Stephen M. Stigler (2007), Statistical Science 22(4) (2007) — ARXIV-0804.2996
Key takeaways
- Maximum likelihood took about two centuries to justify. Fisher’s three proofs (1922, 1925, and an unpublished 1930 characterization) are defensible under his implicit regularity conditions, and the 1950s counterexamples (Hodges’s superefficiency, growing parameter counts, unbounded likelihoods) mark real limits. Yet ML’s useful scope still exceeds anything that can reasonably be proved.
Seeded from the abstract and a skim, before the reading in NOTE-074. What follows is what the work says about itself.
Maximum likelihood looks like an obvious, even prehistoric idea, but its mathematical history is anything but simple. Lagrange, Daniel Bernoulli, Euler, Laplace and Gauss all explored it, not always in ways now accepted. Stigler traces the story from before Fisher to Le Cam’s dissertation. He presents Fisher’s unpublished 1930 conditions for consistency and efficiency, and analyses the mathematical basis of Fisher’s three proofs. He shows that Fisher’s information inequality came out of his analysis-of-variance work, and that the later estimating-function approach came from Euler’s relation for homogeneous functions. He reviews how Fisher’s work was received and draws lessons from it.
Standing in the record
Filed from the survey of 2026-09-25 of work the anthology set aside as out of scope (tier C): 360 seconds of active reading over 2 sessions in the papers-feed tracker. It was filed Deferred, unread. NOTE-074 is the close reading of 2026-09-25, and it placed the work: Active — an authoritative, archive-based history by the leading historian of statistics, and worth reading for exactly which ML optimality claims were proved, under what conditions, and which counterexamples matter. It is a history, not a technical reference: for modern conditions it points to Bahadur (1964), Le Cam (1953, 1990), van der Vaart (1998) and Cox (2006).
When seeded, before the reading in NOTE-074, the entry gave a priority for a deeper reading: low — Historical essay. The skim captures its argument. Moderate owner time (t=360 s over 2 sessions). It makes no practice claim.
What the seeding asked a deeper reading to check (NOTE-074 is that reading):
- It is useful historical grounding for any probabilistic-modeling note that leans on MLE optimality claims. It spells out which efficiency claims are theorems and which are conditions.
- A deeper reading should check the §9–10 treatment of the “nasty” example and the 1930 conditions if a note wants to cite a precise account of when ML is efficient.
Access when seeded: I read the arXiv abstract page (v1 18 Apr 2008, posted via the IMS/VTEX proxy after journal publication) and the full arXiv PDF (24 pp.). I read the section heads, the introduction’s framing (Hodges’s “nasty, ugly little fact”) and §14 (Conclusion). Crossref gives Statistical Science 22(4) with a print date of 2007-11-01. That date is Crossref’s, and the day may be nominal.