A locally correlated process and its applications in Bayesian estimation (Q1323853)

Consider an additive noise signal model \[ y(t)=x(t)+\varepsilon (t),\;t \in T, \tag{1} \] where \(x\) is the signal of interest, \(\varepsilon\) is noise with mean 0 and variance \(\sigma^ 2\) (which might be zero), and \(\varepsilon\) is uncorrelated with \(x\). Moreover, \(\varepsilon (t)\) and \(\varepsilon (s)\) are independent for \(s \neq t\). Given observations \(y(s_ 1)\), \(y(s_ 2),\dots,\) and \(y(s_ n)\), where the \(s_ i\)'s need not be distinct, our aim is to estimate \(x(t)\) for any given \(t\). The domain \(T\) is a subset of \(R^ d\). Although it is interesting to discuss the problem for general dimension \(d\), our primary focus is on one dimension. The organization of this paper is as follows. In Section 2, a class of local correlation functions is constructed from \(B\)-spline bases and the corresponding random processes are obtained through local integration of a Wiener process. Then a Bayes estimate is proposed in Section 3 using this process as a prior. The estimate can be thought of as a kriging with no trend function. Some comparisons between this Bayes estimate and kernel smoothing and between this Bayes estimate and cubic smoothing splines are given. In Section 4, mean square convergence of the Bayes estimate is studied. Moreover, a local convergence is obtained regardless of correct specification of the correlation functions. Some simulations and an application are presented in Section 5 to look at the performance of the Bayes estimate in comparison with that of kernel smoothing or that of smoothing splines. Finally a brief discussion about the local Bayesian estimate on higher dimension is given in Section 6.