On a Refinement of the Central Limit Theorem for Sums of Independent Random Indicators

Mikhailov, Vladimir Gavrilovich

doi:10.1137/1138044

Cited by 16 publications

(8 citation statements)

References 3 publications

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“…However, upon comparing the WTW projection (at the level t = 0.01, for the year 2000) obtained by running the skewness-corrected Gaussian approximation with the projection based on the full Poisson-Binomial distribution, we found that 20%  of the statistically-significant links are lost in the Gaussian-based validated projection. The limitations of the Gaussian approximations are discussed in further detail in [42,43].…”

Section: Appendix B Approximations Of the Poisson-binomial Distributionmentioning

confidence: 99%

Inferring monopartite projections of bipartite networks: an entropy-based approach

et al. 2017

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Bipartite networks are currently regarded as providing a major insight into the organization of many real-world systems, unveiling the mechanisms driving the interactions occurring between distinct groups of nodes. One of the most important issues encountered when modeling bipartite networks is devising a way to obtain a (monopartite) projection on the layer of interest, which preserves as much as possible the information encoded into the original bipartite structure. In the present paper we propose an algorithm to obtain statistically-validated projections of bipartite networks, according to which any two nodes sharing a statistically-significant number of neighbors are linked. Since assessing the statistical significance of nodes similarity requires a proper statistical benchmark, here we consider a set of four null models, defined within the Exponential Random Graph framework. Our algorithm outputs a matrix of link-specific p-values, from which a validated projection is straightforwardly obtainable, upon running a multiple hypothesis testing procedure. Finally, we test our method on an economic network (i.e. the countries-products World Trade Web representation) and a social network (i.e. MovieLens, collecting the users' ratings of a list of movies). In both cases non-trivial communities are detected: while projecting the World Trade Web on the countries layer reveals modules of similarly-industrialized nations, projecting it on the products layer allows communities characterized by an increasing level of complexity to be detected; in the second case, projecting MovieLens on the films layer allows clusters of movies whose affinity cannot be fully accounted for by genre similarity to be individuated.

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Section: Appendix B Approximations Of the Poisson-binomial Distributionmentioning

confidence: 99%

Inferring monopartite projections of bipartite networks: an entropy-based approach

et al. 2017

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“…Subsequently many other proofs of this result and similar ones were given using a range of different techniques; [3,14,24,31] is a sampling of work along these lines, and Steele [50] gives an extensive list of relevant references. Much work has also been done on approximating PBDs by normal distributions (see e.g., [2,29,38,51]) and by Binomial distributions (see e.g., [28,44,48]). These results provide structural information about PBDs that can be well-approximated via simpler distributions, but fall short of our goal of obtaining approximations of an unknown PBD up to arbitrary accuracy.…”

Section: Related Workmentioning

confidence: 99%

Learning Poisson Binomial Distributions

2015

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We consider a basic problem in unsupervised learning: learning an unknown Poisson binomial distribution. A Poisson binomial distribution (PBD) over {0, 1, . . . , n} is the distribution of a sum of n independent Bernoulli random variables which may have arbitrary, potentially non-equal, expectations. These distributions were first studied by Poisson (Recherches sur la Probabilitè des jugements en matié criminelle et en matiére civile. Bachelier, Paris, 1837) and are a natural n-parameter generalization of the familiar Binomial Distribution. Surprisingly, prior to our work this basic learning problem was poorly understood, and known results for it were far from optimal. We essentially settle the complexity of the learning problem for this basic class of distributions. As our first main result we give a highly efficient algorithm which learns to -accuracy (with respect to the total variation distance) usingÕ(1/ 3 ) samples independent of n. The running time of the algorithm is quasilinear in the size 123 Algorithmica of its input data, i.e.,Õ(log(n)/ 3 ) bit-operations (we writeÕ(·) to hide factors which are polylogarithmic in the argument toÕ(·); thus, for example,Õ(a log b) denotes a quantity which is O (a log b · log c (a log b)) for some absolute constant c. Observe that each draw from the distribution is a log(n)-bit string). Our second main result is a proper learning algorithm that learns to -accuracy usingÕ(1/ 2 ) samples, and runs in time (1/ ) poly(log(1/ )) · log n. This sample complexity is nearly optimal, since any algorithm for this problem must use (1/ 2 ) samples. We also give positive and negative results for some extensions of this learning problem to weighted sums of independent Bernoulli random variables.

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“…The distribution of the number of f -recurrent runs in a sequence of independent random variables was studied by Mikhailov [16,17]. Similar results for the number of usual and f -recurrent runs with possible omissions of letters were obtained by Mezhennaya in [14] and [15], respectively.…”

Section: Introductionmentioning

confidence: 58%

On the Number of Event Appearances in a Markov Chain

Mezhennaya¹

2019

Int. J. of Appl. Math.

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The paper presents the estimate for the total variation distance between the distribution of the number of appearances of homogeneous disjoint events in a segment of strongly ergodic Markov chain on the finite state space and accompanying Poisson distribution (i.e., Poisson distribution with a parameter equal to the expectation of the random variable under consideration). For this purpose the Chen-Stein method was used. As a result Poisson and normal limit theorems for the number of events appearances are derived. The considered scheme describes the well-known number of runs on consecutive letters, the number of f -recurrent runs, etc., and can be used for describing the properties of distribution of the special form scan statistic.

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On a Refinement of the Central Limit Theorem for Sums of Independent Random Indicators

Abstract: Explicit and rather tight upper bounds for the distance (in the uniform metric) between the distribution function of a sum of independent random indicators and its asymptotic expansion are obtained.

Cited by 16 publications

References 3 publications

Inferring monopartite projections of bipartite networks: an entropy-based approach

Inferring monopartite projections of bipartite networks: an entropy-based approach

Learning Poisson Binomial Distributions

On the Number of Event Appearances in a Markov Chain

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