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On the second-order random walk model for irregular locations. (English) Zbl 1199.60276
A new model of second-order random walk $$x_1,\dots,x_n$$ is proposed for the case when $$x_i$$ should represent a smooth curve at locations $$s_i$$, $$s_1<\dots<s_n$$. The model is based on the idea to interpret $$x_i$$ as observations of a continuous process $$x(t)$$ at points $$s_i$$ where $$x(t)$$ is a weak solution to $${d^2\over dt^2}x(t)={d\over dt}W(t)$$, $$W(t)$$ being the standard Wienner process. In fact, a Galerkin approximation is used with some simplification to obtain $$x_i$$ as a Gaussian Markov random field with the properties $E(x_i\;| \;x_{i-1},x_{i-2})= \left( {d_{i-1}\over d_{i-2}}+1 \right)x_{i-1} -{d_{i-1}\over d_{i-2}} x_{i-2},$
$\text{Var}(x_i\;| \;x_{i-1},x_{i-2})= {d_{i-1}^2(d_{i-2}-d_{i-1})\over 2}$ The random walk is used as a prior in a semiparametric model for Bayesian analysis of an econometric time series.

##### MSC:
 60J25 Continuous-time Markov processes on general state spaces 62F15 Bayesian inference 91B84 Economic time series analysis
##### Software:
GMRFLib; Fahrmeir
Full Text:
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