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

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.