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Journal articles on the topic 'Probability bounds analysis'

Author: Grafiati

Published: 4 June 2021

Last updated: 5 February 2022

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1

Enszer, Joshua A., D. Andrei Măceș, and Mark A. Stadtherr. "Probability bounds analysis for nonlinear population ecology models." Mathematical Biosciences 267 (September 2015): 97–108. http://dx.doi.org/10.1016/j.mbs.2015.06.012.

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2

Enszer, Joshua A., Youdong Lin, Scott Ferson, George F. Corliss, and Mark A. Stadtherr. "Probability bounds analysis for nonlinear dynamic process models." AIChE Journal 57, no. 2 (January 10, 2011): 404–22. http://dx.doi.org/10.1002/aic.12278.

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3

Cuesta, Juan A., and Carlos Matrán. "Conditional bounds and best L∞-approximations in probability spaces." Journal of Approximation Theory 56, no. 1 (January 1989): 1–12. http://dx.doi.org/10.1016/0021-9045(89)90128-7.

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4

Hughett, Paul. "Error Bounds for Numerical Inversion of a Probability Characteristic Function." SIAM Journal on Numerical Analysis 35, no. 4 (August 1998): 1368–92. http://dx.doi.org/10.1137/s003614299631085x.

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5

Feng, Geng. "Sensitivity Analysis for Systems under Epistemic Uncertainty with Probability Bounds Analysis." International Journal of Computer Applications 179, no. 31 (April 17, 2018): 1–6. http://dx.doi.org/10.5120/ijca2018915892.

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6

Wang, Jing, and Xin Geng. "Theoretical Analysis of Label Distribution Learning." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 5256–63. http://dx.doi.org/10.1609/aaai.v33i01.33015256.

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As a novel learning paradigm, label distribution learning (LDL) explicitly models label ambiguity with the definition of label description degree. Although lots of work has been done to deal with real-world applications, theoretical results on LDL remain unexplored. In this paper, we rethink LDL from theoretical aspects, towards analyzing learnability of LDL. Firstly, risk bounds for three representative LDL algorithms (AA-kNN, AA-BP and SA-ME) are provided. For AA-kNN, Lipschitzness of the label distribution function is assumed to bound the risk, and for AA-BP and SA-ME, rademacher complexity is utilized to give data-dependent risk bounds. Secondly, a generalized plug-in decision theorem is proposed to understand the relation between LDL and classification, uncovering that approximation to the conditional probability distribution function in absolute loss guarantees approaching to the optimal classifier, and also data-dependent error probability bounds are presented for the corresponding LDL algorithms to perform classification. As far as we know, this is perhaps the first research on theory of LDL.

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7

Aydın, Ata Deniz, and Aurelian Gheondea. "Probability Error Bounds for Approximation of Functions in Reproducing Kernel Hilbert Spaces." Journal of Function Spaces 2021 (April 30, 2021): 1–15. http://dx.doi.org/10.1155/2021/6617774.

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We find probability error bounds for approximations of functions f in a separable reproducing kernel Hilbert space H with reproducing kernel K on a base space X , firstly in terms of finite linear combinations of functions of type K x i and then in terms of the projection π x n on span K x i i = 1 n , for random sequences of points x = x i i in X . Given a probability measure P , letting P K be the measure defined by d P K x = K x , x d P x , x ∈ X , our approach is based on the nonexpansive operator L 2 X ; P K ∋ λ ↦ L P , K λ ≔ ∫ X λ x K x d P x ∈ H , where the integral exists in the Bochner sense. Using this operator, we then define a new reproducing kernel Hilbert space, denoted by H P , that is the operator range of L P , K . Our main result establishes bounds, in terms of the operator L P , K , on the probability that the Hilbert space distance between an arbitrary function f in H and linear combinations of functions of type K x i , for x i i sampled independently from P , falls below a given threshold. For sequences of points x i i = 1 ∞ constituting a so-called uniqueness set, the orthogonal projections π x n to span K x i i = 1 n converge in the strong operator topology to the identity operator. We prove that, under the assumption that H P is dense in H , any sequence of points sampled independently from P yields a uniqueness set with probability 1. This result improves on previous error bounds in weaker norms, such as uniform or L p norms, which yield only convergence in probability and not almost certain convergence. Two examples that show the applicability of this result to a uniform distribution on a compact interval and to the Hardy space H 2 D are presented as well.

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8

Block, Hentry W., Tim Costigan, and Allan R. Sampson. "Product-Type Probability Bounds of Higher Order." Probability in the Engineering and Informational Sciences 6, no. 3 (July 1992): 349–70. http://dx.doi.org/10.1017/s0269964800002588.

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GIaz and Johnson [14] introduce ith-order product-type approximations, βi, i =1,…n − 1, for Pn = P(X1 ≤c1, X2 ≤ c2,…Xn≤ cn) and show that Pn≥ βn−1 ≥ βn−2 ≥… ≥ β2 ≥ β1 when X is MTP2. In this article, it is shown thatunder weaker positive dependence conditions. For multivariate normal distributions, these conditions reduce to cov(Xi,Xj) ≥ 0 for 1 ≤ i < j ≤ n and cov(Xi,Xj| Xj−1) ≥ 0 for 1 ≤ i < j − 1, j = 3,…,n. This is applied to group sequential analysis with bivariate normal responses. Conditions for Pn ≥ β3 ≥ β2 ≥β1 are also derived. Bound conditions are also obtained that ensure that product-type approximations are nested lower bounds to upper orthant probabilities P(X1 > C1,…,Xn > cn). It is shown that these conditions are satisfied for the multivariate exponential distribution of Marshall and Olkin [20].

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9

McCormick, William P., and You Sung Park. "Asymptotic analysis of extremes from autoregressive negative binomial processes." Journal of Applied Probability 29, no. 4 (December 1992): 904–20. http://dx.doi.org/10.2307/3214723.

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It is well known that most commonly used discrete distributions fail to belong to the domain of maximal attraction for any extreme value distribution. Despite this negative finding, C. W. Anderson showed that for a class of discrete distributions including the negative binomial class, it is possible to asymptotically bound the distribution of the maximum. In this paper we extend Anderson's result to discrete-valued processes satisfying the usual mixing conditions for extreme value results for dependent stationary processes. We apply our result to obtain bounds for the distribution of the maximum based on negative binomial autoregressive processes introduced by E. McKenzie and Al-Osh and Alzaid. A simulation study illustrates the bounds for small sample sizes.

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10

Chib, Siddhartha, and Ram C. Tiwari. "Extreme Bounds Analysis in the Kalman Filter." American Statistician 45, no. 2 (May 1991): 113. http://dx.doi.org/10.2307/2684370.

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11

Ruggeri, Fabrizio. "Robust bayesian analysis given bounds on the probability of a set." Communications in Statistics - Theory and Methods 22, no. 11 (January 1993): 2983–98. http://dx.doi.org/10.1080/03610929308831198.

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12

LaMotte, Lynn Roy. "Collapsibility hypotheses and diagnostic bounds in regression analysis." Metrika 50, no. 2 (January 20, 2000): 109–19. http://dx.doi.org/10.1007/s001840050038.

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13

Bovier, Anton. "Sharp upper bounds on perfect retrieval in the Hopfield model." Journal of Applied Probability 36, no. 3 (September 1999): 941–50. http://dx.doi.org/10.1239/jap/1032374647.

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We prove a sharp upper bound on the number of patterns that can be stored in the Hopfield model if the stored patterns are required to be fixed points of the gradient dynamics. We also show corresponding bounds on the one-step convergence of the sequential gradient dynamics. The bounds coincide with the known lower bounds and confirm the heuristic expectations. The proof is based on a crucial idea of Loukianova (1997) using the negative association properties of some random variables arising in the analysis.

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14

Lagoa, C. M., F. Dabbene, and R. Tempo. "Hard Bounds on the Probability of Performance With Application to Circuit Analysis." IEEE Transactions on Circuits and Systems I: Regular Papers 55, no. 10 (November 2008): 3178–87. http://dx.doi.org/10.1109/tcsi.2008.923436.

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15

Ruggeri, F. "Bounds on the Prior Probability of a Set and Robust Bayesian Analysis." Theory of Probability & Its Applications 37, no. 2 (January 1993): 358–59. http://dx.doi.org/10.1137/1137079.

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16

Hamilton, Andrew J., Vojtech Novotný, Edward K. Waters, Yves Basset, Kurt K. Benke, Peter S. Grimbacher, Scott E. Miller, et al. "Estimating global arthropod species richness: refining probabilistic models using probability bounds analysis." Oecologia 171, no. 2 (September 12, 2012): 357–65. http://dx.doi.org/10.1007/s00442-012-2434-5.

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17

Kanter, Marek. "Discrimination distance bounds and statistical applications." Probability Theory and Related Fields 86, no. 3 (September 1990): 403–22. http://dx.doi.org/10.1007/bf01208258.

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18

Gao, Y., and J. Culberson. "An Analysis of Phase Transition in NK Landscapes." Journal of Artificial Intelligence Research 17 (October 1, 2002): 309–32. http://dx.doi.org/10.1613/jair.1081.

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In this paper, we analyze the decision version of the NK landscape model from the perspective of threshold phenomena and phase transitions under two random distributions, the uniform probability model and the fixed ratio model. For the uniform probability model, we prove that the phase transition is easy in the sense that there is a polynomial algorithm that can solve a random instance of the problem with the probability asymptotic to 1 as the problem size tends to infinity. For the fixed ratio model, we establish several upper bounds for the solubility threshold, and prove that random instances with parameters above these upper bounds can be solved polynomially. This, together with our empirical study for random instances generated below and in the phase transition region, suggests that the phase transition of the fixed ratio model is also easy.

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19

Mao, Jianfeng, Zhiwu Yu, and Lizhong Jiang. "Stochastic Analysis of Vehicle-Bridge Coupled Interaction and Uncertainty Bounds of Random Responses in Heavy Haul Railways." International Journal of Structural Stability and Dynamics 19, no. 12 (December 2019): 1950144. http://dx.doi.org/10.1142/s021945541950144x.

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The systematic running safety assessment of railway bridges presents lots of challenges, one of which is estimating the uncertainty bounds of the structural responses of bridges under vehicle loads with multisource randomness. In this study, a probability safety assessment method is proposed for evaluating the uncertainty bounds of random time-history responses for the stochastic train-bridge coupled system. First, a refined probabilistic model for the train-bridge coupled system (TBS) in heavy haul railway is established with the multi-excitations of random track irregularities, random vehicle loads and stochastic structural parameters. The probability density evolution method (PDEM) is employed to obtain the solution of the time-varying probability transferred between the stochastic excitations and the output of the dynamic responses. Then, to establish a rapid and straightforward approach for the systematic running safety assessment of the TBS, the quantiles of the probability distribution are used to estimate the time-history uncertainty bounds of random responses of interest distributed in real probability functions. Case studies by the field test and numerical simulation are presented to verify and investigate the accuracy and reliability of the proposed method. The results show that the quantiles of the probability distribution proposed are suitable for the systematic running safety assessment of the TBS.

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20

Embrechts, Paul, and Giovanni Puccetti. "Bounds for functions of multivariate risks." Journal of Multivariate Analysis 97, no. 2 (February 2006): 526–47. http://dx.doi.org/10.1016/j.jmva.2005.04.001.

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21

Morris, B., and Yuval Peres. "Evolving sets, mixing and heat kernel bounds." Probability Theory and Related Fields 133, no. 2 (June 6, 2005): 245–66. http://dx.doi.org/10.1007/s00440-005-0434-7.

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22

Barron, Andrew, Lucien Birgé, and Pascal Massart. "Risk bounds for model selection via penalization." Probability Theory and Related Fields 113, no. 3 (February 28, 1999): 301–413. http://dx.doi.org/10.1007/s004400050210.

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23

TAKARA, Kaoru, and Kaori TOSA. "APPLICATION OF PROBABILITY DISTRIBUTIONS WITH LOWER AND UPPER BOUNDS TO HYDROLOGIC FREQUENCY ANALYSIS." PROCEEDINGS OF HYDRAULIC ENGINEERING 43 (1999): 121–26. http://dx.doi.org/10.2208/prohe.43.121.

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24

Karanki, Durga Rao, Hari Shankar Kushwaha, Ajit Kumar Verma, and Srividya Ajit. "Uncertainty Analysis Based on Probability Bounds (P-Box) Approach in Probabilistic Safety Assessment." Risk Analysis 29, no. 5 (May 2009): 662–75. http://dx.doi.org/10.1111/j.1539-6924.2009.01221.x.

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25

Hall, Peter, and J. S. Marron. "Lower bounds for bandwidth selection in density estimation." Probability Theory and Related Fields 90, no. 2 (June 1991): 149–73. http://dx.doi.org/10.1007/bf01192160.

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26

Panchenko, Dmitry, and Michel Talagrand. "Bounds for diluted mean-fields spin glass models." Probability Theory and Related Fields 130, no. 3 (March 3, 2004): 319–36. http://dx.doi.org/10.1007/s00440-004-0342-2.

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27

Nguyen, Hoi, Terence Tao, and Van Vu. "Random matrices: tail bounds for gaps between eigenvalues." Probability Theory and Related Fields 167, no. 3-4 (January 25, 2016): 777–816. http://dx.doi.org/10.1007/s00440-016-0693-5.

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28

Sinclair, Alistair, Piyush Srivastava, Daniel Štefankovič, and Yitong Yin. "Spatial mixing and the connective constant: optimal bounds." Probability Theory and Related Fields 168, no. 1-2 (July 14, 2016): 153–97. http://dx.doi.org/10.1007/s00440-016-0708-2.

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29

Hitczenko, Pawe? "Upper bounds for theL p -norms of martingales." Probability Theory and Related Fields 86, no. 2 (June 1990): 225–38. http://dx.doi.org/10.1007/bf01474643.

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30

Sheng, Shuyang. "A Structural Econometric Analysis of Network Formation Games Through Subnetworks." Econometrica 88, no. 5 (2020): 1829–58. http://dx.doi.org/10.3982/ecta12558.

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The objective of this paper is to identify and estimate network formation models using observed data on network structure. We characterize network formation as a simultaneous‐move game, where the utility from forming a link depends on the structure of the network, thereby generating strategic interactions between links. With the prevalence of multiple equilibria, the parameters are not necessarily point identified. We leave the equilibrium selection unrestricted and propose a partial identification approach. We derive bounds on the probability of observing a subnetwork, where a subnetwork is the restriction of a network to a subset of the individuals. Unlike the standard bounds as in Ciliberto and Tamer (2009), these subnetwork bounds are computationally tractable in large networks provided we consider small subnetworks. We provide Monte Carlo evidence that bounds from small subnetworks are informative in large networks.

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31

Loganathan, G. V., P. Mattejat, C. Y. Kuo, and M. H. Diskin. "Frequency Analysis of Low Flows: Hypothetical Distribution Methods and a Physically Based Approach." Hydrology Research 17, no. 3 (June 1, 1986): 129–50. http://dx.doi.org/10.2166/nh.1986.0009.

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A mixed log Pearson type III distribution, a double bounded probability density function, partial duration series and a physically based approach are analyzed for frequency estimates of low flows. The mixed log Pearson III involves a point probability mass at zero for intermittent streams. The double bounded probability distribution has lower and upper bounds with a point mass at the lower bound. Two approaches are used in partial duration series i) truncation, and ii) censoring which represent curtailing of the population and the sample respectively. The parameters are estimated by maximum likelihood procedure. Considering low flows as part of the recession limb of stream flow hydrographs a physically based approach is formulated. By using the exponential decay of stream recessions and considering the initial recession flows, recession durations, and recharge due to incoming storms as statistically independent random variables, a first order random coefficient Markov model for initial recession flows is formed. The resulting steady state probability distribution for initial recession flows is combined with the probability distribution of the exponential decay to obtain the probabilities of low flow events. The methods are applied to both perennial and intermittent streams.

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32

Rüschendorf, L. "Analysis of risk bounds in partially specified additive factor models." Insurance: Mathematics and Economics 86 (May 2019): 115–21. http://dx.doi.org/10.1016/j.insmatheco.2019.02.007.

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33

Sprouse, Chadwick. "Integral curvature bounds and bounded diameter." Communications in Analysis and Geometry 8, no. 3 (2000): 531–43. http://dx.doi.org/10.4310/cag.2000.v8.n3.a4.

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34

K�lske, Christof. "Universal bounds on the selfaveraging of random diffraction measures." Probability Theory and Related Fields 126, no. 1 (May 1, 2003): 29–50. http://dx.doi.org/10.1007/s00440-003-0261-7.

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35

Bobkov, Sergey G., Gennadiy P. Chistyakov, and Friedrich Götze. "Berry–Esseen bounds in the entropic central limit theorem." Probability Theory and Related Fields 159, no. 3-4 (June 12, 2013): 435–78. http://dx.doi.org/10.1007/s00440-013-0510-3.

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36

ZHOU, Zhenyu, and Takuro SATO. "Error Probability Bounds Analysis of JMLSE Based Interference Cancellation Algorithms for MQAM-OFDM Systems." IEICE Transactions on Communications E94-B, no. 7 (2011): 2032–42. http://dx.doi.org/10.1587/transcom.e94.b.2032.

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37

Einmahl, John H. J., and David M. Mason. "Bounds for weighted multivariate empirical distribution functions." Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete 70, no. 4 (1985): 563–71. http://dx.doi.org/10.1007/bf00531867.

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38

Chulaevsky, Victor. "Stochastic Regularization and Eigenvalue Concentration Bounds for Singular Ensembles of Random Operators." International Journal of Statistical Mechanics 2013 (November 27, 2013): 1–14. http://dx.doi.org/10.1155/2013/931063.

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We propose a simple approach allowing reducing the eigenvalue concentration analysis of a class of random operator ensembles with singular probability distribution to the analysis of an auxiliary ensemble with bounded probability density. Our results apply to the Wegner- and Minami-type estimates for single- and multiparticle operators.

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39

Smith, Cyril, and Martin Knott. "On Hoeffding-Fréchet bounds and cyclic monotone relations." Journal of Multivariate Analysis 40, no. 2 (February 1992): 328–34. http://dx.doi.org/10.1016/0047-259x(92)90029-f.

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40

Rataj, Jan, and Ivan Saxl. "Analysis of planar anisotropy by means of the Steiner compact." Journal of Applied Probability 26, no. 3 (September 1989): 490–502. http://dx.doi.org/10.2307/3214407.

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A graphical method for the estimation of the anisotropy of planar fibre systems based on the Steiner compact set is proposed and discussed. Upper bounds for the deviation in probability of the graphical estimate of the Steiner compact are given and a consistency theorem is proved.

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41

Heredia-Zavoni, Ernesto, Dante Campos, and Gallegher Ramı´rez. "Reliability Based Assessment of Deck Elevations for Offshore Jacket Platforms." Journal of Offshore Mechanics and Arctic Engineering 126, no. 4 (November 1, 2004): 331–36. http://dx.doi.org/10.1115/1.1834622.

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Structural reliability analyses of fixed marine platforms subjected to storm wave loading are performed to assess deck elevations. Platforms are modeled as a series system consisting of the deck and jacket bays. The structural reliability analyses are carried out assuming dominant failure modes for the system components. Upper and lower bounds of the probability of failure are computed. The variation of the reliability index per bay component as a function of wave height, with a focus on those wave heights that generate forces on the deck, is analyzed. A comparison is given for the deck probability of failure and the lower bound probability of failure of the jacket in order to assess how the deck or the jacket controls the probability of failure of the system. Results are also given for reliability analyses considering different deck elevations. Finally, an analysis of the total probabilities of failure, unconditioned on wave heights, is given.

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42

Bagul, Yogesh, and Christophe Chesneau. "Some sharp circular and hyperbolic bounds of exp(-x2) with applications." Applicable Analysis and Discrete Mathematics 14, no. 1 (2020): 239–54. http://dx.doi.org/10.2298/aadm190123010b.

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This article is devoted to the determination of sharp lower and upper bounds for exp(-x2) over the interval (-?,?). The bounds are of the type [a+f(x)/a+1]? , where f(x) denotes either cosine or hyperbolic cosine. The results are then used to obtain and refine some known Cusa-Huygens type inequalities. In particular, a new simple proof of Cusa-Huygens type inequalities is presented as an application. For other interesting applications of the main results, sharp bounds of the truncated Gaussian sine integral and error functions are established. They can be useful in probability theory.

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43

Seaman, John, and Pat Odell. "On Goldstein’s variance bound." Advances in Applied Probability 17, no. 3 (September 1985): 679–81. http://dx.doi.org/10.2307/1427126.

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Goldstein (1974) derived an upper bound on the variance of certain non-negative functions when the first two moments of the underlying random variable are known. This bound is compared to a simple and fundamental variance bound which requires only that the range of the function be known. It is shown that Goldstein’s bound frequently exceeds the simpler bound. Finally, an interpretation of such bounds in the context of economic risk analysis is given.

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44

Liu, Jia. "The parameterized Gallager’s first bounds based on conditional triplet-wise error probability." Mathematics and Computers in Simulation 163 (September 2019): 32–46. http://dx.doi.org/10.1016/j.matcom.2019.02.005.

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45

Fabian, Marian J., René Henrion, Alexander Y. Kruger, and Jiří V. Outrata. "Error Bounds: Necessary and Sufficient Conditions." Set-Valued and Variational Analysis 18, no. 2 (February 10, 2010): 121–49. http://dx.doi.org/10.1007/s11228-010-0133-0.

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46

Rataj, Jan, and Ivan Saxl. "Analysis of planar anisotropy by means of the Steiner compact." Journal of Applied Probability 26, no. 03 (September 1989): 490–502. http://dx.doi.org/10.1017/s0021900200038092.

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A graphical method for the estimation of the anisotropy of planar fibre systems based on the Steiner compact set is proposed and discussed. Upper bounds for the deviation in probability of the graphical estimate of the Steiner compact are given and a consistency theorem is proved.

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47

Zhang, J., L. M. Zhang, and Wilson H. Tang. "New methods for system reliability analysis of soil slopes." Canadian Geotechnical Journal 48, no. 7 (July 2011): 1138–48. http://dx.doi.org/10.1139/t11-009.

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A slope may have many possible slip surfaces. As sliding along any slip surface can cause slope failure, the system failure probability of a slope is different from the probability of failure along an individual slip surface. In this paper, we first suggest an efficient method for evaluating the system failure probability of a slope that considers a large number of possible slip surfaces. To obtain more insights into the system failure probability of a slope, we also propose a method to identify a few representative slip surfaces most important for system reliability analysis among a large number of potential slip surfaces and to calculate the system failure probability based on these representative slip surfaces. An equation for estimating the bounds of system failure probability based on the failure probability of the most critical slip surface is also suggested. The system failure probability is governed by only a few representative slip surfaces. For a homogenous slope, the failure probability of the most critical slip surface is a good approximation of the system failure probability. For a slope in layered soils, the system failure probability can be significantly larger than the failure probability of the most critical slip surface.

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48

Han, Chun Lei, Mei Lu Lin, and Ye Yang. "Performance Analysis for Hybrid Cooperative Beamforming in Wireless Relay Networks." Applied Mechanics and Materials 543-547 (March 2014): 3435–40. http://dx.doi.org/10.4028/www.scientific.net/amm.543-547.3435.

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This paper considers a relay-assisted hybrid cooperative beamforming system in independent and identically Rayleigh fading channels. In this system, each relay can adaptively employ amplify-and-forward (AF) or decode-and-forward (DF) protocol for maximizing the instantaneous signal-to-noise ratio (SNR). Tight upper and lower bounds for the outage probability have been derived in closed form, based on which we investigate the achievable diversity of the hybrid cooperative beamforming system. We also present the corresponding upper and lower bounds of the average channel capacity. Numerical results are provided to validate our theoretic analyses.

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49

Ouyang, Yu Hua, and Xiang Dong Jia. "Asymptotic Performance Analysis of Two-Way Opportunistic Relaying Channels." Applied Mechanics and Materials 347-350 (August 2013): 1419–24. http://dx.doi.org/10.4028/www.scientific.net/amm.347-350.1419.

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The work interests in the two-way opportunistic relaying (TWOR) system in which the analog network coding (ANC) employed. For the TWOR-ANC, we first present the traditional max-min best relay selection scheme as well as the corresponding closed-form analytical expressions for the cumulative distribution function (CDF) and the probability density function (PDF) of the end-to-end SNR. Then, to simplify the analysis of system performance, by using the approximation of the modified Bessel function of the second kind, , we obtain the performance bounds of the TWOR-ANC system. The analytical results show that e-z/z determines the lower bound of system performance, and the exact solution of system performance is very close to the lower bound in low SNRs. At the same time, 1/z does the upper bound, and the exact solution is very close to the upper bound in high SNRs. Thus, in high SNRs or lower SNRs, the system performance can be measured approximately by the upper bound or lower bound.

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50

Uzun, Hasan B., and Kenneth S. Alexander. "Lower bounds for boundary roughness for droplets in Bernoulli percolation." Probability Theory and Related Fields 127, no. 1 (September 1, 2003): 62–88. http://dx.doi.org/10.1007/s00440-003-0276-0.

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