Stochastic Convergence of Persistence Landscapes and Silhouettes

Chazal, Frédéric; Fasy, Brittany Terese; Lecci, Fabrizio; Rinaldo, Alessandro; Wasserman, Larry

doi:10.1145/2582112.2582128

“…3 Some applications of this result can be found in Chetverikov (2011Chetverikov ( , 2012, Wasserman, Kolar and Rinaldo (2013), and Chazal, Fasy, Lecci, Rinaldo, and Wasserman (2013).…”

Section: Construct the Multiplier Bootstrap Test Statisticmentioning

confidence: 89%

Testing Many Moment Inequalities

Chernozhukov¹,

Chetverikov²,

Kato³

2013

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Abstract. This paper considers the problem of testing many moment inequalities where the number of moment inequalities, denoted by p, is possibly much larger than the sample size n. There are variety of economic applications where the problem of testing many moment inequalities appears; a notable example is the entry model of Ciliberto and Tamer (2009) where p = 2 m+1 with m being the number of firms. We consider the test statistic given by the maximum of p Studentized (or t-type) statistics, and analyze various ways to compute critical values for the test. Specifically, we consider critical values based upon (i) the union (Bonferroni) bound combined with a moderate deviation inequality for self-normalized sums, (ii) the multiplier bootstrap. We also consider two step variants of (i) and (ii) by incorporating moment selection. We prove validity of these methods, showing that under mild conditions, they lead to tests with error in size decreasing polynomially in n while allowing for p being much larger than n; indeed p can be of order exp(n c ) for some c > 0. Importantly, all these results hold without any restriction on correlation structure between p Studentized statistics, and also hold uniformly with respect to suitably wide classes of underlying distributions. We also show that all the tests developed in this paper are asymptotically minimax optimal when p grows with n.

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“…14 Some applications of this result can be found in Chetverikov (2011Chetverikov ( , 2012, Wasserman, Kolar and Rinaldo (2013), and Chazal, Fasy, Lecci, Rinaldo, and Wasserman (2013).…”

Section: Construct the Empirical Bootstrap Test Statisticmentioning

confidence: 89%

Testing many moment inequalities

Chernozhukov¹,

Chetverikov²,

Kato³

2014

2

0

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Abstract. This paper considers the problem of testing many moment inequalities where the number of moment inequalities, denoted by p, is possibly much larger than the sample size n. There are a variety of economic applications where the problem of testing many moment inequalities appears; a notable example is a market structure model of Ciliberto and Tamer (2009) where p = 2 m+1 with m being the number of firms. We consider the test statistic given by the maximum of p Studentized (or t-type) statistics, and analyze various ways to compute critical values for the test statistic. Specifically, we consider critical values based upon (i) the union bound combined with a moderate deviation inequality for self-normalized sums, (ii) the multiplier and empirical bootstraps, and (iii) two-step and three-step variants of (i) and (ii) by incorporating selection of uninformative inequalities that are far from being binding and novel selection of weakly informative inequalities that are potentially binding but do not provide first order information. We prove validity of these methods, showing that under mild conditions, they lead to tests with error in size decreasing polynomially in n while allowing for p being much larger than n; indeed p can be of order exp(n c ) for some c > 0. Importantly, all these results hold without any restriction on correlation structure between p Studentized statistics, and also hold uniformly with respect to suitably large classes of underlying distributions. Moreover, when p grows with n, we show that all of our tests are (minimax) optimal in the sense that they are uniformly consistent against alternatives whose "distance" from the null is larger than the threshold (2(log p)/n) 1/2 , while any test can only have trivial power in the worst case when the distance is smaller than the threshold. Finally, we show validity of a test based on block multiplier bootstrap in the case of dependent data under some general mixing conditions. Date: First public version: December 2013 (arXiv:1312.7614v1). This version: December 16, 2014.

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“…sets of the empirical distance function, with the upper level sets of a density estimator. This approach has been suggested by Phillips et al (2015); Chazal et al (2014a); Bobrowski et al (2014);Chung et al (2009);Bubenik (2015). The idea is to consider the upper level sets L t = {x : p h (x) > t}.…”

Section: Distance Functions and Persistent Homologymentioning

confidence: 99%

“…It is essentially a smooth, probabilistic version of the distance function. The properties of the DTM are discussed in Chazal et al (2011Chazal et al ( , 2014aChazal et al ( , 2015.…”

Section: Distance Functions and Persistent Homologymentioning

confidence: 99%