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 dose-response 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 dose-response relationship can be modeled by a shape-restricted
normal regression. We develop an iterative two-step 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 radioim-munological 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. / published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy
| Identifer | oai:union.ndltd.org:HKU/oai:hub.hku.hk:10722/174559 |
| Date | January 2012 |
| Creators | Xue, Huitian., 薛惠天. |
| Contributors | Ng, KW, Tian, G |
| Publisher | The University of Hong Kong (Pokfulam, Hong Kong) |
| Source Sets | Hong Kong University Theses |
| Language | English |
| Detected Language | English |
| Type | PG_Thesis |
| Source | http://hub.hku.hk/bib/B47850012 |
| Rights | The author retains all proprietary rights, (such as patent rights) and the right to use in future works., Creative Commons: Attribution 3.0 Hong Kong License |
| Relation | HKU Theses Online (HKUTO) |
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