Misspecification in Statistical Models

Last update: 24 Apr 2026 14:34
First version: 24 April 2026

A statistical model is a family of probability distributions, indexed by either a finite vector ("parametric") or an infinite-dimensional object like a function ("nonparametric"). (Yes yes conditional distributions, conditional expectations, etc., if you can make those quibbles you can make the modifications.) The model is well-specified if the true data-generating distribution is somewhere in the family. If the true distribution is not in the family, then the model is mis-specified, at least for that problem.

Obviously there are very few situations where we should think our models are actually well-specified, but mis-specified models are much harder to wrap our heads around.

Parameter estimation in mis-specified models. (The sandwich variance matrix is our friend, but I feel like things need to go beyond that.)
Hypothesis testing and confidence sets for mis-specified models, which feels like it has to tie in to ---
Parameter interpretation for mis-specified models.
Mis-specification testing and model checking.
Specification searches.

Statistics

Halbert White, Estimation, Inference and Specification Analysis

Bruce Lindsay and Liawei Liu, "Model Assessment Tools for a Model False World", Statistical Science 24 (2009): 303--318, arxiv:1010.0304 [Their model-adequacy index is, essentially, the number of samples needed to detect the falsity of the model with some reasonable, pre-set level of power, with fixed size/significance level. This is a very natural quantity. In fact, by results which go back to Kullback's book, the power grows exponentially, with a rate equal to the Kullback-Leibler divergence rate. (More exactly, one minus the power goes to zero exponentially at that rate, but you know what I meant.) Large deviations theory includes generalizations of this result. Many statisticians, I'd guess, would prefer the Lindsay-Liu index because will feel it more natural to them to gauge error in terms of a sample size rather than bits, but to each their own.]

Peter J. Huber
- "On the Non-Optimality of Optimal Procedures", pp. 31--46 in Javier Rojo (ed.), Optimality: The Third Erich L. Lehmann Symposium
- "The behavior of maximum likelihood estimates under nonstandard conditions", Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1 (Univ. of Calif. Press, 1967), pp. 221--233

CRS, "Dynamics of Bayesian Updating with Dependent Data and Misspecified Models", Electronic Journal of Statistics 3 (2009): 1039--1074, arxiv:0901.1342
Sabina J. Sloman, Daniel M. Oppenheimer, Stephen B. Broomell, and CRS, "Characterizing the robustness of Bayesian adaptive experimental designs to active learning bias", arxiv:2205.13698

Cuimin Ba, "Robust Misspecified Models", American Economic Review 116 (2026): 1340--1379
Pierpaolo De Blasi and Stephen G. Walker, "Bayesian Estimation of the Discrepancy with Misspecified Parametric Models", Bayesian Analysis 8 (2013): 781--800
Oscar Kempthorne, "The classical problem of inference--goodness of fit", Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, 235--249
Vladimir N. Minin, John D. O'Brien, Arseni Seregin, "Empirically corrected estimation of complete-data population summaries under model misspecification", arxiv:0911.0930
William Perkins, Mark Tygert, Rachel Ward, "Significance testing without truth", arxiv:1301.1208