RANDOMIZED URN MODELS REVISITED USING STOCHASTIC APPROXIMATION
This paper presents the link between stochastic approximation and clinical trials based on randomized urn models investigated by Bai and Hu [Stochastic Process. Appl. 80 (1999) 87–101], Bai and Hu [Ann. Appl. Probab. 15 (2005) 914–940] and Bai, Hu and Shen [J. Multivariate Anal. 81 (2002) 1–18]. We...
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| Veröffentlicht in: | The Annals of applied probability 2013-08, Vol.23 (4), p.1409-1436 |
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| creator | Laruelle, Sophie Pagès, Gilles |
| description | This paper presents the link between stochastic approximation and clinical trials based on randomized urn models investigated by Bai and Hu [Stochastic Process. Appl. 80 (1999) 87–101], Bai and Hu [Ann. Appl. Probab. 15 (2005) 914–940] and Bai, Hu and Shen [J. Multivariate Anal. 81 (2002) 1–18]. We reformulate the dynamics of both the urn composition and the assigned treatments as standard stochastic approximation (SA) algorithms with remainder. Then, we derive the a.s. convergence and the asymptotic normality [central limit theorem (CLT)] of the normalized procedure under less stringent assumptions by calling upon the ODE and SDE methods. As a second step, we investigate a more involved family of models, known as multi-arm clinical trials, where the urn updating depends on the past performances of the treatments. By increasing the dimension of the state vector, our SA approach provides this time a new asymptotic normality result. |
| doi_str_mv | 10.1214/12-AAP875 |
| format | Article |
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Appl. 80 (1999) 87–101], Bai and Hu [Ann. Appl. Probab. 15 (2005) 914–940] and Bai, Hu and Shen [J. Multivariate Anal. 81 (2002) 1–18]. We reformulate the dynamics of both the urn composition and the assigned treatments as standard stochastic approximation (SA) algorithms with remainder. Then, we derive the a.s. convergence and the asymptotic normality [central limit theorem (CLT)] of the normalized procedure under less stringent assumptions by calling upon the ODE and SDE methods. As a second step, we investigate a more involved family of models, known as multi-arm clinical trials, where the urn updating depends on the past performances of the treatments. By increasing the dimension of the state vector, our SA approach provides this time a new asymptotic normality result.</description><subject>62E20</subject><subject>62F12</subject><subject>62L05</subject><subject>62L20</subject><subject>62P10</subject><subject>adaptive asset allocation</subject><subject>Approximation</subject><subject>Asymptotic methods</subject><subject>asymptotic normality</subject><subject>Central limit theorem</subject><subject>Clinical trials</subject><subject>Eigenvalues</subject><subject>extended Pólya urn models</subject><subject>Martingales</subject><subject>Mathematical theorems</subject><subject>Mathematics</subject><subject>Matrices</subject><subject>multi-arm clinical trials</subject><subject>nonhomogeneous generating matrix</subject><subject>Odes</subject><subject>Perceptron convergence procedure</subject><subject>Probability</subject><subject>Randomized algorithms</subject><subject>Stochastic approximation</subject><subject>Stochastic models</subject><subject>strong consistency</subject><subject>Theorems</subject><subject>Vector space</subject><issn>1050-5164</issn><issn>2168-8737</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2013</creationdate><recordtype>article</recordtype><recordid>eNpVkU1Lw0AQhhdRsH4c_AFCwJOH6M5-ZHcvwpLGNpAmJUlFvIRNmmBDNTVpBf-9KSkVLzPw8s4zzLwI3QB-AALsEYit9VwKfoJGBBxpS0HFKRoB5tjm4LBzdNF1NcZYMSVG6CnW4Tia-W_e2FrEoTWLxl6QWLH34id-uhcTP5xYSRq5U52kvmvp-TyOXv2ZTv0ovEJnlVl35fWhX6LFs5e6UzuIJr6rA7ugArb2klaUidxIQ5xCGlOpnHJgjilMrhjHSoCSZQWCMJAMWJmDUU7JCsny3CFLeon0wN20TV0W23JXrFfLbNOuPkz7kzVmlbmL4KAemmnMJoN-v6QMU9oz7gfGu1n_m5zqINtrGHPOBSffTu-9O-772pXdNqubXfvZn9gTlQLFKMF_xKJtuq4tqyMWcLaPoy_ZEEfvvR28dbdt2qOR9H9hEnP6C5wUfxg</recordid><startdate>20130801</startdate><enddate>20130801</enddate><creator>Laruelle, Sophie</creator><creator>Pagès, Gilles</creator><general>Institute of Mathematical Statistics</general><general>Institute of Mathematical Statistics (IMS)</general><general>The Institute of Mathematical Statistics</general><scope>AAYXX</scope><scope>CITATION</scope><scope>JQ2</scope><scope>1XC</scope><scope>VOOES</scope><orcidid>https://orcid.org/0000-0001-6487-3079</orcidid></search><sort><creationdate>20130801</creationdate><title>RANDOMIZED URN MODELS REVISITED USING STOCHASTIC APPROXIMATION</title><author>Laruelle, Sophie ; Pagès, Gilles</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c371t-d3f347ba8a26c8aaf9b35146acab945097198ef172418414eb1a96e4c84bb62d3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2013</creationdate><topic>62E20</topic><topic>62F12</topic><topic>62L05</topic><topic>62L20</topic><topic>62P10</topic><topic>adaptive asset allocation</topic><topic>Approximation</topic><topic>Asymptotic methods</topic><topic>asymptotic normality</topic><topic>Central limit theorem</topic><topic>Clinical trials</topic><topic>Eigenvalues</topic><topic>extended Pólya urn models</topic><topic>Martingales</topic><topic>Mathematical theorems</topic><topic>Mathematics</topic><topic>Matrices</topic><topic>multi-arm clinical trials</topic><topic>nonhomogeneous generating matrix</topic><topic>Odes</topic><topic>Perceptron convergence procedure</topic><topic>Probability</topic><topic>Randomized algorithms</topic><topic>Stochastic approximation</topic><topic>Stochastic models</topic><topic>strong consistency</topic><topic>Theorems</topic><topic>Vector space</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Laruelle, Sophie</creatorcontrib><creatorcontrib>Pagès, Gilles</creatorcontrib><collection>CrossRef</collection><collection>ProQuest Computer Science Collection</collection><collection>Hyper Article en Ligne (HAL)</collection><collection>Hyper Article en Ligne (HAL) (Open Access)</collection><jtitle>The Annals of applied probability</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Laruelle, Sophie</au><au>Pagès, Gilles</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>RANDOMIZED URN MODELS REVISITED USING STOCHASTIC APPROXIMATION</atitle><jtitle>The Annals of applied probability</jtitle><date>2013-08-01</date><risdate>2013</risdate><volume>23</volume><issue>4</issue><spage>1409</spage><epage>1436</epage><pages>1409-1436</pages><issn>1050-5164</issn><eissn>2168-8737</eissn><abstract>This paper presents the link between stochastic approximation and clinical trials based on randomized urn models investigated by Bai and Hu [Stochastic Process. Appl. 80 (1999) 87–101], Bai and Hu [Ann. Appl. Probab. 15 (2005) 914–940] and Bai, Hu and Shen [J. Multivariate Anal. 81 (2002) 1–18]. We reformulate the dynamics of both the urn composition and the assigned treatments as standard stochastic approximation (SA) algorithms with remainder. Then, we derive the a.s. convergence and the asymptotic normality [central limit theorem (CLT)] of the normalized procedure under less stringent assumptions by calling upon the ODE and SDE methods. As a second step, we investigate a more involved family of models, known as multi-arm clinical trials, where the urn updating depends on the past performances of the treatments. By increasing the dimension of the state vector, our SA approach provides this time a new asymptotic normality result.</abstract><cop>Hayward</cop><pub>Institute of Mathematical Statistics</pub><doi>10.1214/12-AAP875</doi><tpages>28</tpages><orcidid>https://orcid.org/0000-0001-6487-3079</orcidid><oa>free_for_read</oa></addata></record> |
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| subjects | 62E20 62F12 62L05 62L20 62P10 adaptive asset allocation Approximation Asymptotic methods asymptotic normality Central limit theorem Clinical trials Eigenvalues extended Pólya urn models Martingales Mathematical theorems Mathematics Matrices multi-arm clinical trials nonhomogeneous generating matrix Odes Perceptron convergence procedure Probability Randomized algorithms Stochastic approximation Stochastic models strong consistency Theorems Vector space |
| title | RANDOMIZED URN MODELS REVISITED USING STOCHASTIC APPROXIMATION |
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