := [−1, 1], and c > 0 is a suitable absolute constant. Both of our trigonometric and algebraic results are sharp up to the factor c 1+1/p. In fact, we prove our results for the much wider classes of generalized trigonometric and algebraic polynomials.
Tamkang Journal of Mathematics, 1998
We prove a generalizatin of Cadson's inequality. We improve a previous equivalent inequality of Carlson's inequality and give some examples.
Springer INdAM Series, 2013
2013
We propose a new approach for deriving probabilistic inequalities based on bounding likelihood ratios. We demonstrate that this approach is more general and powerful than the classical method frequently used for deriving concentration inequalities such as Chernoff bounds. We discover that the proposed approach is inherently related to statistical concepts such as monotone likelihood ratio, maximum likelihood, and the method of moments for parameter estimation. A connection between the proposed approach and the large deviation theory is also established. We show that, without using moment generating functions, tightest possible concentration inequalities may be readily derived by the proposed approach. We have derived new concentration inequalities using the proposed approach, which cannot be obtained by the classical approach based on moment generating functions.
2016
In this paper, we propose a new approach for deriving probabilistic inequalities. Our main idea is to exploit the information of underlying distributions by virtue of the monotone likelihood ratio property and Berry-Essen inequality. Unprecedentedly sharp bounds for the tail probabilities of some common distributions are established. The applications of the probabilistic inequalities in parameter estimation are discussed.
Probability Theory and Related Fields, 2007
We introduce a version of Stein’s method for proving concentration and moment inequalities in problems with dependence. Simple illustrative examples from combinatorics, physics, and mathematical statistics are provided.
Journal of Mathematical Inequalities, 2007
Journal of Approximation Theory, 2002
A sharpened version of Carleman's inequality is proved. This result unifies and generalizes some recent results of this type. Also the "ordinary" sum that serves as the upper bound is replaced by the corresponding Cesaro sum. Moreover, a Carleman type inequality with a more general measure is proved and this result may also be seen as a generalization of a continuous variant of Carleman's inequality, which is usually referred to as Knopp's inequality. A new elementary proof of (Carleman-)Knopp's inequality and a new inequality of Hardy-Knopp type is pointed out.
In this paper, we propose a new approach for deriving probabilistic inequalities. Our main idea is to exploit the information of underlying distributions by virtue of the monotone likelihood ratio property and Berry-Essen inequality. Unprecedentedly sharp bounds for the tail probabilities of some common distributions are established.
Theory of Probability and Mathematical Statistics
The Diaz-Metcalf and Pólya-Szegő inequalities are proved in the probabilistic setting. These results generalize the classical case for both sums and integrals. Using these results we obtain some other well-known inequalities in the probabilistic setting, namely the Kantorovich, Rennie, and Schweitzer inequalities.
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