Typical decision problems in the first-order autoregressive time series

Abstract

Applied statistical decision theory has wide applications in decision-making fields of studies, such as economic, business management and industrial managements. In this work, following Pratt et al.’s [Introduction to statistical decision theory. 3rd ed. Cambridge, MA: The MIT Press; 2001] approach, we provide theoretical and practical formulations for the calculations of the key decision-making indices expected value of perfect information and expected value of sample information, whenever the unknown state appears to be the first-order autoregressive (AR) time series parameter that assumes a normal prior distribution. A practical procedure is furnished for calculating the decision-making indices. We treat the finite and infinite state spaces for the linear value functions and the quadratic opportunity losses. Interestingly our investigations on the distribution of the mean of the posterior distribution lead us to a general form for the corresponding statistic and its distribution, discussed by Reeves [The distribution of the maximum likelihood estimator of the parameter in the first-order AR series. Biometrika. 1972;59:387–394], Moschopoulos and Canada [The distribution function of a linear combination of chi-squares. Comput Math Appl. 1984;10:383–386], and Roychowdhury and Bhattacharya [On the performance of estimators of parameter in AR model of order one and optimal prediction under asymmetric loss. Model Assist Stat Appl. 2008;3:225–232].