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A Personal Celebration of Dr. D. Basu with Emphasis on Examples-Counterexamples-Clarifications

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  • Nitis Mukhopadhyay

    (University of Connecticut)

Abstract

Preparing this centennial tribute to Dr. D. Basu (5 July, 1924 – 24 March, 2001) created an opportunity to selectively revisit a number of core notions in statistical inference. We explored intrinsic beauty, ingenuity and power of Basu’s (1955) theorem by (i) weaving through the process of Rao-Blackwellization, (ii) looking back at the uniformly minimum variance unbiased estimator (UMVUE), and (iii) proposing a randomly stopped version of Stein’s identity. In doing so, we have (i) suggested a layman’s interpretation of the notion of completeness on its own, (ii) appealed to symmetry that often remains hidden within Rao-Blackwellization, and (iii) considered notions of approximate sufficiency-ancillarity-independence via intuitive understanding. We have confronted the notion of minimal sufficiency in a common mean ( $$\mu )$$ μ ) estimation problem from a bivariate normal population (with known variances and correlation coefficient), and to our surprise we find that we can do away with only one observation to construct the UMVUE or an UMP test or a likelihood ratio test for $$\mu $$ μ in some situations. We wrap up with constructions of interesting ratios X/Z and Y/Z made up of independent or dependent random variables (X, Y, Z), where X/Z and Y/Z would be overwhelmingly favored to be dependent, but surprisingly they are not. It should not be a surprise that Basu’s theorem, and how Dr. Basu influenced statistical inference ever so gently, have acted as a common thread holding together much of the discussions included in this celebratory piece.

Suggested Citation

  • Nitis Mukhopadhyay, 2024. "A Personal Celebration of Dr. D. Basu with Emphasis on Examples-Counterexamples-Clarifications," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 86(1), pages 134-159, November.
    • Handle: RePEc:spr:sankha:v:86:y:2024:i:1:d:10.1007_s13171-024-00359-5
    DOI: 10.1007/s13171-024-00359-5
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    References listed on IDEAS

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    1. Pal, Nabendu & Lin, Jyh-Jiuan & Chang, Ching-Hui & Kumar, Somesh, 2007. "A revisit to the common mean problem: Comparing the maximum likelihood estimator with the Graybill-Deal estimator," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 5673-5681, August.
    2. Nitis Mukhopadhyay & Swarnali Banerjee, 2013. "Fisher information, sufficiency, and ancillarity: some clarifications," METRON, Springer;Sapienza Università di Roma, vol. 71(1), pages 33-38, June.
    3. Philip Yu & Yijun Sun & Bimal Sinha, 2002. "Estimation of the Common Mean of a Bivariate Normal Population," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 54(4), pages 861-878, 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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    Cited by:

    1. Dipak Dey & Subhashis Ghosal & Tapas Samanta, 2024. "Editorial Article: Remembering D. Basu’s Legacy in Statistics," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 86(1), pages 1-7, November.

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