Characterization of the least squares estimator: Mis-specified multivariate isotonic regression model with dependent errors

Bagchi, Pramita; Dhar, Subhra

doi:10.1090/tpms/1210

Authors: Pramita Bagchi and Subhra Sankar Dhar
Journal: Theor. Probability and Math. Statist. 110 (2024), 143-158
MSC (2020): Primary 62G08, 62G05, 60B10
DOI: https://doi.org/10.1090/tpms/1210
Published electronically: May 10, 2024
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This article investigates some nice properties of the least squares estimator of multivariate isotonic regression function (denoted as LSEMIR), when the model is mis-specified, and the errors are $\beta$-mixing stationary random variables. Under mild conditions, it is observed that the least squares estimator converges uniformly to a certain monotone function, which is closest to the original function in an appropriate sense.

References

Jason Abrevaya and Jian Huang, On the bootstrap of the maximum score estimator, Econometrica 73 (2005), no. 4, 1175–1204. MR 2149245, DOI 10.1111/j.1468-0262.2005.00613.x
D. Anevski and O. Hössjer, A general asymptotic scheme for inference under order restrictions, Ann. Statist. 34 (2006), no. 4, 1874–1930. MR 2283721, DOI 10.1214/009053606000000443
Pramita Bagchi, Moulinath Banerjee, and Stilian A. Stoev, Inference for monotone functions under short- and long-range dependence: confidence intervals and new universal limits, J. Amer. Statist. Assoc. 111 (2016), no. 516, 1634–1647. MR 3601723, DOI 10.1080/01621459.2015.1100622
Pramita Bagchi and Subhra Sankar Dhar, A study on the least squares estimator of multivariate isotonic regression function, Scand. J. Stat. 47 (2020), no. 4, 1192–1221. MR 4178191, DOI 10.1111/sjos.12459
Moulinath Banerjee and Jon A. Wellner, Likelihood ratio tests for monotone functions, Ann. Statist. 29 (2001), no. 6, 1699–1731. MR 1891743, DOI 10.1214/aos/1015345959
R. E. Barlow, Statistical inference under order restrictions; the theory and application of isotonic regression, Tech. report, 1972.
R. Berk, L. Brown, A. Buja, E. George, and L. Zhao, Working with misspecified regression models, Journal of Quantitative Criminology 34 (2018), 633–655.
Michael J. Best and Nilotpal Chakravarti, Active set algorithms for isotonic regression; a unifying framework, Math. Programming 47 (1990), no. 3, (Ser. A), 425–439. MR 1068274, DOI 10.1007/BF01580873
F. Boussama, Ergodicity, mixing and estimation in garch models, Unpublished Ph. D. Dissertation, University of Paris 7 (1998).
H. D. Brunk, Conditional expectation given a $\sigma$-lattice and applications, Ann. Math. Statist. 36 (1965), 1339–1350. MR 185629, DOI 10.1214/aoms/1177699895
Sabyasachi Chatterjee, Adityanand Guntuboyina, and Bodhisattva Sen, On matrix estimation under monotonicity constraints, Bernoulli 24 (2018), no. 2, 1072–1100. MR 3706788, DOI 10.3150/16-BEJ865
K. Chatzikokolakis and K. Martin, A monotonicity principle for information theory, Electron. Notes Theor. Comput. Sci. 218 (2008), 111–129.
Ju. A. Davydov, Mixing conditions for Markov chains, Teor. Verojatnost. i Primenen. 18 (1973), 321–338 (Russian, with English summary). MR 321183
Subhra Sankar Dhar, Trimmed mean isotonic regression, Scand. J. Stat. 43 (2016), no. 1, 202–212. MR 3467002, DOI 10.1111/sjos.12173
G. Ellison and S. Ellison, Strategic entry deterrence and the behavior of pharmaceutical incumbents prior to patent expiration, American Economic Journal: Microeconomics (2011), 1–36.
Arnaud Guyader, Nick Hengartner, Nicolas Jégou, and Eric Matzner-Løber, Iterative isotonic regression, ESAIM Probab. Stat. 19 (2015), 1–23. MR 3374866, DOI 10.1051/ps/2014012
Qiyang Han, Tengyao Wang, Sabyasachi Chatterjee, and Richard J. Samworth, Isotonic regression in general dimensions, Ann. Statist. 47 (2019), no. 5, 2440–2471. MR 3988762, DOI 10.1214/18-AOS1753
L. Hoffmann, Multivariate isotonic regression and its algorithms, Ph.D. thesis, Wichita State University, 2009.
Ming-Hong Liu and Vasant A. Ubhaya, Integer isotone optimization, SIAM J. Optim. 7 (1997), no. 4, 1152–1159. MR 1479619, DOI 10.1137/S1052623494272302
Gary G. Makowski, Consistency of an estimator of doubly nondecreasing regression functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 39 (1977), no. 4, 263–268. MR 652723, DOI 10.1007/BF01877494
Rosa L. Matzkin, Restrictions of economic theory in nonparametric methods, Handbook of econometrics, Vol. IV, Handbooks in Econom., vol. 2, North-Holland, Amsterdam, 1994, pp. 2523–2558. MR 1315977
D. L. McLeish, A maximal inequality and dependent strong laws, Ann. Probability 3 (1975), no. 5, 829–839. MR 400382, DOI 10.1214/aop/1176996269
Paul Milgrom and Chris Shannon, Monotone comparative statics, Econometrica 62 (1994), no. 1, 157–180. MR 1258667, DOI 10.2307/2951479
Abdelkader Mokkadem, Mixing properties of ARMA processes, Stochastic Process. Appl. 29 (1988), no. 2, 309–315. MR 958507, DOI 10.1016/0304-4149(88)90045-2
E. T. Moolchan, D. L. Hudson, J. R. Schroeder, and S. S. Sehnert, Heart rate and blood pressure responses to tobacco smoking among african-american adolescents, Journal of the National Medical Association 96 (2004), no. 6, 767.
W. F. Ramirez, Computational methods for process simulation, Elsevier Science, Oxford, 1989.
Emmanuel Rio, Asymptotic theory of weakly dependent random processes, Probability Theory and Stochastic Modelling, vol. 80, Springer, Berlin, 2017. Translated from the 2000 French edition [ MR2117923]. MR 3642873, DOI 10.1007/978-3-662-54323-8
Tim Robertson and F. T. Wright, Multiple isotonic median regression, Ann. Statist. 1 (1973), 422–432. MR 378224
Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
Ou Zhao and Michael Woodroofe, Estimating a monotone trend, Statist. Sinica 22 (2012), no. 1, 359–378. MR 2933180, DOI 10.5705/ss.2009.153