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In this paper, within the framework of a Bayesian model, we consider the problem of sequentially estimating the intensity parameter of a homogeneous Poisson process with a linear exponential (LINEX) loss function and a fixed cost per unit time. An asymptotically pointwise optimal (APO) rule is proposed. It is shown to be asymptotically optimal for the arbitrary priors and asymptotically non-deficient for the conjugate priors in a similar sense of Bickel and Yahav [Asymptotically pointwise optimal procedures in sequential analysis, in 442-456] and Woodroofe [A.P.O. rules are asymptotically non-deficient for estimation with squared error loss, Z. Wahrsch. verw. Gebiete 58 (1981), pp. 331-341], respectively. The proposed APO rule is illustrated using a real data set.

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The problem of Bayes sequential estimation of the unknown parameter in a particular exponential family of distributions with relative LINEX loss and fixed cost for each observation is considered in this article. Optimal, nearly optimal, and asymptotically pointwise optimal procedures with deterministic stopping rules are derived, and the approximate optimal procedures are shown to be asymptotically nondeficient in the sense of Woodroofe (1981). In addition, a robust procedure with a deterministic stopping rule, which does not depend on the parameters of the prior distribution, is proposed, and the asymptotic second-order expansion of the corresponding Bayes risk is obtained.