Hostname: page-component-f7d5f74f5-wqfsk Total loading time: 0 Render date: 2023-10-05T03:44:52.853Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "coreDisableSocialShare": false, "coreDisableEcommerceForArticlePurchase": false, "coreDisableEcommerceForBookPurchase": false, "coreDisableEcommerceForElementPurchase": false, "coreUseNewShare": true, "useRatesEcommerce": true } hasContentIssue false


Published online by Cambridge University Press:  29 January 2014

Kyungchul Song*
University of British Columbia
*Address Correspondence to Kyungchul Song, Department of Economics, University of British Columbia, 997-1873 East Mall, Vancouver, BC, Canada, V6T 1Z1; e-mail:


This paper considers a decision maker who prefers to make a point decision when the object of interest is interval-identified with regular bounds. When the bounds are just identified along with known interval length, the local asymptotic minimax decision with respect to a symmetric convex loss function takes an obvious form: an efficient lower bound estimator plus the half of the known interval length. However, when the interval length or any nontrivial upper bound for the length is not known, the minimax approach suffers from triviality because the maximal risk is associated with infinitely long identified intervals. In this case, this paper proposes a local asymptotic minimax regret approach and shows that the midpoint between semiparametrically efficient bound estimators is optimal.

Copyright © Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)



Ball, K. (1997) An elementary introduction to modern convex geometry. In Levy, S., (ed.) Flavors of Geometry, MSRI Publications 31, pp. 158. Cambridge University Press.Google Scholar
Berger, J.O. (1985) Statistical Decision Theory and Bayesian Analysis. Springer-Verlag.CrossRefGoogle Scholar
Bickel, P.J., Klaassen, A.J., Ritov, Y., and Wellner, J.A. (1993) Efficient and Adaptive Estimation for Semiparametric Models, Springer Verlag.
Blackwell, D. (1951) Comparison of experiments. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 93102. University of California Press.Google Scholar
Chow, Y.S. & Teicher, H. (2003) Probability Theory: Independence, Interchangeability, Martingales. 3rd ed. Springer-Verlag.Google Scholar
Fan, Y. & Park, S. (2009) Partial identification of the distribution of treatment effects and its confidence sets (with Sang Soo Park). In Fomby, T.B. & Hill, R.C. (eds.), Nonparametric Econometric Methods, (Advances in Econometrics 25) pp. 370. Emerald Group Publishing.CrossRefGoogle Scholar
Fukuda, K. & Uno, T. (2007) Polynomial time algorithms for maximizing the intersection volume of polytopes. Pacific Journal of Optimization 3, 3752.Google Scholar
Hájek, J. (1972) Local asymptotic minimax and admissibility in estimation. In Le Cam, L., Neyman, J., & Scott, E.L. (eds.), Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 175194. University of California Press.Google Scholar
Hirano, K. & Porter, J. (2009) Asymptotics for statistical treatment rules. Econometrica 77, 16831701.Google Scholar
Horowitz, J. & Manski, C.F. (1998) Censoring of outcomes and regressors due to survey nonresponse: Identification and estimation using weights and imputations. Journal of Econometrics 84, 3758.CrossRefGoogle Scholar
Imbens, G. & Manski, C. (2004) Confidence intervals for partially identified parameters. Econometrica 72, 18451857.CrossRefGoogle Scholar
Kitagawa, T. (2012) Estimation and Inference for Set-Identified Parameters Using Posterior Lower Probability. Working paper, University College London.
Le Cam, L. (1960) Locally asymptotically normal families of distributions. University of California Publications in Statistics 3, 332.Google Scholar
Le Cam, L. (1979) On a theorem of J. Hájek. In Jurečková, J. (ed.), Contributions to Statistics—Hájek Memorial Volume, pp. 119135. Akademian.CrossRefGoogle Scholar
Manski, C.F. (1990) Nonparametric bounds on treatment effects. American Economic Review: Papers and Proceedings 80, 319323.Google Scholar
Manski, C.F. (2003) Partial Identification of Probability Distributions. Springer-Verlag.Google Scholar
Manski, C.F. (2004) Statistical treatment rules for heterogeneous populations. Econometrica 72, 12211246.CrossRefGoogle Scholar
Molinari, F. (2010) Missing treatments. Journal of Business & Economic Statistics 28, 8295.CrossRefGoogle Scholar
Stoye, J. (2009a) More on confidence intervals for partially identified models. Econometrica 77, 12991315.Google Scholar
Stoye, J. (2009b) Minimax regret treatment choice with finite samples. Journal of Econometrics 151, 7081.CrossRefGoogle Scholar
Strasser, H. (1985) Mathematical Theory of Statistics. Walter de Gruyter.CrossRefGoogle Scholar
Tetenov, A. (2012a) Statistical Treatment Choice Based on Asymmetric Minmax Regret Criteria. Journal of Econometrics 166, 157165.CrossRefGoogle Scholar
Tetenov, A. (2012b) Measuring Precision of Statistical Inference on Partially Identified Parameters. Working paper. Collegio Carlo Alberto.
van der Vaart, A.W. (1989) On the asymptotic information bound. Annals of Statistics 17, 14871500.CrossRefGoogle Scholar
van der Vaart, A.W. (1991) On differentiable functionals. Annals of Statistics 19, 178204.CrossRefGoogle Scholar
van der Vaart, A.W. (1998) Asymptotic Statistics. Cambridge University Press.CrossRefGoogle Scholar
van der Vaart, A.W. & Wellner, J.A. (1996) Weak Convergence and Empirical Processes. Springer-Verlag.CrossRefGoogle Scholar
Winter, B.B. (1975) A portmanteau theorem for vague convergence. Studia Scientiarum Mathematicarum Hungarica 10, 247253.Google Scholar