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A General Theory of Three-Stage Estimation Strategy with Second-Order Asymptotics and Its Applications

Author

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  • Nitis Mukhopadhyay

    (University of Connecticut-Storrs)

  • Soumik Banerjee

    (University of Connecticut-Storrs)

Abstract

We begin with a generic expression of an optimal fixed-sample-size nβˆ— which has an expression Ξ»g(πœƒ) with Ξ» > 0 and g(πœƒ) > 0 where πœƒ is an unknown parameter. A consistent estimator of πœƒ is a sample mean of independent and identically distributed (i.i.d.) random variables. Under fairly relaxed set of conditions on g(.), we have developed a general theory of three-stage sampling strategy detailing requisite mathematical techniques for proving both asymptotic (as Ξ» β†’ ∞ $\lambda \rightarrow \infty $ ) first-order and second-order analyses. We believe that this theory is broad and rich especially since the technicalities developed are not tailored to fit a specific inference problem of choice. We have validated this sentiment with the help of illustrations which cannot be handled satisfactorily by improvising upon some of the existing methodologies. For example, (i) Illustration 1 proposes a three-stage strategy under Linex loss in a recently developed inference problem; (ii) Illustration 2 handles estimation of πœƒ in a Uniform(0,πœƒ) distribution which obviously stays outside an exponential family; and (iii) Illustration 3 incorporates expressions of g(.) functions which no existing paper’s analyses could treat.

Suggested Citation

  • Nitis Mukhopadhyay & Soumik Banerjee, 2023. "A General Theory of Three-Stage Estimation Strategy with Second-Order Asymptotics and Its Applications," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 401-440, February.
    • Handle: RePEc:spr:sankha:v:85:y:2023:i:1:d:10.1007_s13171-021-00253-4
    DOI: 10.1007/s13171-021-00253-4
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    References listed on IDEAS

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    1. H. Hamdy & N. Mukhopadhyay & M. Costanza & M. Son, 1988. "Triple stage point estimation for the exponential location parameter," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 40(4), pages 785-797, December.
    2. Masafumi Akahira & Kei Takeuchi, 2001. "Information Inequalities in a Family of Uniform Distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(3), pages 427-435, September.
    3. N. Mukhopadhyay, 1980. "A consistent and asymptotically efficient two-stage procedure to construct fixed width confidence intervals for the mean," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 27(1), pages 281-284, December.
    4. N. Mukhopadhyay & T. Solanky, 1997. "Estimation after sequential selection and ranking," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 45(1), pages 95-106, January.
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