Depth Induced Regression Medians and Uniqueness
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 noti...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-04-01
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| Series: | Stats |
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| Online Access: | https://www.mdpi.com/2571-905X/3/2/9 |
| _version_ | 1797571002237976576 |
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| author | Yijun Zuo |
| author_facet | Yijun Zuo |
| author_sort | Yijun Zuo |
| collection | DOAJ |
| description | 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 <inline-formula> <math display="inline"> <semantics> <msub> <mi>D</mi> <mi>C</mi> </msub> </semantics> </math> </inline-formula> 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 <inline-formula> <math display="inline"> <semantics> <msub> <mi>D</mi> <mi>C</mi> </msub> </semantics> </math> </inline-formula> 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 <inline-formula> <math display="inline"> <semantics> <msub> <mi>D</mi> <mi>C</mi> </msub> </semantics> </math> </inline-formula> cases. These and other findings indicate that the PRD and its induced median are highly favorable among their leading competitors. |
| first_indexed | 2024-03-10T20:33:25Z |
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| institution | Directory Open Access Journal |
| issn | 2571-905X |
| language | English |
| last_indexed | 2024-03-10T20:33:25Z |
| publishDate | 2020-04-01 |
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| spelling | doaj.art-e4fd4bec70464ac8a921604782b650b02023-11-19T21:15:35ZengMDPI AGStats2571-905X2020-04-01329410610.3390/stats3020009Depth Induced Regression Medians and UniquenessYijun Zuo0Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, USAThe 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 <inline-formula> <math display="inline"> <semantics> <msub> <mi>D</mi> <mi>C</mi> </msub> </semantics> </math> </inline-formula> 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 <inline-formula> <math display="inline"> <semantics> <msub> <mi>D</mi> <mi>C</mi> </msub> </semantics> </math> </inline-formula> 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 <inline-formula> <math display="inline"> <semantics> <msub> <mi>D</mi> <mi>C</mi> </msub> </semantics> </math> </inline-formula> cases. These and other findings indicate that the PRD and its induced median are highly favorable among their leading competitors.https://www.mdpi.com/2571-905X/3/2/9uniquenessregression depthmaximum depth estimatorregression medianrobustness |
| spellingShingle | Yijun Zuo Depth Induced Regression Medians and Uniqueness Stats uniqueness regression depth maximum depth estimator regression median robustness |
| title | Depth Induced Regression Medians and Uniqueness |
| title_full | Depth Induced Regression Medians and Uniqueness |
| title_fullStr | Depth Induced Regression Medians and Uniqueness |
| title_full_unstemmed | Depth Induced Regression Medians and Uniqueness |
| title_short | Depth Induced Regression Medians and Uniqueness |
| title_sort | depth induced regression medians and uniqueness |
| topic | uniqueness regression depth maximum depth estimator regression median robustness |
| url | https://www.mdpi.com/2571-905X/3/2/9 |
| work_keys_str_mv | AT yijunzuo depthinducedregressionmediansanduniqueness |