Semiparametric Bayesian analysis of transformation linear mixed models
type: Item
instance of: scholarly article
title: Semiparametric Bayesian analysis of transformation linear mixed models
zbMATH Open document ID: 1414.62097
author: Ying Wu
author: Dan Chen
author: N. S. Tang
published in: Journal of Multivariate Analysis
publication date: 2018-05-17T00:00:00Z
review text: They are called semi-parametric models, those that combine random variables with parameterized distributions with others whose distributions are not assumed to be dependent on parameters. One such model, also called a mixed linear model, is as follows: \(n,p\), and \(q\) are positive integers. For each \((i=1,\dots ,n)\), \(T_{i}\) is a positive integer; \(b_{i}=(b_{i1},\dots ,b_{iq})^{\prime}\) is a random \((q\times 1)\)-matrix. For each \(i=1,\dots ,n\), and \((j=1,\dots ,T_{i})\), \(X_{ij}=(X_{ij1},\dots ,X_{ijp})^{\prime}\) is a random \((p\times 1)\)-matrix; \(W_{ij}=(W_{ij1},\ldots ,W_{ijq})^{\prime}\) is a random \((q\times 1)\)-matrix; \(Y_{ij}\) is a random variable; \(\varepsilon_{ij}\) is a random variable; \(f:\mathbb{R}\rightarrow\mathbb{R}\) is a strictly monotone and differentiable function; and \[ \begin{aligned} f(Y_{ij}) &=X_{ij}^{\prime}\gamma +W_{ij}^{\prime}b_{i}+\varepsilon_{ij} \\ &=\sum_{r=1}^{p}X_{ijr}\gamma_{r}+\sum_{s=1}^{q}W_{ijs}b_{is}, \end{aligned} \] where \(\gamma =(\gamma_{1},\ldots ,\gamma_{p})^{\prime}\) is a \((p\times 1)\)-matrix. The matrix form of this model is given by \[ \mathbf{f}(\mathbf{Y}_{i})=\mathbf{X}_{i}\gamma +\mathbf{W}_{i} \mathbf{b}_{i}+\varepsilon_{i}, \quad i=1,\ldots ,n , \] where \[ \begin{aligned} \mathbf{f}(\mathbf{Y}_{i})&=(f(Y_{i1}) ,\ldots ,f(Y_{iT_{i}}))^{\prime}, \quad i=1,\ldots ,n , \\ \mathbf{W}_{i} &= \begin{pmatrix} W_{i11} & \cdots & W_{i1q} \\ \vdots & & \vdots \\ W_{iT_{i}1} & \cdots & W_{iT_{i}q} \end{pmatrix}, \quad i=1,\ldots ,n, \\ \varepsilon_i &=(\varepsilon_{i1},\ldots ,\varepsilon_{iT_{i}})^{\prime},\quad i=1,\ldots ,n. \end{aligned} \] To ensure the identifiability of the parameter \(\gamma\), and also for reasons of simplicity, in this paper it is assumed that \[ \varepsilon _{11},\ldots ,\varepsilon _{1T_{1}},\ldots ,\varepsilon _{n1},\ldots ,\varepsilon _{nT_{n}} \] are independent and have the same distribution \(N(0,1)\). It is usually assumed that \(b_{1},\ldots ,b_{q}\) are i.i…
Mathematics Subject Classification ID: 62F15
Mathematics Subject Classification ID: 62H12
Mathematics Subject Classification ID: 62J20
Mathematics Subject Classification ID: 62P10
zbMATH DE Number: 6869761
zbMATH Keywords: Bayesian local influence analysis
zbMATH Keywords: Dirichlet process prior
zbMATH Keywords: linear mixed model
zbMATH Keywords: P-spline
zbMATH Keywords: transformation
Wikidata QID: Q57535635
reviewed by: Oscar Bustos
describes a project that uses: SemiPar
describes a project that uses: mixAK
MaRDI profile type: Publication
full work available at URL: j.jmva.2018.03.007
OpenAlex ID: W2789401640
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DBLP publication ID: journals/ma/TangWC18
DOI: 10.1016/J.JMVA.2018.03.007
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description: scientific article; zbMATH DE number 6869761