scholarly journals Depth Induced Regression Medians and Uniqueness

Stats ◽  
2020 ◽  
Vol 3 (2) ◽  
pp. 94-106
Author(s):  
Yijun Zuo
Keyword(s):  
Least Squares ◽  
Nonparametric Statistics ◽  
Data Depth ◽  
Limiting Distribution ◽  
Maximum Depth ◽  
Least Squares Estimator ◽  
One Dimension ◽  
Uniqueness Property ◽  
Regression Depth

The notion of median in one dimension is a foundational element in nonparametric statistics. It has been extended to multi-dimensional cases both in location and in regression via notions of data depth. Regression depth (RD) and projection regression depth (PRD) represent the two most promising notions in regression. Carrizosa depth D C is another depth notion in regression. Depth-induced regression medians (maximum depth estimators) serve as robust alternatives to the classical least squares estimator. The uniqueness of regression medians is indispensable in the discussion of their properties and the asymptotics (consistency and limiting distribution) of sample regression medians. Are the regression medians induced from RD, PRD, and D C unique? Answering this question is the main goal of this article. It is found that only the regression median induced from PRD possesses the desired uniqueness property. The conventional remedy measure for non-uniqueness, taking average of all medians, might yield an estimator that no longer possesses the maximum depth in both RD and D C cases. These and other findings indicate that the PRD and its induced median are highly favorable among their leading competitors.

scholarly journals Regression Medians and Uniqueness

2020 ◽  
Author(s):  
Yijun Zuo
Keyword(s):  
Least Squares ◽  
Nonparametric Statistics ◽  
Data Depth ◽  
Limiting Distribution ◽  
Maximum Depth ◽  
Least Squares Estimator ◽  
One Dimension ◽  
Uniqueness Property ◽  
Regression Depth

Notion of median in one dimension is a foundational element in nonparametric statistics. It has been extended to multi-dimensional cases both in location and in regression via notions of data depth. Regression depth (RD) and projection regression depth (PRD) represent the two most promising notions in regression. Carrizosa depth DC is another depth notion in regression. Depth induced regression medians (maximum depth estimators) serve as robust alternatives to the classical least squares estimator. The uniqueness of regression medians is indispensable in the discussion of their properties and the asymptotics (consistency and limiting distribution) of sample regression medians. Are the regression medians induced from RD, PRD, and DC unique? Answering this question is the main goal of this article. It is found that only the regression median induced from PRD possesses the desired uniqueness property. The conventional remedy measure for non-uniqueness, taking average of all medians, might yield an estimator that no longer possesses the maximum depth in both RD and DC cases. These and other findings indicate that the PRD and its induced median are highly favorable among their leading competitors.

scholarly journals Asymptotic inference for nearly unstable INAR(1) models

2003 ◽  
Vol 40 (3) ◽  
pp. 750-765 ◽  
Author(s):  
M. Ispány ◽  
G. Pap ◽  
M. C. A. van Zuijlen
Keyword(s):  
Least Squares ◽  
Rate Of Convergence ◽  
Limiting Distribution ◽  
Least Squares Estimator ◽  
Asymptotic Inference ◽  
First Order ◽  
Conditional Least Squares ◽  
Conditional Least Squares Estimator

A sequence of first-order integer-valued autoregressive (INAR(1)) processes is investigated, where the autoregressive-type coefficient converges to 1. It is shown that the limiting distribution of the conditional least squares estimator for this coefficient is normal and the rate of convergence is n3/2. Nearly critical Galton–Watson processes with unobservable immigration are also discussed.

The limiting distribution of the least-squares estimator in nearly integrated seasonal models

1992 ◽  
Vol 20 (2) ◽  
pp. 121-134 ◽  
Author(s):  
Pierre Perron
Keyword(s):  
Least Squares ◽  
Limiting Distribution ◽  
Least Squares Estimator

scholarly journals Asymptotic inference for nearly unstable INAR(1) models

2003 ◽  
Vol 40 (03) ◽  
pp. 750-765 ◽  
Author(s):  
M. Ispány ◽  
G. Pap ◽  
M. C. A. van Zuijlen
Keyword(s):  
Least Squares ◽  
Rate Of Convergence ◽  
Limiting Distribution ◽  
Least Squares Estimator ◽  
Asymptotic Inference ◽  
First Order ◽  
Conditional Least Squares ◽  
Conditional Least Squares Estimator

A sequence of first-order integer-valued autoregressive (INAR(1)) processes is investigated, where the autoregressive-type coefficient converges to 1. It is shown that the limiting distribution of the conditional least squares estimator for this coefficient is normal and the rate of convergence is n 3/2. Nearly critical Galton–Watson processes with unobservable immigration are also discussed.

scholarly journals An overview of 38 least squares–based frameworks for structural damage tomography

2019 ◽  
Vol 19 (1) ◽  
pp. 215-239 ◽  
Author(s):  
Danny Smyl ◽  
Sven Bossuyt ◽  
Waqas Ahmad ◽  
Anton Vavilov ◽  
Dong Liu
Keyword(s):  
Structural Health Monitoring ◽  
Least Squares ◽  
Health Monitoring ◽  
Structural Damage ◽  
Tomographic Imaging ◽  
Inverse Methods ◽  
One Dimension ◽  
Distributed Data ◽  
Structural Health ◽  
Static Elasticity

The ability to reliably detect damage and intercept deleterious processes, such as cracking, corrosion, and plasticity are central themes in structural health monitoring. The importance of detecting such processes early on lies in the realization that delays may decrease safety, increase long-term repair/retrofit costs, and degrade the overall user experience of civil infrastructure. Since real structures exist in more than one dimension, the detection of distributed damage processes also generally requires input data from more than one dimension. Often, however, interpretation of distributed data—alone—offers insufficient information. For this reason, engineers and researchers have become interested in stationary inverse methods, for example, utilizing distributed data from stationary or quasi-stationary measurements for tomographic imaging structures. Presently, however, there are barriers in implementing stationary inverse methods at the scale of built civil structures. Of these barriers, a lack of available straightforward inverse algorithms is at the forefront. To address this, we provide 38 least-squares frameworks encompassing single-state, two-state, and joint tomographic imaging of structural damage. These regimes are then applied to two emerging structural health monitoring imaging modalities: Electrical Resistance Tomography and Quasi-Static Elasticity Imaging. The feasibility of the regimes are then demonstrated using simulated and experimental data.

Consistency of the least squares estimator of an infinite-dimensional parameter

1992 ◽  
Vol 33 (4) ◽  
pp. 603-607
Author(s):  
A. Ya. Dorogovtsev
Keyword(s):  
Least Squares ◽  
Least Squares Estimator ◽  
Dimensional Parameter ◽  
Infinite Dimensional
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