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This article revisits the problem of sequential point estimation of the autogressive parameter in an autoregressive model of order 1, where the errors are independent and identically distributed with mean 0 and unknown variance. This problem was originally considered in Sriram (1988), where first-order efficiency properties and a second-order expansion for the expected value of a stopping rule were established. Here, we obtain an asymptotic expression for the so-called regret due to not knowing σ, as the cost of estimation error tends to infinity. Under suitable assumptions, our extensive analysis shows that all but one term in the regret are asymptotically bounded. If the errors have a bounded support, however, then the regret remains asymptotically bounded. Finally, we illustrate the performance of our sequential procedure and the associated regret for well-known blowfly data (Nicholson, 1950) and Internet traffic data using the residual bootstrap method for autoregressive models. [ABSTRACT FROM AUTHOR]

This article contains our responses to the eight discussion pieces written by experts, commenting on the results presented in Sriram and Iaci (2014). The majority of the discussants described complementary sequential problems arising in time series and other models involving dependent data, while two of the discussants provided thoughtful views on the phenomenon ofnegative regretinvestigated in our article. Prompted by an excellent observation made in one of the discussion pieces, we have attempted to shed further light on the phenomenon ofnegative regretin the context of sequential estimation of the autoregressive parameter. [ABSTRACT FROM AUTHOR]

This article revisits sequential estimation of the autoregressive parameter β in a first-order autoregressive (AR(1)) model and construction of a sequential confidence region for a parameter vector θ in a first-order threshold autoregressive (TAR(1)) model. To resolve a theoretical conjecture raised in Sriram (1986), we provide a comprehensive numerical study that strongly suggests that theregretin using a sequential estimator of β can be significantlynegativefor many heavy-tailed error distributions and even for normal errors. Secondly, to investigate yet another conjecture about the limiting distribution of a sequential pivotal quantity for θ in a TAR(1) model, we conduct an extensive numerical study that strongly suggests that the sequential confidence region has much better coverage probability than that of a fixed sample counterpart, regardless of whether the θ values are inside or on or near the boundary of the ergodic region of the series. These highlight the usefulness of sequential sampling methods in fitting linear and nonlinear time series models. [ABSTRACT FROM AUTHOR]