Testing for multimodality with dependent data

Chan, Kung‐Sik; Tong, Hongwei

doi:10.1093/biomet/91.1.113

Cited by 13 publications

(12 citation statements)

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“…In the presence of sudden changes or jumps in time-series, which may be indicative of a phase transition, the frequency distribution of key population variables should be multimodal (Scheffer & Carpenter 2003). In this study, we tested whether the analysed time series of pollock recruitment followed a multimodal distribution using an extended application of the Silverman test (Chan & Tong 2004). To increase the power of the test we adopted the Hall & York (2001) adjustment under the null hypothesis of unimodality (i.e.…”

Section: Methodsmentioning

confidence: 99%

“…The null hypothesis of unimodality is rejected (at 0.05 significance level) up to the year 1995. The reported p-values were computed based on the Silverman test assuming independent and identically distributed data, with an adjustment due to Hall & York (2001) and 1000 bootstrap replications; these p-values were almost identical under the assumption that the data were generated from a first-order or second-order Markov process (Chan & Tong 2004). Also shown, on the right side of the plot, is the frequency distribution of the entire (i.e.…”

Section: Terminology Preamble: Phase and Regimementioning

confidence: 99%

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Climate change causing phase transitions of walleye pollock ( Theragra chalcogramma ) recruitment dynamics

et al. 2005

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In 1976 the North Pacific climate shifted, resulting in an average increase of the water temperature. In the Gulf of Alaska the climate shift was followed (i.e. early 1980s) by a gradual but dramatic increase in the abundance of groundfish species that typically prey on pre-recruitment stages of walleye pollock. In the present study we used a previously parameterized model to investigate the effect of these climate and biological changes on the recruitment dynamics of walleye pollock in the Gulf of Alaska. Simulations covered the 1970-2000 time frame and emphasized the medium-to-long temporal scale (i.e. about 5-10 years) of environmental variability. Results showed that during periods characterized by high sea surface temperature and high predation on juvenile pollock stages, recruitment variability and magnitude were below average, and recruitment control was delayed to stages older than the 0-group. Opposite dynamics (i.e. high abundance and variability, and early recruitment control) occurred during periods characterized by low temperature and predation. These results are in general agreement with empirical observations, and allowed us to formulate causal explanations for their occurrence. We interpreted the delay of recruitment control and the reduction of variability as an effect of increased constraint on the abundance of post age-0 stages, in turn imposed by high density dependence and predation mortality. On the other hand, low density-dependence and predation favoured post age-0 survival, and allowed for an unconstrained link between larval and recruitment abundance. Our findings demonstrate that the dominant mechanisms of pollock survival change over contrasting climate regimes. Such changes may in turn cause a phase transition of recruitment dynamics with profound implications for the management of the entire stock.

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Section: Methodsmentioning

confidence: 99%

Section: Terminology Preamble: Phase and Regimementioning

confidence: 99%

Climate change causing phase transitions of walleye pollock ( Theragra chalcogramma ) recruitment dynamics

et al. 2005

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“…Hansen (1999) proposed using an empirical likelihood estimator of the Markov transition probability but did not prove that the resulting version of the MB is consistent or provides asymptotic refinements. Chan and Tong (1998) proposed using the MB in a test for multimodality in the distribution of dependent data. Paparoditis and Politis (2001) proposed estimating the Markov transition probability by resampling the data in a suitable way.…”

Section: The Bootstrap For Markov Processesmentioning

confidence: 99%

Bootstrap Methods for Time Series

Kreiß¹,

Lahiri²

2012

Handbook of Statistics

101

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The bootstrap is a method for estimating the distribution of an estimator or test statistic by resampling one's data or a model estimated from the data. The methods that are available for implementing the bootstrap and the accuracy of bootstrap estimates depend on whether the data are a random sample from a distribution or a time series. This paper is concerned with the application of the bootstrap to time-series data when one does not have a finite-dimensional parametric model that reduces the data generation process to independent random sampling. We review the methods that have been proposed for implementing the bootstrap in this situation and discuss the accuracy of these methods relative to that of first-order asymptotic approximations. We argue that methods for implementing the bootstrap with time-series data are not as well understood as methods for data that are sampled randomly from a distribution. Moreover, the performance of the bootstrap as measured by the rate of convergence of estimation errors tends to be poorer with time series than with random samples. This is an important problem for applied research because first-order asymptotic approximations are often inaccurate and misleading with time-series data and samples of the sizes encountered in applications. We conclude that there is a need for further research in the application of the bootstrap to time series, and we describe some of the important unsolved problems.

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“…See [9] for a survey of classical results in the area. Many natural structural assumptions have been considered in statistics and learning theory, such as monotonicity [10,41,42,50], monotone hazard rate [15,20,46], unimodality [39,69,82], convexity and concavity [43,55], logconcavity [6,37,81], k-modality [7,13,40], smoothness [12,35,36,53], and mixtures of structured distributions [3, 5, 19, 21-26, 31-33, 38, 51, 64, 70, 80]. The reader is referred to [28,65] for a more extensive review of this vast literature.…”

Section: Related Workmentioning

confidence: 99%

Efficiently learning structured distributions from untrusted batches

Chen

Moitra

2020

Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

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We study the problem, introduced by Qiao and Valiant, of learning from untrusted batches. Here, we assume m users, all of whom have samples from some underlying distribution p over 1,. .. ,n. Each user sends a batch of k i.i.d. samples from this distribution; however an ϵ-fraction of users are untrustworthy and can send adversarially chosen responses. The goal of the algorithm is to learn p in total variation distance. When k = 1 this is the standard robust univariate density estimation setting and it is well-understood that Ω(ϵ) error is unavoidable. Suprisingly, Qiao and Valiant gave an estimator which improves upon this rate when k is large. Unfortunately, their algorithms run in time which is exponential in either n or k. We first give a sequence of polynomial time algorithms whose estimation error approaches the information-theoretically optimal bound for this problem. Our approach is based on recent algorithms derived from the sum-of-squares hierarchy, in the context of high-dimensional robust estimation. We show that algorithms for learning from untrusted batches can also be cast in this framework, but by working with a more complicated set of test functions. It turns out that this abstraction is quite powerful, and can be generalized to incorporate additional problem specific constraints. Our second and main result is to show that this technology can be leveraged to build in prior knowledge about the shape of the distribution. Crucially, this allows us to reduce the sample complexity of learning from untrusted batches to polylogarithmic in n for most natural classes of distributions, which is important in many applications. To do so, we demonstrate that these sum-of-squares algorithms for robust mean estimation can be made to handle complex combinatorial constraints (e.g. those arising from VC theory), which may be of independent technical interest.

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Testing for multimodality with dependent data

Cited by 13 publications

References 0 publications

Climate change causing phase transitions of walleye pollock ( Theragra chalcogramma ) recruitment dynamics

Climate change causing phase transitions of walleye pollock ( Theragra chalcogramma ) recruitment dynamics

Bootstrap Methods for Time Series

Efficiently learning structured distributions from untrusted batches

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