Semiparametric Bayesian analysis of transformation linear mixed models
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.d. random vectors following the same distribution \(N_{q}(\mathbf{0},\Sigma)\), where \(\Sigma\) is a \((q\times q)\)-covariance matrix. In order to make the proposed model conform to more common situations in applications, the authors assume that \(b_{1},\ldots ,b_{q}\) are i.i.d. random vectors following a Dirichlet distribution. Since the function \(f\) is not supposed to be explicitly known, it is not possible to make statistical inferences about the parameter \(\gamma \). So in this paper \(f\) is replaced by an approximation of the form \[ \sum_{k=1}^{n}\beta _{k}B_{k}, \] where the \(B_{k}^{\prime}\)s are B-spline functions, and the \(\beta _{k}^{\prime}\)s are real constants to be estimated. No results are formally demonstrated on properties of the proposed estimator, nor is its asymptotic behavior mentioned. The performance of the suggested technique is analyzed through three simulation experiments, changing the function \(f\) each time. All of them are developed with detail and rigor. Finally, the results of applying the proposed estimator in a real case are shown. Remarkable is the detailed description of the algorithm that implements the so-called Gibb's sampler. It is an interesting paper but somewhat incomplete in terms of the study of the proposed estimator.
scientific article; zbMATH DE number 6869761
scientific article