arXiv 20 Jan 2020 · Mathematics — Statistics Theory · 7 citations (OpenAlex)
arXiv:2001.07042 · PDF · DOI · OpenAlex · Extracted main text
The Multinomial Logit (MNL) model and the axiom it satisfies, the Independence of Irrelevant Alternatives (IIA), are together the most widely used tools of discrete choice. The MNL model serves as the workhorse model for a variety of fields, but is also widely criticized, with a large body of experimental literature claiming to document real-world settings where IIA fails to hold. Statistical tests of IIA as a modelling assumption have been the subject of many practical tests focusing on specific deviations from IIA over the past several decades, but the formal size properties of hypothesis testing IIA are still not well understood. In this work we replace some of the ambiguity in this literature with rigorous pessimism, demonstrating that any general test for IIA with low worst-case error would require a number of samples exponential in the number of alternatives of the choice problem. A major benefit of our analysis over previous work is that it lies entirely in the finite-sample domain, a feature crucial to understanding the behavior of tests in the common data-poor settings of discrete choice. Our lower bounds are structure-dependent, and as a potential cause for optimism, we find that if one restricts the test of IIA to violations that can occur in a specific collection of choice sets (e.g., pairs), one obtains structure-dependent lower bounds that are much less pessimistic. Our analysis of this testing problem is unorthodox in being highly combinatorial, counting Eulerian orientations of cycle decompositions of a particular bipartite graph constructed from a data set of choices. By identifying fundamental relationships between the comparison structure of a given testing problem and its sample efficiency, we hope these relationships will help lay the groundwork for a rigorous rethinking of the IIA testing problem as well as other testing problems in discrete choice.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Yuting Wei and Martin J Wainwright (2016) Sharp minimax bounds for testing discrete monotone distributions. In Information Theory (ISIT), 2016 IEEE International Symposiu… | 1.000 | 5 | 4 | 100% |
| 2 | Liam Paninski (2008) A coincidence-based test for uniformity given very sparsely sampled discrete data | 0.928 | 4 | 4 | 100% |
| 3 | Gregory Valiant and Paul Valiant (2017) An automatic inequality prover and instance optimal identity testing | 0.928 | 4 | 3 | 100% |
| 4 | Jean-Claude Falmagne (1978) A representation theorem for finite random scale systems | 0.843 | 3 | 3 | 100% |
| 5 | Jerry Hausman and Daniel McFadden (1984) Specification tests for the multinomial logit model | 0.811 | 4 | 2 | 100% |
| 6 | Daniel McFadden, William B Tye, and Kenneth Train (1977) An application of diagnostic tests for the independence from irrelevant alternatives property of the multinomial logit model | 0.811 | 4 | 2 | 100% |
| 7 | Constantinos Daskalakis, Gautam Kamath, and John Wright (2018) Which distribution distances are sublinearly testable?. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete… | 0.737 | 3 | 2 | 100% |
| 8 | R.. Ducan Luce (1959) Individual Choice Behavior a Theoretical Analysis | 0.737 | 3 | 2 | 100% |
| 9 | Jayadev Acharya, Constantinos Daskalakis, and Gautam Kamath (2015) Optimal testing for properties of distributions. In Advances in Neural Information Processing Systems. 3591–3599 | 0.644 | 2 | 2 | 100% |
| 10 | Austin R Benson, Ravi Kumar, and Andrew Tomkins (2016) On the relevance of irrelevant alternatives. In Proceedings of the 25th International Conference on World Wide Web. Internationa… | 0.644 | 2 | 2 | 100% |
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