Asymptotics of a clustering criterion for smooth distributions

Bharath, Karthik; Pozdnyakov, Vladimir; Dey, Dipak K.

doi:10.1214/13-ejs801

Cited by 3 publications

(38 citation statements)

References 25 publications

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“…case. A pleasant by-product of the proposed framework is that the proofs of the limit results, in some cases, follow along similar lines to the ones in Bharath et al [2013].…”

Section: Introductionmentioning

confidence: 74%

“…A value p 0 ∈ (0, 1) which maximizes the split function is called the split point. When F is invertible (Q is the unique inverse), Bharath et al [2013] considered a different criterion function defined as, for 0 < p < 1,…”

Section: Preliminariesmentioning

confidence: 99%

“…, n are order statistics corresponding to i.i.d. real-valued observations X i obtained from F , then the empirical counterpart of G, christened, the Empirical Cross-over Function (ECF), was defined in Bharath et al [2013] as…”

Section: Preliminariesmentioning

confidence: 99%

“…At this juncture the hard part has been done and we are in a convenient position of using arguments from Bharath et al [2013] directly, with minimal changes, in the limit theorems for t n . This is so since the proofs of results for the empirical split point p n in Bharath et al [2013] are based solely on the weak convergence and uniform convergence of the sample criterion function; the results in the preceding section play a similar role in that regard and we can therefore repeat arguments from their proofs with very little change. We are interested in showing consistency of t n and proving a CLT.…”

Section: Finite Dimensional Convergencementioning

confidence: 99%

“…The purpose of this article is two-fold: we propose a clustering criterion for observations from a smooth, invertible distribution function which is noticeably simpler than the one recently proposed by Bharath et al [2013]; we then demonstrate the power of its simplicity by proving limit theorems for clustering constructs under a dependence setup extending their work from the i.i.d. case.…”

Section: Introductionmentioning

confidence: 99%

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On a clustering criterion for dependent observations

Bharath

2014

Journal of Statistical Planning and Inference

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A univariate clustering criterion for stationary processes satisfying a β-mixing condition is proposed extending the work of Bharath et al.[2013] to the dependent setup. The approach is characterized by an alternative sample criterion function based on truncated partial sums which renders the framework amenable to various interesting extensions for which limit results for partial sums are available. Techniques from empirical process theory for mixing sequences play a vital role in the arguments employed in the proofs of the limit theorems.

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“…case. A pleasant by-product of the proposed framework is that the proofs of the limit results, in some cases, follow along similar lines to the ones in Bharath et al [2013].…”

Section: Introductionmentioning

confidence: 74%

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Finite Dimensional Convergencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On a clustering criterion for dependent observations

Bharath

2014

Journal of Statistical Planning and Inference

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Asymptotics of the Empirical Cross-over Function

2013

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We consider a combination of heavily trimmed sums and sample quantiles which arises when examining properties of clustering criteria and prove limit theorems. The object of interest, which we call the Empirical Cross-over Function, is an L-statistic whose weights do not comply with the requisite regularity conditions for usage of existing limit results. The law of large numbers, CLT and a functional CLT are proven.

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Can Tests for Jumps be Viewed as Tests for Clusters?

2015

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We investigate the utility in employing asymptotic results related to a clustering criterion to the problem of testing for the presence of jumps in financial models. We consider the Jump Diffusion model for option pricing and demonstrate how the testing problem can be reduced to the problem of testing for the presence of clusters in the increments data. The overarching premise behind the proposed approach is in the isolation of the increments with considerably larger mean pertaining to the jumps from the ones which arise from the diffusion component. Empirical verification is provided via simulations and the test is applied to financial datasets. component? The problem is of obvious importance when prediction is the primary concern.The ramifications, while constructing a model for asset pricing, of not incorporating a jump component when the underlying process which generates the data indeed does possess one, can be quite severe. The problem has received appreciable attention over the years based on several techniques; we refer to a few articles from an exhaustive list: Ait-Sahalia [2002], Ait-Sahalia and Jacod [2009], Carr andWu [2003], Barndoff-Nielsen and Shephard [2003], Podolskij and Ziggel [2010] and Lee and Mykland [2008].The problem can be viewed as a deconstruction problem wherein the observed series of returns are deconstructed back to their continuous and jump components. This taxonomy between the continuous and the jump components of the purported model assists us in seeking 'typical' behavior of statistics based on observations under the presence and absence of the jump components. Intuitively, by constructing a test statistic which would eventually isolate the jump component under the presence of jumps in the underlying process, a suitable asymptotic hypothesis test can be developed. To elaborate, for simplicity, suppose that jumps are all positive valued and the number of jumps are finite in [0, T ]. Based on a sampling frequency, suppose we consider the increments (difference between successive observations); we would then expect to see, primarily, two groups of data: one centered around a value which is considerably larger than the other corresponding, respectively, to the jumps and the nonjumps. Such behavior is the motivation behind the test statistics based on truncated power variations employed in Ait-Sahalia [2002] and Ait-Sahalia and Jacod [2009]. In this article, we approach the problem of constructing a suitable test statistic through a different route: we ask if the isolation of the jumps can be viewed as a model-based clustering behavior wherein the distributional properties of the model, for large samples, leads to the formation of two clusters with cluster centers far apart. Under this setup, this article ought to be viewed as a first step towards providing a general answer applicable to a broad class of semimartingale models; while the alternative hypothesis of 'jumps' encompasses a large class of models, we will focus primarily on the Merton-type model wherein the jump component i...

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Asymptotics of a clustering criterion for smooth distributions

Cited by 3 publications

References 25 publications

On a clustering criterion for dependent observations

On a clustering criterion for dependent observations

Asymptotics of the Empirical Cross-over Function

Can Tests for Jumps be Viewed as Tests for Clusters?

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