On general asymptotically second-order efficient purely sequential fixed-width confidence interval (FWCI) and minimum ri
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On general asymptotically second-order efficient purely sequential fixed-width confidence interval (FWCI) and minimum risk point estimation (MRPE) strategies for a normal mean and optimality Nitis Mukhopadhyay1 · Srawan Kumar Bishnoi1 Received: 20 May 2020 / Accepted: 29 August 2020 © Sapienza Università di Roma 2020
Abstract We develop a generalized class of purely sequential sampling strategies associated with both fixed-width confidence interval (FWCI) and minimum risk point estimation (MRPE) problems for the unknown mean μ of a normally distributed population having its variance σ 2 also unknown. Under this newly proposed general class of associated estimation strategies, we develop a variety of asymptotic first-order and asymptotic second-order properties such as asymptotic consistency, first-order efficiency, first-order risk efficiency, second-order efficiency, and second-order regret analysis. Next, we proceed to locate an optimal strategy within our newly built large class of possibilities. Such optimality is defined as having been associated with the minimal second-order asymptotic variance of a stopping time within the general class of proposed strategies. We follow through by exploring both the FWCI and MRPE problems with the help of data analysis from simulations. Keywords Asymptotic consistency · Asymptotic first-order efficiency · Asymptotic first-order risk efficiency · Asymptotically optimal strategy · Asymptotic second-order efficiency · Data analysis · Fixed-width confidence interval (FWCI) · Minimum risk point estimation (MRPE) · Optimality · Purely sequential methodologies · Regret analysis · Simulations · Unified theory Mathematics Subject Classification 62L10 · 62L05 · 62L12
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Nitis Mukhopadhyay [email protected] Srawan Kumar Bishnoi [email protected]
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Department of Statistics, University of Connecticut, Austin Building U4120, 215 Glenbrook Rd, Storrs, CT 06269-4120, USA
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N. Mukhopadhyay, S. K. Bishnoi
1 Introduction In this paper, we revisit both fixed-width confidence interval (FWCI) and the minimum risk point estimation (MRPE) problems under a generalized class of purely sequential stopping rules for an unknown normal mean μ(∈ ) when the population variance σ 2 (∈ + ) remains unknown. We propose replacing the customary sample standard deviation with rather nonconventional estimators of σ or σ 2 coming from an appropriately defined large class of choices while defining the purely sequential stopping boundaries. Every purely sequential estimation strategy from this newly defined class will be shown to enjoy the customarily accepted and crucially important characteristics involving both asymptotic first-order and second-order properties. That said, it makes good sense for us to embark upon finding a way to identify a stopping rule within this large class that may be designated as the best or optimal in some appropriate sense. More specifics will be developed.
1.1 A brief literature review Dantzig [5] showed that the FWCI and the MRPE problems for an unknown n
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