Existence and uniqueness of weighted generalized $ψ$-estimators
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866918140053553152 |
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| author | Barczy, Matyas Páles, Zsolt |
| author_facet | Barczy, Matyas Páles, Zsolt |
| contents | We introduce the notions of generalized and weighted generalized $ψ$-estimators as unique points of sign change of some appropriate functions, and we give necessary as well as sufficient conditions for their existence. We also derive a set of sufficient conditions under which the so-called $ψ$-expectation function has a unique point of sign change. We present several examples from statistical estimation theory, where our results are well-applicable. For example, we consider the cases of empirical quantiles, empirical expectiles, some $ψ$-estimators that are important in robust statistics, and some examples from maximum likelihood theory as well. Further, we introduce Bajraktarević-type (in particular, quasi-arithmetic-type) $ψ$-estimators. Our results specialized to $ψ$-estimators with a function $ψ$ being continuous in its second variable provide new results for (usual) $ψ$-estimators (also called Z-estimators). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_06026 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Existence and uniqueness of weighted generalized $ψ$-estimators Barczy, Matyas Páles, Zsolt Statistics Theory Classical Analysis and ODEs 62F10, 62F35, 26E60, 26A48 We introduce the notions of generalized and weighted generalized $ψ$-estimators as unique points of sign change of some appropriate functions, and we give necessary as well as sufficient conditions for their existence. We also derive a set of sufficient conditions under which the so-called $ψ$-expectation function has a unique point of sign change. We present several examples from statistical estimation theory, where our results are well-applicable. For example, we consider the cases of empirical quantiles, empirical expectiles, some $ψ$-estimators that are important in robust statistics, and some examples from maximum likelihood theory as well. Further, we introduce Bajraktarević-type (in particular, quasi-arithmetic-type) $ψ$-estimators. Our results specialized to $ψ$-estimators with a function $ψ$ being continuous in its second variable provide new results for (usual) $ψ$-estimators (also called Z-estimators). |
| title | Existence and uniqueness of weighted generalized $ψ$-estimators |
| topic | Statistics Theory Classical Analysis and ODEs 62F10, 62F35, 26E60, 26A48 |
| url | https://arxiv.org/abs/2211.06026 |