10  Appendices

10.1 A Short List of Contributors to Statistical Inference Based On Likelihood

  1. R.A. Fisher (1922, 1925)
    • Affiliation: Rothamsted / University College London (UCL)
    • Connection: The “father” of the field; his work at UCL influenced the entire next generation, including Rao and Pearson.
    • Contribution: Fisher formally defined Likelihood as a function distinct from probability (\(L(\theta; x)\) vs \(f(x; \theta)\)). He introduced the Maximum Likelihood Estimator (MLE), derived the Fisher Information measure, and established the asymptotic efficiency of MLEs.
  2. J. Neyman & E.S. Pearson (1928, 1933)
    • Affiliations: UC Berkeley (Neyman) / UCL (Pearson)
    • Connection: Egon Pearson was Karl Pearson’s son (Fisher’s rival). Neyman founded the Berkeley Statistics department, which later hosted Lehmann, Scheffé, and Le Cam.
    • Contribution: They introduced the Likelihood Ratio criterion (\(\lambda\)) as a general method for hypothesis testing. Their 1933 Lemma proved that for simple hypotheses, the Likelihood Ratio Test is the Uniformly Most Powerful (UMP) test, establishing the optimality of likelihood-based methods.
  3. S.S. Wilks (1938)
    • Affiliation: Princeton University
    • Connection: A student of Henry Rietz, Wilks spent time at UCL with Pearson and Wishart before establishing the Princeton program.
    • Contribution: Wilks derived the asymptotic distribution of the Likelihood Ratio statistic for composite hypotheses. He proved that \(-2 \ln \Lambda\) converges to a Chi-square distribution with degrees of freedom equal to the difference in the number of free parameters.
  4. M.S. Bartlett (1937)
    • Affiliation: UCL / University of Manchester
    • Connection: A student of Fisher and colleague of Pearson at UCL; famously debated with Fisher on conditional inference.
    • Contribution: Bartlett improved the accuracy of the Likelihood Ratio Test for small samples. He introduced the Bartlett Correction, a scaling factor for the test statistic that aligns its expected value with that of the limiting Chi-square distribution.
  5. A. Wald (1943)
    • Affiliation: Columbia University
    • Connection: A student of Karl Menger in Vienna; at Columbia, he worked with Wolfowitz and influenced the decision-theoretic approach adopted by Stein and Karlin.
    • Contribution: Wald developed the Wald Test, which tests hypotheses based on the distance between the unrestricted MLE and the hypothesized value. He demonstrated the asymptotic equivalence of the Likelihood Ratio, Wald, and Score tests for standard models.
  6. H. Cramér (1946)
    • Affiliation: Stockholm University
    • Connection: Hosted Rao in Stockholm; his textbook synthesized the British (Fisher/Pearson) and American (Wilks/Wald) schools into one rigorous framework.
    • Contribution: In his influential text Mathematical Methods of Statistics, Cramér provided rigorous proofs for the consistency and asymptotic normality of the MLE. Independently of Rao, he established the Cramér-Rao Lower Bound.
  7. C.R. Rao (1948)
    • Affiliation: Indian Statistical Institute (ISI)
    • Connection: A PhD student of Fisher at Cambridge; he unified the testing theories of Fisher, Neyman, and Wald.
    • Contribution: Rao introduced the Score Test (or Lagrange Multiplier Test), which allows for hypothesis testing using only the restricted MLE (under the null hypothesis). He also independently derived the lower bound for estimator variance.
  8. E.L. Lehmann & H. Scheffé (1950, 1955)
    • Affiliation: UC Berkeley
    • Connection: Lehmann was Neyman’s student at Berkeley; Scheffé was a colleague. They solidified the “Berkeley School” of optimality.
    • Contribution: Building on the concept of Sufficiency, they established the Lehmann-Scheffé Theorem. This result connects complete sufficient statistics to Uniformly Minimum Variance Unbiased Estimators (UMVUE).
  9. S. Karlin & H. Rubin (1956)
    • Affiliation: Stanford University
    • Connection: Part of the “Stanford School” (along with Stein) that focused on decision theory, heavily influenced by Wald’s earlier work.
    • Contribution: They formalized the property of Monotone Likelihood Ratio (MLR) for families of distributions. The Karlin-Rubin Theorem extended the Neyman-Pearson Lemma to one-sided composite hypotheses.
  10. W. James & C. Stein (1961)
    • Affiliation: Stanford University
    • Connection: Stein was a student of Wald at Columbia before moving to Stanford.
    • Contribution: They discovered Stein’s Paradox, showing that the MLE (sample mean) is inadmissible for estimating the mean of a multivariate Normal distribution (\(p \ge 3\)). They introduced the James-Stein Estimator, a shrinkage estimator that dominates the MLE.
  11. L. Le Cam (1960s)
    • Affiliation: UC Berkeley
    • Connection: A student of Neyman at Berkeley; he took the asymptotic torch from Wald and Wilks and placed it on a rigorous topological foundation.
    • Contribution: Le Cam modernized asymptotic theory by introducing Local Asymptotic Normality (LAN). He showed that under general conditions, the likelihood ratio of a complex experiment behaves asymptotically like that of a Normal shift experiment.