scholarly journals Random Permutations, Non-Decreasing Subsequences and Statistical Independence

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1415
Author(s):  
Jesús E. García ◽  
Verónica A. González-López
Keyword(s):  
Random Variables ◽  
Continuous Data ◽  
Random Permutations ◽  
Statistical Independence ◽  
Independence Test ◽  
Independence Tests ◽  
Present Procedure ◽  
Longest Increasing Subsequence

In this paper, we show how the longest non-decreasing subsequence, identified in the graph of the paired marginal ranks of the observations, allows the construction of a statistic for the development of an independence test in bivariate vectors. The test works in the case of discrete and continuous data. Since the present procedure does not require the continuity of the variables, it expands the proposal introduced in Independence tests for continuous random variables based on the longest increasing subsequence (2014). We show the efficiency of the procedure in detecting dependence in real cases and through simulations.

On statistical independence and zero correlation in several dimensions

1960 ◽  
Vol 1 (4) ◽  
pp. 492-496 ◽  
Author(s):  
H. O. Lancaster
Keyword(s):  
Canonical Form ◽  
Arbitrary Number ◽  
Random Variables ◽  
Summable Function ◽  
Statistical Independence ◽  
Bivariate Distributions ◽  
Canonical Correlations ◽  
Zero Correlation ◽  
Maximal Correlation

Bivariate distributions, subject to a condition of φ2 boundedness to be defined later, can be written in a canonical form. Sarmanov [4] used such a form to deduce that two random variables are independent if and only if the maximal correlation of any square summable function, ξ (x1), of the first variable with any square summable function, η(x2), of the second variable is zero. This is equivalent to the condition that the canonical correlations are all zero. The theorem of Sarmanov [4] was proved without any restriction in Lancaster [2] and the proof is now extended to an arbitrary number of dimensions.

scholarly journals On Some Densities in the Set of Permutations

10.37236/372 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Eugenijus Manstavičius
Keyword(s):  
Saddle Point ◽  
Cycle Length ◽  
Random Variables ◽  
Point Method ◽  
Saddle Point Method ◽  
Asymptotic Density ◽  
Independent Random Variables ◽  
Random Permutations ◽  
Set Of Permutations

The asymptotic density of random permutations with given properties of the $k$th shortest cycle length is examined. The approach is based upon the saddle point method applied for appropriate sums of independent random variables.

scholarly journals Efficient Markov Network Structure Discovery Using Independence Tests

2009 ◽  
Vol 35 ◽  
pp. 449-484 ◽  
Author(s):  
F. Bromberg ◽  
D. Margaritis ◽  
V. Honavar
Keyword(s):  
Conditional Independence ◽  
Structure Learning ◽  
Statistical Tests ◽  
Likelihood Estimation ◽  
Data Sets ◽  
Statistical Independence ◽  
Real World Data ◽  
Independence Tests ◽  
Markov Network ◽  
Experimental Comparisons

We present two algorithms for learning the structure of a Markov network from data: GSMN* and GSIMN. Both algorithms use statistical independence tests to infer the structure by successively constraining the set of structures consistent with the results of these tests. Until very recently, algorithms for structure learning were based on maximum likelihood estimation, which has been proved to be NP-hard for Markov networks due to the difficulty of estimating the parameters of the network, needed for the computation of the data likelihood. The independence-based approach does not require the computation of the likelihood, and thus both GSMN* and GSIMN can compute the structure efficiently (as shown in our experiments). GSMN* is an adaptation of the Grow-Shrink algorithm of Margaritis and Thrun for learning the structure of Bayesian networks. GSIMN extends GSMN* by additionally exploiting Pearl's well-known properties of the conditional independence relation to infer novel independences from known ones, thus avoiding the performance of statistical tests to estimate them. To accomplish this efficiently GSIMN uses the Triangle theorem, also introduced in this work, which is a simplified version of the set of Markov axioms. Experimental comparisons on artificial and real-world data sets show GSIMN can yield significant savings with respect to GSMN*, while generating a Markov network with comparable or in some cases improved quality. We also compare GSIMN to a forward-chaining implementation, called GSIMN-FCH, that produces all possible conditional independences resulting from repeatedly applying Pearl's theorems on the known conditional independence tests. The results of this comparison show that GSIMN, by the sole use of the Triangle theorem, is nearly optimal in terms of the set of independences tests that it infers.

Records, permutations and greatest convex minorants

1989 ◽  
Vol 106 (1) ◽  
pp. 169-177 ◽  
Author(s):  
Charles M. Goldie
Keyword(s):  
Random Walk ◽  
Random Variables ◽  
Random Permutations ◽  
Record Times ◽  
Distribution Free ◽  
Standard Representation ◽  
Bernoulli Random Variables ◽  
Convex Minorants ◽  
Probability Spaces

AbstractTheorems on random permutations are translated into distribution-free results about record times and greatest convex minorants, by defining them together on appropriate probability spaces. The Bernoulli random variables that appear in the standard representation of the number of sides of the greatest convex minorant of a random walk are identified.

scholarly journals Jackknife approach to the estimation of mutual information

2018 ◽  
Vol 115 (40) ◽  
pp. 9956-9961 ◽  
Author(s):  
Xianli Zeng ◽  
Yingcun Xia ◽  
Howell Tong
Keyword(s):  
Data Analysis ◽  
Mutual Information ◽  
Random Variables ◽  
Kernel Estimation ◽  
Continuous Data ◽  
Fundamental Issue ◽  
Kernel Estimate ◽  
Kernel Estimates ◽  
Unresolved Problem

Quantifying the dependence between two random variables is a fundamental issue in data analysis, and thus many measures have been proposed. Recent studies have focused on the renowned mutual information (MI) [Reshef DN, et al. (2011)Science334:1518–1524]. However, “Unfortunately, reliably estimating mutual information from finite continuous data remains a significant and unresolved problem” [Kinney JB, Atwal GS (2014)Proc Natl Acad Sci USA111:3354–3359]. In this paper, we examine the kernel estimation of MI and show that the bandwidths involved should be equalized. We consider a jackknife version of the kernel estimate with equalized bandwidth and allow the bandwidth to vary over an interval. We estimate the MI by the largest value among these kernel estimates and establish the associated theoretical underpinnings.

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