ADAPTATION TO LOWEST DENSITY REGIONS WITH APPLICATION TO SUPPORT RECOVERY
A scheme for locally adaptive bandwidth selection is proposed which sensitively shrinks the bandwidth of a kernel estimator at lowest density regions such as the support boundary which are unknown to the statistician. In case of a Holder continuous density, this locally minimax-optimal bandwidth is...
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| Veröffentlicht in: | The Annals of statistics 2016-02, Vol.44 (1), p.255-287 |
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| container_title | The Annals of statistics |
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| creator | Patschkowski, Tim Rohde, Angelika |
| description | A scheme for locally adaptive bandwidth selection is proposed which sensitively shrinks the bandwidth of a kernel estimator at lowest density regions such as the support boundary which are unknown to the statistician. In case of a Holder continuous density, this locally minimax-optimal bandwidth is shown to be smaller than the usual rate, even in case of homogeneous smoothness. Some new type of risk bound with respect to a densitydependent standardized loss of this estimator is established. This bound is fully nonasymptotic and allows to deduce convergence rates at lowest density regions that can be substantially faster than n-1/2. It is complemented by a weighted minimax lower bound which splits into two regimes depending on the value of the density. The new estimator adapts into the second regime, and it is shown that simultaneous adaptation into the fastest regime is not possible in principle as long as the Holder exponent is unknown. Consequences on plug-in rules for support recovery are worked out in detail. In contrast to those with classical density estimators, the plug-in rules based on the new construction are minimax-optimal, up to some logarithmic factor. |
| doi_str_mv | 10.1214/15-AOS1366 |
| format | Article |
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In case of a Holder continuous density, this locally minimax-optimal bandwidth is shown to be smaller than the usual rate, even in case of homogeneous smoothness. Some new type of risk bound with respect to a densitydependent standardized loss of this estimator is established. This bound is fully nonasymptotic and allows to deduce convergence rates at lowest density regions that can be substantially faster than n-1/2. It is complemented by a weighted minimax lower bound which splits into two regimes depending on the value of the density. The new estimator adapts into the second regime, and it is shown that simultaneous adaptation into the fastest regime is not possible in principle as long as the Holder exponent is unknown. Consequences on plug-in rules for support recovery are worked out in detail. In contrast to those with classical density estimators, the plug-in rules based on the new construction are minimax-optimal, up to some logarithmic factor.</description><identifier>ISSN: 0090-5364</identifier><identifier>EISSN: 2168-8966</identifier><identifier>DOI: 10.1214/15-AOS1366</identifier><language>eng</language><publisher>Hayward: Institute of Mathematical Statistics</publisher><subject>62G07 ; Adaptation ; adaptation to lowest density regions ; Anisotropic density estimation ; Asymptotic methods ; bandwidth selection ; Bandwidths ; Decision analysis ; Density ; density dependent minimax optimality ; Density estimation ; Estimating techniques ; Estimation bias ; Estimators ; Mathematical theorems ; Minimax ; Oracles ; Statistical variance ; Studies ; support estimation ; Truncation ; White noise</subject><ispartof>The Annals of statistics, 2016-02, Vol.44 (1), p.255-287</ispartof><rights>Copyright © 2016 Institute of Mathematical Statistics</rights><rights>Copyright Institute of Mathematical Statistics Feb 2016</rights><rights>Copyright 2016 Institute of Mathematical Statistics</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c374t-5c5f4cee25055c5338519f222c874899dc30ee68ff6c18823a9cd3a2d130f86a3</citedby><cites>FETCH-LOGICAL-c374t-5c5f4cee25055c5338519f222c874899dc30ee68ff6c18823a9cd3a2d130f86a3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/43818906$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/43818906$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>230,314,780,784,803,832,885,926,27924,27925,58017,58021,58250,58254</link.rule.ids></links><search><creatorcontrib>Patschkowski, Tim</creatorcontrib><creatorcontrib>Rohde, Angelika</creatorcontrib><title>ADAPTATION TO LOWEST DENSITY REGIONS WITH APPLICATION TO SUPPORT RECOVERY</title><title>The Annals of statistics</title><description>A scheme for locally adaptive bandwidth selection is proposed which sensitively shrinks the bandwidth of a kernel estimator at lowest density regions such as the support boundary which are unknown to the statistician. In case of a Holder continuous density, this locally minimax-optimal bandwidth is shown to be smaller than the usual rate, even in case of homogeneous smoothness. Some new type of risk bound with respect to a densitydependent standardized loss of this estimator is established. This bound is fully nonasymptotic and allows to deduce convergence rates at lowest density regions that can be substantially faster than n-1/2. It is complemented by a weighted minimax lower bound which splits into two regimes depending on the value of the density. The new estimator adapts into the second regime, and it is shown that simultaneous adaptation into the fastest regime is not possible in principle as long as the Holder exponent is unknown. Consequences on plug-in rules for support recovery are worked out in detail. In contrast to those with classical density estimators, the plug-in rules based on the new construction are minimax-optimal, up to some logarithmic factor.</description><subject>62G07</subject><subject>Adaptation</subject><subject>adaptation to lowest density regions</subject><subject>Anisotropic density estimation</subject><subject>Asymptotic methods</subject><subject>bandwidth selection</subject><subject>Bandwidths</subject><subject>Decision analysis</subject><subject>Density</subject><subject>density dependent minimax optimality</subject><subject>Density estimation</subject><subject>Estimating techniques</subject><subject>Estimation bias</subject><subject>Estimators</subject><subject>Mathematical theorems</subject><subject>Minimax</subject><subject>Oracles</subject><subject>Statistical variance</subject><subject>Studies</subject><subject>support estimation</subject><subject>Truncation</subject><subject>White noise</subject><issn>0090-5364</issn><issn>2168-8966</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><recordid>eNo9kM1Lw0AQxRdRsFYv3oWANyG639m9GdLYBkITmq2lpyVsNtBQTc2mB_97Iw05zTDzm_eGB8Ajgq8II_qGmB9mBSKcX4EZRlz4QnJ-DWYQSugzwuktuHOugRAySckMJOEizFWokmztqcxLs11cKG8Rr4tE7b1NvBwWhbdL1MoL8zxNogkttnmebdTARNlnvNnfg5u6PDr7MNY52H7EKlr5abYczlLfkID2PjOspsZazCAbekIEQ7LGGBsRUCFlZQi0lou65gYJgUkpTUVKXCECa8FLMgfvF91T1zbW9PZsjodKn7rDV9n96rY86GibjtOxlK3TiFIZMCY5GSSeJ4mfs3W9btpz9z18rVEgAkg4HIzn4OVCma51rrP15IGg_k9bI6bHtAf46QI3rm-7iaREICEhJ3-MhHR5</recordid><startdate>20160201</startdate><enddate>20160201</enddate><creator>Patschkowski, Tim</creator><creator>Rohde, Angelika</creator><general>Institute of Mathematical Statistics</general><general>The Institute of Mathematical Statistics</general><scope>AAYXX</scope><scope>CITATION</scope><scope>JQ2</scope></search><sort><creationdate>20160201</creationdate><title>ADAPTATION TO LOWEST DENSITY REGIONS WITH APPLICATION TO SUPPORT RECOVERY</title><author>Patschkowski, Tim ; Rohde, Angelika</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c374t-5c5f4cee25055c5338519f222c874899dc30ee68ff6c18823a9cd3a2d130f86a3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2016</creationdate><topic>62G07</topic><topic>Adaptation</topic><topic>adaptation to lowest density regions</topic><topic>Anisotropic density estimation</topic><topic>Asymptotic methods</topic><topic>bandwidth selection</topic><topic>Bandwidths</topic><topic>Decision analysis</topic><topic>Density</topic><topic>density dependent minimax optimality</topic><topic>Density estimation</topic><topic>Estimating techniques</topic><topic>Estimation bias</topic><topic>Estimators</topic><topic>Mathematical theorems</topic><topic>Minimax</topic><topic>Oracles</topic><topic>Statistical variance</topic><topic>Studies</topic><topic>support estimation</topic><topic>Truncation</topic><topic>White noise</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Patschkowski, Tim</creatorcontrib><creatorcontrib>Rohde, Angelika</creatorcontrib><collection>CrossRef</collection><collection>ProQuest Computer Science Collection</collection><jtitle>The Annals of statistics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Patschkowski, Tim</au><au>Rohde, Angelika</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>ADAPTATION TO LOWEST DENSITY REGIONS WITH APPLICATION TO SUPPORT RECOVERY</atitle><jtitle>The Annals of statistics</jtitle><date>2016-02-01</date><risdate>2016</risdate><volume>44</volume><issue>1</issue><spage>255</spage><epage>287</epage><pages>255-287</pages><issn>0090-5364</issn><eissn>2168-8966</eissn><abstract>A scheme for locally adaptive bandwidth selection is proposed which sensitively shrinks the bandwidth of a kernel estimator at lowest density regions such as the support boundary which are unknown to the statistician. In case of a Holder continuous density, this locally minimax-optimal bandwidth is shown to be smaller than the usual rate, even in case of homogeneous smoothness. Some new type of risk bound with respect to a densitydependent standardized loss of this estimator is established. This bound is fully nonasymptotic and allows to deduce convergence rates at lowest density regions that can be substantially faster than n-1/2. It is complemented by a weighted minimax lower bound which splits into two regimes depending on the value of the density. The new estimator adapts into the second regime, and it is shown that simultaneous adaptation into the fastest regime is not possible in principle as long as the Holder exponent is unknown. Consequences on plug-in rules for support recovery are worked out in detail. In contrast to those with classical density estimators, the plug-in rules based on the new construction are minimax-optimal, up to some logarithmic factor.</abstract><cop>Hayward</cop><pub>Institute of Mathematical Statistics</pub><doi>10.1214/15-AOS1366</doi><tpages>33</tpages><oa>free_for_read</oa></addata></record> |
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| ispartof | The Annals of statistics, 2016-02, Vol.44 (1), p.255-287 |
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| language | eng |
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| source | JSTOR Mathematics & Statistics; JSTOR Archive Collection A-Z Listing; EZB-FREE-00999 freely available EZB journals; Project Euclid Complete |
| subjects | 62G07 Adaptation adaptation to lowest density regions Anisotropic density estimation Asymptotic methods bandwidth selection Bandwidths Decision analysis Density density dependent minimax optimality Density estimation Estimating techniques Estimation bias Estimators Mathematical theorems Minimax Oracles Statistical variance Studies support estimation Truncation White noise |
| title | ADAPTATION TO LOWEST DENSITY REGIONS WITH APPLICATION TO SUPPORT RECOVERY |
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