## Abstract

Under the squared error loss plus linear cost of sampling, we revisit the *minimum risk point estimation* (MRPE) problem for an unknown normal mean \(\mu\) when the variance \(\sigma ^{2}\) also remains unknown. We begin by defining a new class of purely sequential MRPE methodologies based on a general estimator \(W_{n}\) for \(\sigma\) satisfying a set of conditions in proposing the requisite stopping boundary. Under such appropriate set of sufficient conditions on \(W_{n}\) and a properly constructed associated stopping variable, we show that (i) the normalized stopping time converges in law to a normal distribution (Theorem 3.3), and (ii) the square of such a normalized stopping time is uniformly integrable (Theorem 3.4). These results subsequently lead to an * asymptotic second-order* expansion of the associated *regret* function in general (Theorem 4.1). After such general considerations, we include a number of substantial illustrations where we respectively substitute appropriate multiples of *Gini’s mean difference* and the *mean absolute deviation * in the place of the general estimator \(W_{n}\). These illustrations show a number of desirable asymptotic *first-order* and * second-order* properties under the resulting purely sequential MRPE strategies. We end this discourse by highlighting selected summaries obtained via simulations.

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## Acknowledgements

The comments received from two anonymous reviewers, the Associate Editor, and the Executive Editor on our earlier version have genuinely helped us in preparing this revised manuscript. We express our gratitude to all of them and thank them.

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Hu, J., Mukhopadhyay, N. Second-order asymptotics in a class of purely sequential minimum risk point estimation (MRPE) methodologies.
*Jpn J Stat Data Sci* **2, **81–104 (2019). https://doi.org/10.1007/s42081-018-0028-0

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### Keywords

- Asymptotic first-order properties
- Asymptotic second-order properties
- Linear cost
- Regret
- Risk efficiency
- Sequential strategy
- Simulations
- Squared error loss

### Mathematics Subject Classification

- 62L10
- 62L12
- 62G05
- 62G20