Bayesian calibration under a Student-\(t\) model
The linear calibration problem is given by \[ Y_i=\alpha+\beta x_i+e_i,\quad Y_0=\alpha +\beta x_0+e_i, \] where \((X_i,Y_i)\), \(i=1,\dots,n\), is an observed sample, \(\alpha\) and \(\beta\) are unknown parameters, \(e_0\), \(e_1\), \dots, \(e_n\) are independent errors, \(Y_0\) is an observed value and \(x_0\) is to be predicted (estimated). The authors consider a Bayesian approach to this problem in following two settings: (i) \(e_i\) are i.i.d. with Student-t distribution, (ii) \(e_i\) are independent \(N(0,w_i\sigma^2)\) distributed r.v.s where \((w_0,\dots, w_n)\) are unknown parameters which are considered to be independent with distribution proportional to an inverted \(\chi^2\) distribution. The authors prove the equivalence of (i) and (ii) in the Bayesian setting, obtain conditional distributions of the unknown parameters for known priors and propose a Gibbs sampler procedure for \(x_0\) estimation. A simulation example and an application to measuring Marathon courses are given.
scientific article; zbMATH DE number 1336632