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Purely sequential bounded-risk point estimation of the negative binomial mean under various loss functions: one-sample problem

Author

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  • Nitis Mukhopadhyay

    (University of Connecticut)

  • Sudeep R. Bapat

    (University of Connecticut
    University of California)

Abstract

A negative binomial (NB) distribution is useful to model over-dispersed count data arising from agriculture, health, and pest control. We design purely sequential bounded-risk methodologies to estimate an unknown NB mean $$\mu (>0)$$ μ ( > 0 ) under different forms of loss functions including customary and modified Linex loss as well as squared error loss. We handle situations when the thatch parameter $$\tau (>0)$$ τ ( > 0 ) may be assumed known or unknown. Our proposed methodologies are shown to satisfy properties including first-order asymptotic efficiency and first-order asymptotic risk efficiency. Summaries are provided from extensive sets of simulations showing encouraging performances of the proposed methodologies for small and moderate sample sizes. We follow with illustrations obtained by implementing estimation strategies using real data from statistical ecology: (1) weed count data of different species from a field in Netherlands and (2) count data of migrating woodlarks at the Hanko bird sanctuary in Finland.

Suggested Citation

  • Nitis Mukhopadhyay & Sudeep R. Bapat, 2018. "Purely sequential bounded-risk point estimation of the negative binomial mean under various loss functions: one-sample problem," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(5), pages 1049-1075, October.
    • Handle: RePEc:spr:aistmt:v:70:y:2018:i:5:d:10.1007_s10463-017-0620-2
    DOI: 10.1007/s10463-017-0620-2
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    References listed on IDEAS

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    1. Makoto Aoshima & Kazuyoshi Yata, 2010. "Asymptotic second-order consistency for two-stage estimation methodologies and its applications," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(3), pages 571-600, June.
    2. Mukhopadhyay N. & Duggan W.T., 2001. "A Two-Stage Point Estimation Procedure For The Mean Of An Exponential Distribution And Second-Order Results," Statistics & Risk Modeling, De Gruyter, vol. 19(2), pages 155-172, February.
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    Cited by:

    1. Jun Hu, 2021. "Improving Hall’s Accelerated Sequential Procedure: Generalized Multistage Fixed-Width Confidence Intervals for a Normal Mean," Methodology and Computing in Applied Probability, Springer, vol. 23(3), pages 823-835, September.
    2. 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).

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