Reading list. Part 2.
👉 Andrews, Askey, Roy — Special Functions A broad and serious reference on special functions. Useful if you want to see the q-binomial theorem as part of the larger world of special functions.
👉 Kathleen O’Hara — Unimodality of Gaussian coefficients: A constructive proof, 1990 A famous paper on the unimodality of Gaussian coefficients. Difficult, but important if you are interested in the shape of the distribution: why the coefficients rise toward the middle and then fall.
👉 Doron Zeilberger — Kathy O’Hara’s Constructive Proof of the Unimodality of the Gaussian Polynomials, 1989 A more explanatory bridge to O’Hara’s proof. Still not easy, but easier than starting with the original paper.
👉 Pak, Panova — Strict unimodality of q-binomial coefficients, 2013 A serious paper on strict unimodality of q-binomial coefficients. Interesting if you want to go beyond “there is a maximum near the centre” and understand finer shape properties of the distribution.
👉 Dhand — A combinatorial proof of strict unimodality for q-binomial coefficients, 2014 A combinatorial proof of strict unimodality in large cases. Hard, but closer in spirit to the combinatorial nature of our problem.
Sources connecting this to ROC AUC
👉 Donald Bamber — The area above the ordinal dominance graph and the area below the receiver operating characteristic graph, 1975 A very important bridge between ROC AUC and the Mann–Whitney U statistic. A good reference for the claim that the area under the ROC curve is a pairwise ranking statistic.
👉 Hanley, McNeil — The meaning and use of the area under a receiver operating characteristic curve, 1982 A classic applied paper on the meaning of ROC AUC. Useful for the interpretation of AUC as the probability that a randomly chosen positive example receives a higher score than a randomly chosen negative example.
👉 Green, Swets — Signal Detection Theory and Psychophysics, 1966 Classic background on signal detection theory and ROC curves. Less about discrete tied score blocks, more about the historical and conceptual origins of ROC analysis.
Suggested reading order for the stochastic ROC project
First:
👉 Wikipedia — Gaussian binomial coefficient
👉 An Invitation to Enumeration — q-analogues
Then:
👉 Andrews, Eriksson — Integer Partitions
👉 Bamber — area below ROC and the Mann–Whitney connection
Then, for a more serious foundation:
👉 Stanley — Enumerative Combinatorics
👉 Mann, Whitney, 1947
And only then, if you want to go deeper into the shape of the distribution:
👉 O’Hara — Unimodality of Gaussian coefficients
👉 Pak, Panova — Strict unimodality of q-binomial coefficients