IDEAS home Printed from https://ideas.repec.org/a/spr/sankha/v85y2023i1d10.1007_s13171-021-00253-4.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s13171-021-00253-4
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s13171-021-00253-4?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Nitis Mukhopadhyay & Tumulesh K. S. Solanky, 2012. "On Two-Stage Comparisons with a Control Under Heteroscedastic Normal Distributions," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 501-522, September.
    2. Hokwon Cho, 2019. "Two-Stage Procedure of Fixed-Width Confidence Intervals for the Risk Ratio," Methodology and Computing in Applied Probability, Springer, vol. 21(3), pages 721-733, September.
    3. Bandyopadhyay, Uttam & Sarkar, Suman & Biswas, Atanu, 2022. "Sequential confidence interval for comparing two Bernoulli distributions in a non-conventional set-up," Statistics & Probability Letters, Elsevier, vol. 181(C).
    4. Kazuyoshi Yata & Makoto Aoshima, 2012. "Inference on High-Dimensional Mean Vectors with Fewer Observations Than the Dimension," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 459-476, September.
    5. Nitis Mukhopadhyay & Srawan Kumar Bishnoi, 2021. "An Unusual Application of CramΓ©r-Rao Inequality to Prove the Attainable Lower Bound for a Ratio of Complicated Gamma Functions," Methodology and Computing in Applied Probability, Springer, vol. 23(4), pages 1507-1517, December.
    6. Sudeep R. Bapat, 2023. "A novel sequential approach to estimate functions of parameters of two gamma populations," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 86(6), pages 627-641, August.
    7. M. Son & L. Haugh & H. Hamdy & M. Costanza, 1997. "Controlling Type II Error While Constructing Triple Sampling Fixed Precision Confidence Intervals for the Normal Mean," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 49(4), pages 681-692, December.
    8. Sen, Pranab K. & Tsai, Ming-Tien, 1999. "Two-Stage Likelihood Ratio and Union-Intersection Tests for One-Sided Alternatives Multivariate Mean with Nuisance Dispersion Matrix," Journal of Multivariate Analysis, Elsevier, vol. 68(2), pages 264-282, February.
    9. Hwang, Leng-Cheng, 2020. "A robust two-stage procedure for the Poisson process under the linear exponential loss function," Statistics & Probability Letters, Elsevier, vol. 163(C).
    10. Christopher S. Withers & Saralees Nadarajah, 2022. "Cornish-Fisher Expansions for Functionals of the Weighted Partial Sum Empirical Distribution," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1791-1804, September.
    11. Bhargab Chattopadhyay & Nitis Mukhopadhyay, 2019. "Constructions of New Classes of One- and Two-Sample Nonparametric Location Tests," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1229-1249, December.
    12. Leng-Cheng Hwang & Chia-Chen Yang, 2015. "A robust two-stage procedure in Bayes sequential estimation of a particular exponential family," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(2), pages 145-159, February.
    13. Nitis Mukhopadhyay, 2010. "On a Sharper Lower Bound for a Percentile of a Student’s t Distribution with an Application," Methodology and Computing in Applied Probability, Springer, vol. 12(4), pages 609-622, December.
    14. Makoto Aoshima & Kazuyoshi Yata, 2015. "Asymptotic Normality for Inference on Multisample, High-Dimensional Mean Vectors Under Mild Conditions," Methodology and Computing in Applied Probability, Springer, vol. 17(2), pages 419-439, June.
    15. Ajit Chaturvedi & Sudeep R. Bapat & Neeraj Joshi, 2020. "Purely Sequential and k-Stage Procedures for Estimating the Mean of an Inverse Gaussian Distribution," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 1193-1219, September.
    16. Ali Yousef & Hosny Hamdy, 2019. "Three-Stage Estimation of the Mean and Variance of the Normal Distribution with Application to an Inverse Coefficient of Variation with Computer Simulation," Mathematics, MDPI, vol. 7(9), pages 1-15, September.
    17. Aoshima, Makoto & Mukhopadhyay, Nitis, 1998. "Fixed-Width Simultaneous Confidence Intervals for Multinormal Means in Several Intraclass Correlation Models," Journal of Multivariate Analysis, Elsevier, vol. 66(1), pages 46-63, July.
    18. S. Zacks, 2007. "Review of Some Functionals of Compound Poisson Processes and Related Stopping Times," Methodology and Computing in Applied Probability, Springer, vol. 9(2), pages 343-356, June.
    19. Nitis Mukhopadhyay & William Duggan, 1999. "On a Two-Stage Procedure Having Second-Order Properties with Applications," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 51(4), pages 621-636, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:sankha:v:85:y:2023:i:1:d:10.1007_s13171-021-00253-4. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.