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Nonparametric Estimation of a Mixing Distribution with Applications.

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Nonparametric Estimation of a Mixing Distribution with Applications./
作者:	Dixit, Vaidehi Ulhas.
面頁冊數:	1 online resource (103 pages)
附註:	Source: Dissertations Abstracts International, Volume: 84-01, Section: B.
Contained By:	Dissertations Abstracts International84-01B.
標題:	Probability. -
電子資源:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=29176826click for full text (PQDT)
ISBN:	9798835547043

Nonparametric Estimation of a Mixing Distribution with Applications.
Dixit, Vaidehi Ulhas.

Nonparametric Estimation of a Mixing Distribution with Applications. - 1 online resource (103 pages)

Source: Dissertations Abstracts International, Volume: 84-01, Section: B.

Thesis (Ph.D.)--North Carolina State University, 2022.

Includes bibliographical references

DIXIT, VAIDEHI ULHAS. Nonparametric Estimation of aMixing Distribution with Applications. (Under the direction of Ryan Martin).Problems where a signal of interest is corrupted by noise are ubiquitous in science, business, and engineering applications. They are also quite challenging, so these have attracted considerable attention from both the statistics and applied mathematics communities. Mixture models are commonly used to formulate these problems in a statistical setting, where the unobservable signal of interest is characterized by an unknown mixing distribution. Then the separate-signal-from-noise problem amounts to estimating that mixing distribution based on data observed from the corresponding mixture. This problem is especially challenging because it boils down to estimation of a distribution based on only noise-corrupted samples from it.Likelihood-based methods tend to focus-either by choice or by certain mathematical consequences-on mixing distributions that are discrete. This includes pure likelihood and Bayesian approaches. In addition to various computational challenges, this tendency towards finite/discrete mixing distributions implies that likelihood-based methods do not provide a fully satisfactory solution to the problem of separating signal from noise. Therefore, other methods, which are not likelihood-based, deserve consideration. One such alternative is the so-called predictive recursion (PR) algorithm that provides a computationally efficient and genuinely nonparametric mixing distribution estimate, first introduced in 1998. An important feature of PR is that it is capable of producing a mixing distribution estimate having a smooth density function. While the PR algorithm itself is not new, its unconventional form implies that there are still a number of unanswered questions and opportunities for new developments. This dissertation explores some of these. With nonparametric estimation of mixing distributions via the PR algorithm as the primary focus, this dissertation makes the following three contributions. First, we explore an application of nonparametric estimation of a mixing distribution supported on the unit sphere, which is relevant for modeling directional data, as often encountered in meteorology, material science, and neuroscience. Estimating a mixing distribution on a sphere has not been explored in the literature and implementation of the PR algorithm in multivariate settings is also largely unexplored. We show that PR gives a fast, robust estimate of the mixing distribution on a sphere. Additionally, this estimation technique can be used for comparing simple parametric models for directional data to more complex nonparametric mixtures.Second, the currently available consistency results for PR make rather strong assumptions about certain aspects of the mixture model. So strong that they are not satisfied in certain practically relevant mixture models. We prove new consistency results that are based on weaker assumptions, thus expanding the scope of PR. In particular, these new developments make PR applicable in the important application of monotone density estimation. The resulting PR estimator is also shown to have strong finite-sample performance compared to other standard methods in the literature. Third, there is a major computational bottleneck to the application of the PR algorithm in cases where the underlying signal is multivariate. That is, each iteration of the PR algorithm requires evaluation of a normalizing constant, an integral, and in problems where the mixing distribution support is of three or more dimensions, numerical integration or quadrature methods become inefficient/infeasible.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2023

Mode of access: World Wide Web

ISBN: 9798835547043Subjects--Topical Terms:

518898
Probability.
Index Terms--Genre/Form:

542853
Electronic books.

Nonparametric Estimation of a Mixing Distribution with Applications.
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502 $a Thesis (Ph.D.)--North Carolina State University, 2022.
504 $a Includes bibliographical references
520 $a DIXIT, VAIDEHI ULHAS. Nonparametric Estimation of aMixing Distribution with Applications. (Under the direction of Ryan Martin).Problems where a signal of interest is corrupted by noise are ubiquitous in science, business, and engineering applications. They are also quite challenging, so these have attracted considerable attention from both the statistics and applied mathematics communities. Mixture models are commonly used to formulate these problems in a statistical setting, where the unobservable signal of interest is characterized by an unknown mixing distribution. Then the separate-signal-from-noise problem amounts to estimating that mixing distribution based on data observed from the corresponding mixture. This problem is especially challenging because it boils down to estimation of a distribution based on only noise-corrupted samples from it.Likelihood-based methods tend to focus-either by choice or by certain mathematical consequences-on mixing distributions that are discrete. This includes pure likelihood and Bayesian approaches. In addition to various computational challenges, this tendency towards finite/discrete mixing distributions implies that likelihood-based methods do not provide a fully satisfactory solution to the problem of separating signal from noise. Therefore, other methods, which are not likelihood-based, deserve consideration. One such alternative is the so-called predictive recursion (PR) algorithm that provides a computationally efficient and genuinely nonparametric mixing distribution estimate, first introduced in 1998. An important feature of PR is that it is capable of producing a mixing distribution estimate having a smooth density function. While the PR algorithm itself is not new, its unconventional form implies that there are still a number of unanswered questions and opportunities for new developments. This dissertation explores some of these. With nonparametric estimation of mixing distributions via the PR algorithm as the primary focus, this dissertation makes the following three contributions. First, we explore an application of nonparametric estimation of a mixing distribution supported on the unit sphere, which is relevant for modeling directional data, as often encountered in meteorology, material science, and neuroscience. Estimating a mixing distribution on a sphere has not been explored in the literature and implementation of the PR algorithm in multivariate settings is also largely unexplored. We show that PR gives a fast, robust estimate of the mixing distribution on a sphere. Additionally, this estimation technique can be used for comparing simple parametric models for directional data to more complex nonparametric mixtures.Second, the currently available consistency results for PR make rather strong assumptions about certain aspects of the mixture model. So strong that they are not satisfied in certain practically relevant mixture models. We prove new consistency results that are based on weaker assumptions, thus expanding the scope of PR. In particular, these new developments make PR applicable in the important application of monotone density estimation. The resulting PR estimator is also shown to have strong finite-sample performance compared to other standard methods in the literature. Third, there is a major computational bottleneck to the application of the PR algorithm in cases where the underlying signal is multivariate. That is, each iteration of the PR algorithm requires evaluation of a normalizing constant, an integral, and in problems where the mixing distribution support is of three or more dimensions, numerical integration or quadrature methods become inefficient/infeasible.
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