Stein's Method and Birth-Death Processes
Barbour introduced a probabilistic view of Stein's method for estimating the error in probability approximations. However, in the case of approximations by general distributions on the integers, there have been no purely probabilistic proofs of Stein bounds till this paper. Furthermore, the met...
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| Veröffentlicht in: | The Annals of probability 2001-07, Vol.29 (3), p.1373-1403 |
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| container_title | The Annals of probability |
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| creator | Brown, Timothy C. Xia, Aihua |
| description | Barbour introduced a probabilistic view of Stein's method for estimating the error in probability approximations. However, in the case of approximations by general distributions on the integers, there have been no purely probabilistic proofs of Stein bounds till this paper. Furthermore, the methods introduced here apply to a very large class of approximating distributions on the non-negative integers, among which there is a natural class for higher-order approximations by probability distributions rather than signed measures (as previously). The methods also produce Stein magic factors for process approximations which do not increase with the window of observation and which are simpler to apply than those in Brown, Weinberg and Xia. |
| doi_str_mv | 10.1214/aop/1015345606 |
| format | Article |
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However, in the case of approximations by general distributions on the integers, there have been no purely probabilistic proofs of Stein bounds till this paper. Furthermore, the methods introduced here apply to a very large class of approximating distributions on the non-negative integers, among which there is a natural class for higher-order approximations by probability distributions rather than signed measures (as previously). The methods also produce Stein magic factors for process approximations which do not increase with the window of observation and which are simpler to apply than those in Brown, Weinberg and Xia.</description><identifier>ISSN: 0091-1798</identifier><identifier>EISSN: 2168-894X</identifier><identifier>DOI: 10.1214/aop/1015345606</identifier><identifier>CODEN: APBYAE</identifier><language>eng</language><publisher>Hayward, CA: Institute of Mathematical Statistics</publisher><subject>60E05 ; 60E15 ; 60F05 ; 60G55 ; Approximation ; Binomial distributions ; Binomials ; Birth rates ; birth-death process ; compound Poisson distribution ; Distribution theory ; distributional approximation ; Exact sciences and technology ; Integers ; Limit theorems ; Markov processes ; Mathematics ; Mortality ; negative binomial distribution ; Poisson process ; Poisson process approximation ; polynomial birth-death distribution ; Polynomials ; Probability and statistics ; Probability theory and stochastic processes ; Random variables ; Sciences and techniques of general use ; Stein's method ; Stochastic processes ; total variation distance ; Wasserstein distance</subject><ispartof>The Annals of probability, 2001-07, Vol.29 (3), p.1373-1403</ispartof><rights>Copyright 2001 Institute of Mathematical Statistics</rights><rights>2002 INIST-CNRS</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c385t-573a9dc627d585fb916e8af57a9d035a98b14351058d5895449b51cd965183e83</citedby><cites>FETCH-LOGICAL-c385t-573a9dc627d585fb916e8af57a9d035a98b14351058d5895449b51cd965183e83</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/2692035$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/2692035$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>230,314,776,780,799,828,881,921,27903,27904,57996,58000,58229,58233</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=13438116$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Brown, Timothy C.</creatorcontrib><creatorcontrib>Xia, Aihua</creatorcontrib><title>Stein's Method and Birth-Death Processes</title><title>The Annals of probability</title><description>Barbour introduced a probabilistic view of Stein's method for estimating the error in probability approximations. However, in the case of approximations by general distributions on the integers, there have been no purely probabilistic proofs of Stein bounds till this paper. Furthermore, the methods introduced here apply to a very large class of approximating distributions on the non-negative integers, among which there is a natural class for higher-order approximations by probability distributions rather than signed measures (as previously). The methods also produce Stein magic factors for process approximations which do not increase with the window of observation and which are simpler to apply than those in Brown, Weinberg and Xia.</description><subject>60E05</subject><subject>60E15</subject><subject>60F05</subject><subject>60G55</subject><subject>Approximation</subject><subject>Binomial distributions</subject><subject>Binomials</subject><subject>Birth rates</subject><subject>birth-death process</subject><subject>compound Poisson distribution</subject><subject>Distribution theory</subject><subject>distributional approximation</subject><subject>Exact sciences and technology</subject><subject>Integers</subject><subject>Limit theorems</subject><subject>Markov processes</subject><subject>Mathematics</subject><subject>Mortality</subject><subject>negative binomial distribution</subject><subject>Poisson process</subject><subject>Poisson process approximation</subject><subject>polynomial birth-death distribution</subject><subject>Polynomials</subject><subject>Probability and statistics</subject><subject>Probability theory and stochastic processes</subject><subject>Random variables</subject><subject>Sciences and techniques of general use</subject><subject>Stein's method</subject><subject>Stochastic processes</subject><subject>total variation distance</subject><subject>Wasserstein distance</subject><issn>0091-1798</issn><issn>2168-894X</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2001</creationdate><recordtype>article</recordtype><recordid>eNplUD1PwzAUtBBIlMLKxJAFwZLWL_Zz7A0In1IQSFCJLXIdR01V6so2A_-eVInagemke3f3dEfIOdAJZMCn2m2mQAEZR0HFARllIGQqFf86JCNKFaSQK3lMTkJYUkpFnvMRuf6Itl1fheTVxoWrE72uk7vWx0V6b3VcJO_eGRuCDafkqNGrYM8GHJPZ48Nn8ZyWb08vxW2ZGiYxppgzrWojsrxGic1cgbBSN5h3LGWolZwDZwgUZSdQyLmaI5haCQTJrGRjctPnbrxbWhPtj1m1dbXx7bf2v5XTbVXMyoEdoGte7Zt3EZM-wngXgrfNzg202k7133A5_NTB6FXj9dq0Ye9inEmAre6i1y1DdH53z4TKum7sDyO5cNQ</recordid><startdate>20010701</startdate><enddate>20010701</enddate><creator>Brown, Timothy C.</creator><creator>Xia, Aihua</creator><general>Institute of Mathematical Statistics</general><general>The Institute of Mathematical Statistics</general><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20010701</creationdate><title>Stein's Method and Birth-Death Processes</title><author>Brown, Timothy C. ; Xia, Aihua</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c385t-573a9dc627d585fb916e8af57a9d035a98b14351058d5895449b51cd965183e83</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2001</creationdate><topic>60E05</topic><topic>60E15</topic><topic>60F05</topic><topic>60G55</topic><topic>Approximation</topic><topic>Binomial distributions</topic><topic>Binomials</topic><topic>Birth rates</topic><topic>birth-death process</topic><topic>compound Poisson distribution</topic><topic>Distribution theory</topic><topic>distributional approximation</topic><topic>Exact sciences and technology</topic><topic>Integers</topic><topic>Limit theorems</topic><topic>Markov processes</topic><topic>Mathematics</topic><topic>Mortality</topic><topic>negative binomial distribution</topic><topic>Poisson process</topic><topic>Poisson process approximation</topic><topic>polynomial birth-death distribution</topic><topic>Polynomials</topic><topic>Probability and statistics</topic><topic>Probability theory and stochastic processes</topic><topic>Random variables</topic><topic>Sciences and techniques of general use</topic><topic>Stein's method</topic><topic>Stochastic processes</topic><topic>total variation distance</topic><topic>Wasserstein distance</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Brown, Timothy C.</creatorcontrib><creatorcontrib>Xia, Aihua</creatorcontrib><collection>Pascal-Francis</collection><collection>CrossRef</collection><jtitle>The Annals of probability</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Brown, Timothy C.</au><au>Xia, Aihua</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Stein's Method and Birth-Death Processes</atitle><jtitle>The Annals of probability</jtitle><date>2001-07-01</date><risdate>2001</risdate><volume>29</volume><issue>3</issue><spage>1373</spage><epage>1403</epage><pages>1373-1403</pages><issn>0091-1798</issn><eissn>2168-894X</eissn><coden>APBYAE</coden><abstract>Barbour introduced a probabilistic view of Stein's method for estimating the error in probability approximations. However, in the case of approximations by general distributions on the integers, there have been no purely probabilistic proofs of Stein bounds till this paper. Furthermore, the methods introduced here apply to a very large class of approximating distributions on the non-negative integers, among which there is a natural class for higher-order approximations by probability distributions rather than signed measures (as previously). The methods also produce Stein magic factors for process approximations which do not increase with the window of observation and which are simpler to apply than those in Brown, Weinberg and Xia.</abstract><cop>Hayward, CA</cop><pub>Institute of Mathematical Statistics</pub><doi>10.1214/aop/1015345606</doi><tpages>31</tpages><oa>free_for_read</oa></addata></record> |
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| ispartof | The Annals of probability, 2001-07, Vol.29 (3), p.1373-1403 |
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| source | JSTOR Mathematics & Statistics; Jstor Complete Legacy; EZB-FREE-00999 freely available EZB journals; Project Euclid Complete |
| subjects | 60E05 60E15 60F05 60G55 Approximation Binomial distributions Binomials Birth rates birth-death process compound Poisson distribution Distribution theory distributional approximation Exact sciences and technology Integers Limit theorems Markov processes Mathematics Mortality negative binomial distribution Poisson process Poisson process approximation polynomial birth-death distribution Polynomials Probability and statistics Probability theory and stochastic processes Random variables Sciences and techniques of general use Stein's method Stochastic processes total variation distance Wasserstein distance |
| title | Stein's Method and Birth-Death Processes |
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