We consider the process dYt = ut dt + dWt , where u is a processnot necessarily adapted to F Y (the filtration generated by the process Y)and W is a Brownian motion. We obtain a general representation for thelikelihood ratio of the law of the Y process relative to Brownian measure.This representation involves only one basic filter (expectation of u conditionalon observed process Y). This generalizes the result of Kailath and Zakai[Ann.Math. Statist. 42 (1971) 130â"140] where it is assumed that the process uis adapted to F Y . In particular, we consider the model in which u is afunctional of Y and of a random element X which is independent of theBrownian motion W. For example, X could be a diffusion or a Markov chain.This result can be applied to the estimation of an unknown multidimensionalparameter I¸ appearing in the dynamics of the process u based on continuousobservation of Y on the time interval [0,T ]. For a specific hidden diffusionfinancial model in which u is an unobserved mean-reverting diffusion, wegive an explicit form for the likelihood function of I¸. For this model we alsodevelop a computationally explicit Eâ"M algorithm for the estimation of I¸. Incontrast to the likelihood ratio, the algorithm involves evaluation of a numberof filtered integrals in addition to the basic filter.
{"title":"Maximum Likelihood Estimation of Hidden Markov Processes","authors":"H. Frydman, P. Lakner","doi":"10.1214/AOAP/1069786500","DOIUrl":"https://doi.org/10.1214/AOAP/1069786500","url":null,"abstract":"We consider the process dYt = ut dt + dWt , where u is a processnot necessarily adapted to F Y (the filtration generated by the process Y)and W is a Brownian motion. We obtain a general representation for thelikelihood ratio of the law of the Y process relative to Brownian measure.This representation involves only one basic filter (expectation of u conditionalon observed process Y). This generalizes the result of Kailath and Zakai[Ann.Math. Statist. 42 (1971) 130â\"140] where it is assumed that the process uis adapted to F Y . In particular, we consider the model in which u is afunctional of Y and of a random element X which is independent of theBrownian motion W. For example, X could be a diffusion or a Markov chain.This result can be applied to the estimation of an unknown multidimensionalparameter I¸ appearing in the dynamics of the process u based on continuousobservation of Y on the time interval [0,T ]. For a specific hidden diffusionfinancial model in which u is an unobserved mean-reverting diffusion, wegive an explicit form for the likelihood function of I¸. For this model we alsodevelop a computationally explicit Eâ\"M algorithm for the estimation of I¸. Incontrast to the likelihood ratio, the algorithm involves evaluation of a numberof filtered integrals in addition to the basic filter.","PeriodicalId":309676,"journal":{"name":"NYU: IOMS: Statistics Working Papers (Topic)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126669658","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1999-03-01DOI: 10.1080/10618600.1999.10474799
J. Simonoff, Chih-Ling Tsai
Abstract An improved AIC-based criterion is derived for model selection in general smoothing-based modeling, including semiparametric models and additive models. Examples are provided of applications to goodness-of-fit, smoothing parameter and variable selection in an additive model and semiparametric models, and variable selection in a model with a nonlinear function of linear terms.
{"title":"Semiparametric and Additive Model Selection Using an Improved Akaike Information Criterion","authors":"J. Simonoff, Chih-Ling Tsai","doi":"10.1080/10618600.1999.10474799","DOIUrl":"https://doi.org/10.1080/10618600.1999.10474799","url":null,"abstract":"Abstract An improved AIC-based criterion is derived for model selection in general smoothing-based modeling, including semiparametric models and additive models. Examples are provided of applications to goodness-of-fit, smoothing parameter and variable selection in an additive model and semiparametric models, and variable selection in a model with a nonlinear function of linear terms.","PeriodicalId":309676,"journal":{"name":"NYU: IOMS: Statistics Working Papers (Topic)","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1999-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121795228","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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