Asymptotics in Statistics

Cam, Lucien Le; Yang, Grace L.

doi:10.1007/978-1-4612-1166-2

Cited by 205 publications

(48 citation statements)

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“…The similarities measures we used are based on several algorithms, including: Hellinger distance, 23 weighted Jaccard coefficient, 19 cosine similarity, 18 and scholar-defined weights. The measures are transformed through Gaussian kernels and other standard kernels 20,30 that assign to every pair of stories that share at least a minimum set of attributes a corresponding similarity weight.…”

Section: Connectionsmentioning

confidence: 99%

Computational folkloristics

2012

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Japan). By exploring the traditional expressions of group members across time and space, scholars can develop a sophisticated understanding of the complex processes of culture development and change.Since inception of the field in the 19 th century, folkloristics has been based on a three-part process: ˲ Fieldwork consisting of the identification of a folk group and the collection of its traditions; ˲ Archiving, editing, and publishing these collections; and ˲ Analyzing the collected folklore based on a single or combination of multiple theoretical perspectives.For the most part, folklorists limit their studies to small, well-defined collections or subsets of much larger collections. A study corpus is often selected based on criteria like genre or topic. The "variants" in the resulting study corpus are then subjected to "close readings"; from an ethnographic perspective, close readings focus on analysis of the symbolic aspects of the expression as an important meaningmaking process for storytellers and their audiences. Conclusions drawn from close readings are subsequently abstracted to make more general comments about the changing contours of the cultural ideology of the group in question. This approach has worked well for generations of folklorists, particularly in the context of the relatively small size of accessible collections and the general alignment of research The sTUdY of folklore, or folkloristics, is predicated on two premises: traditional expressive culture circulates across and within social networks, and discrete "performances" of traditional expressive forms serve an important role as the locus for the ongoing negotiation of cultural ideology (norms, beliefs, and values). The underlying assumption is that folklore emerges from the dialectic tension between individual members of a cultural group on the one hand and the "traditions" of that group on the other. This ongoing tug-of-war ensures that traditional expressive culture is constantly changing, adjusting to the needs of the individuals within a group.The goal of any folkloristic investigation is to understand how traditional expressions create meaning for the people who create and receive them; studies range from a consideration of fairy-tale telling by 19 th -century peasants (such as in the classic works of the Grimm brothers) to analysis of rumor propagation in the aftermath of disasters (such as Hurricane Katrina in 2005 and the 2011 tsunami in key insights the field of computational folkloristics weds algorithmic approaches to classic interpretive problems from the field of folklore. a multimodal network representation of a folklore corpus (hypergraphs) liberates folklore exploration from the limitations of existing classification schemes.imagine a system in which the complexities of a folklore corpus can be explored at different levels of resolution, from the broad perspective of "distant reading" down to the narrow perspective of traditional "close reading."

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Section: Connectionsmentioning

confidence: 99%

Computational folkloristics

2012

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“…j PROPOSITION 5. The two families of processes (L T ) T and ( Z T ) T are contiguous (see Le Cam andYang, 1990, Ch. 3.1, pp.…”

Section: Tightnessmentioning

confidence: 99%

Nonparametric Tests of Change‐Points with Tapered Data

Rozenholc

2001

Journal Time Series Analysis

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In this work we build two families of nonparametric tests using tapered data for the off-line detection of change-points in the spectral characteristics of a stationary Gaussian process. This is done using the Kolmogorov±Smirnov statistics based on integrated tapered periodograms. Convergence is obtained under the null hypothesis by means of a double indexed (frequency±time) process together with some extensions of Dirichlet and Fejer kernels. Consistency is proved using these statistics under the alternative. Then, using numerical simulations, we observe that the use of tapered data signi®cantly improves the properties of the test, especially in the case of small samples. Giratis et al. (1996) for linear processes and Lavielle (1993) for multidimensional Gaussian processes.On the other hand, it is well known that, for estimating the spectral measure of a stationary process, the use of the periodogram requires a large number of data. To bypass this problem of sample sizes, Dahlhaus has shown, in a series of papers (Dahlhaus, 1983(Dahlhaus, , 1988a(Dahlhaus, , 1988b(Dahlhaus, , 1990, that the use of tapered data improves spectral estimation. The increase in the asymptotic variance is balanced by a reduction of the bias, which leads to better results for small sample sizes. This is the old remedy to reduce leakage effects pointed out by Tukey (1967) or more recently in the papers of Zhang (1991), von Sachs (1994 and Janas and von Sachs (1995).In this paper, we extend the results on change-point detection (Picard, 1985) to tapered data. For this, following Dahlhaus, we introduce tapers, and then build two families of test statistics that are related to the Kolmogorov±Smirnov test statistics. Assuming that the process is Gaussian, we show the asymptotic normality of a double indexed (frequency±time) process. This results allows us to de®ne the distribution of our tests under the null hypothesis where no change occurs. By numerical simulation, we show that use of an appropriate taper depending on the sample size signi®cantly improves the detection for small sample size.Section 2 contains the assumptions, notation and the de®nition of the auxiliary process Z T on which the statistical studies rely. We prove that the test statistics derived from Z T converge under the null hypothesis to a known distribution that only depends on the chosen taper. Therefore, it is possible to tabulate this limiting distribution according to the taper, and obtain in practice the associated tests. The result relies on a central limit theorem for Z T stated in Theorem 2. Section 3 contains some preliminary results on the kernels used here. The functional central limit theorem for Z T is proved in Section 4. Section 5 is devoted to the study of the asymptotic distributions of our statistics under the null hypothesis. The main point is to show that the two statistics have the same asymptotic distribution (Theorem 3). In Section 6, we study the asymptotic con®dence region (Corollary 1) and the asymptotic consistency of the tests. Finally, w...

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“…First, we obtain convergence rates from properties of the parameterized problem, rather than generic properties of parametric spaces of densities, facilitating application. Second, we avoid a restrictive assumption that estimators be integrable, so that our analysis encompasses locally asymptotically quadratic (LAQ) problems that typically can only be shown to be stochastically bounded (see LeCam (1986); LeCam and Yang (2000); Hajek (1970)). Third, we relax a Hölder assumption on the Hellinger distance to allow cases where the best convergence rate is not necessarily a power of sample size (LeCam and Yang (2000); Prakasa Rao (1968)).…”

Section: Introductionmentioning

confidence: 99%

Maximal Uniform Convergence Rates in Parametric Estimation Problems

Beckert¹,

McFadden²

2009

Econom. Theory

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This paper considers parametric estimation problems with independent, identically, non-regularly distributed data. It focuses on rate-efficiency, in the sense of maximal possible convergence rates of stochastically bounded estimators, as an optimality criterion, largely unexplored in parametric estimation. Under mild conditions, the Hellinger metric, defined on the space of parametric probability measures, is shown to be an essentially universally applicable tool to determine maximal possible convergence rates. These rates are shown to be attainable in general classes of parametric estimation problems.JEL Classification: C13, C16

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Asymptotics in Statistics

Cited by 205 publications

References 0 publications

Computational folkloristics

Computational folkloristics

Nonparametric Tests of Change‐Points with Tapered Data

Maximal Uniform Convergence Rates in Parametric Estimation Problems

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