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postgraduate thesis: Maximum likelihood estimation of parameters with constraints in normaland multinomial distributions
Title  Maximum likelihood estimation of parameters with constraints in normaland multinomial distributions 

Authors  
Advisors  
Issue Date  2012 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Xue, H. [薛惠天]. (2012). Maximum likelihood estimation of parameters with constraints in normal and multinomial distributions. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b4785001 
Abstract  Motivated by problems in medicine, biology, engineering and economics, con
strained parameter problems arise in a wide variety of applications. Among them
the application to the doseresponse of a certain drug in development has attracted
much interest. To investigate such a relationship, we often need to conduct a dose
response experiment with multiple groups associated with multiple dose levels of
the drug. The doseresponse relationship can be modeled by a shaperestricted
normal regression. We develop an iterative twostep ascent algorithm to estimate
normal means and variances subject to simultaneous constraints. Each iteration
consists of two parts: an expectation{maximization (EM) algorithm that is utilized
in Step 1 to compute the maximum likelihood estimates (MLEs) of the restricted
means when variances are given, and a newly developed restricted De Pierro algorithm that is used in Step 2 to find the MLEs of the restricted variances when
means are given. These constraints include the simple order, tree order, umbrella
order, and so on. A bootstrap approach is provided to calculate standard errors of
the restricted MLEs. Applications to the analysis of two real datasets on radioimmunological assay of cortisol and bioassay of peptides are presented to illustrate
the proposed methods.
Liu (2000) discussed the maximum likelihood estimation and Bayesian estimation in a multinomial model with simplex constraints by formulating this
constrained parameter problem into an unconstrained parameter problem in the
framework of missing data. To utilize the EM and data augmentation (DA) algorithms, he introduced latent variables {Zil;Yil} (to be defined later). However,
the proposed DA algorithm in his paper did not provide the necessary individual
conditional distributions of Yil given (the observed data and) the updated parameter estimates. Indeed, the EM algorithm developed in his paper is based on the
assumption that{ Yil} are fixed given values. Fortunately, the EM algorithm is
invariant under any choice of the value of Yil, so the final result is always correct.
We have derived the aforesaid conditional distributions and hence provide a valid
DA algorithm. A real data set is used for illustration. 
Degree  Master of Philosophy 
Subject  Estimation theory. Parameter estimation. 
Dept/Program  Statistics and Actuarial Science 
Persistent Identifier  http://hdl.handle.net/10722/174559 
HKU Library Item ID  b4785001 
DC Field  Value  Language 

dc.contributor.advisor  Ng, KW   
dc.contributor.advisor  Tian, G   
dc.contributor.author  Xue, Huitian.   
dc.contributor.author  薛惠天.   
dc.date.issued  2012   
dc.identifier.citation  Xue, H. [薛惠天]. (2012). Maximum likelihood estimation of parameters with constraints in normal and multinomial distributions. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b4785001   
dc.identifier.uri  http://hdl.handle.net/10722/174559   
dc.description.abstract  Motivated by problems in medicine, biology, engineering and economics, con strained parameter problems arise in a wide variety of applications. Among them the application to the doseresponse of a certain drug in development has attracted much interest. To investigate such a relationship, we often need to conduct a dose response experiment with multiple groups associated with multiple dose levels of the drug. The doseresponse relationship can be modeled by a shaperestricted normal regression. We develop an iterative twostep ascent algorithm to estimate normal means and variances subject to simultaneous constraints. Each iteration consists of two parts: an expectation{maximization (EM) algorithm that is utilized in Step 1 to compute the maximum likelihood estimates (MLEs) of the restricted means when variances are given, and a newly developed restricted De Pierro algorithm that is used in Step 2 to find the MLEs of the restricted variances when means are given. These constraints include the simple order, tree order, umbrella order, and so on. A bootstrap approach is provided to calculate standard errors of the restricted MLEs. Applications to the analysis of two real datasets on radioimmunological assay of cortisol and bioassay of peptides are presented to illustrate the proposed methods. Liu (2000) discussed the maximum likelihood estimation and Bayesian estimation in a multinomial model with simplex constraints by formulating this constrained parameter problem into an unconstrained parameter problem in the framework of missing data. To utilize the EM and data augmentation (DA) algorithms, he introduced latent variables {Zil;Yil} (to be defined later). However, the proposed DA algorithm in his paper did not provide the necessary individual conditional distributions of Yil given (the observed data and) the updated parameter estimates. Indeed, the EM algorithm developed in his paper is based on the assumption that{ Yil} are fixed given values. Fortunately, the EM algorithm is invariant under any choice of the value of Yil, so the final result is always correct. We have derived the aforesaid conditional distributions and hence provide a valid DA algorithm. A real data set is used for illustration.   
dc.language  eng   
dc.publisher  The University of Hong Kong (Pokfulam, Hong Kong)   
dc.relation.ispartof  HKU Theses Online (HKUTO)   
dc.rights  The author retains all proprietary rights, (such as patent rights) and the right to use in future works.   
dc.rights  Creative Commons: Attribution 3.0 Hong Kong License   
dc.source.uri  http://hub.hku.hk/bib/B47850012   
dc.subject.lcsh  Estimation theory.   
dc.subject.lcsh  Parameter estimation.   
dc.title  Maximum likelihood estimation of parameters with constraints in normaland multinomial distributions   
dc.type  PG_Thesis   
dc.identifier.hkul  b4785001   
dc.description.thesisname  Master of Philosophy   
dc.description.thesislevel  Master   
dc.description.thesisdiscipline  Statistics and Actuarial Science   
dc.description.nature  published_or_final_version   
dc.identifier.doi  10.5353/th_b4785001   
dc.date.hkucongregation  2012   