Mean Square Convergence Rates for Maximum Quasi-Likelihood Estimators

Published Online:https://doi.org/10.1287/12-SSY086

References

  • Anderson, T. W. and Taylor, J. B., Some experimental results on the statistical properties of least squares estimates in control problems. Econometrica, 44(6): 1289–1302, 1976.Google Scholar
  • Araman, V. F. and Caldentey, R., Revenue Management with Incomplete Demand Information. In J. J. Cochran, editor, Encyclopedia of Operations Research. Wiley, 2011.Google Scholar
  • Bartlett, M. S., An inverse matrix adjustment arising in discriminant analysis. The Annals of Mathematical Statistics, 22(1): 107–111, 1951. MR0040068Google Scholar
  • Besbes, O. and Zeevi, A., Dynamic pricing without knowing the demand function: risk bounds and near-optimal algorithms. Operations Research, 57(6): 1407–1420, 2009. MR2597918LinkGoogle Scholar
  • Bhatia, R., Positive Definite Matrices. Princeton University Press, Princeton, 2007. MR2284176Google Scholar
  • Broder, J. and Rusmevichientong, P., Dynamic pricing under a general parametric choice model. Operations Research, 60(4): 965–980, 2012. MR2979434LinkGoogle Scholar
  • Chang, Y. I., Strong consistency of maximum quasi-likelihood estimate in generalized linear models via a last time. Statistics & Probability Letters, 45(3): 237–246, 1999. MR1718035Google Scholar
  • Chen, K., Hu, I., and Ying, Z., Strong consistency of maximum quasi-likelihood estimators in generalized linear models with fixed and adaptive designs. The Annals of Statistics, 27(4): 1155–1163, 1999. MR1740117Google Scholar
  • Chow, Y. S. and Teicher, H., Probability Theory: Independence, Interchangeability, Martingales. Springer Verlag, New York, third edition, 2003.Google Scholar
  • de la Peña, V. H., Lai, T. L., and Shao, Q. M., Self-Normalized Processes: Limit Theory and Statistical Applications. Springer Series in Probability and its Applications. Springer, New York, first edition, 2009. MR2488094Google Scholar
  • den Boer, A. V., Dynamic pricing with multiple products and partially specified demand distribution. Mathematics of Operations Research, Forthcoming, 2013.Google Scholar
  • den Boer, A. V. and Zwart, B., Simultaneously learning and optimizing using controlled variance pricing. Management Science, Forthcoming, 2013.Google Scholar
  • Dugundji, J., Topology. Allyn and Bacon, Boston, 1966. MR0193606Google Scholar
  • Fahrmeir, L. and Kaufmann, H., Consistency and asymptotic normality of the maximum likelihood estimator in generalized linear models. The Annals of Statistics, 13(1): 342–368, 1985. MR0773172Google Scholar
  • Gill, J., Generalized Linear Models: A Unified Approach. Sage Publications, Thousand Oaks, CA, 2001.Google Scholar
  • Goldenshluger, A. and Zeevi, A., Woodroofe’s one-armed bandit problem revisited. The Annals of Applied Probability, 19(4): 1603–1633, 2009. MR2538082Google Scholar
  • Heyde, C. C., Quasi-Likelihood and Its Application. Springer Series in Statistics. Springer Verlag, New York, 1997. MR1461808Google Scholar
  • Keskin, N. B. and Zeevi, A., Dynamic pricing with an unknown linear demand model: asymptotically optimal semi-myopic policies. Working paper, University of Chicago, http://faculty.chicagobooth.edu/bora.keskin/pdfs/DynamicPricingUnknownDemandModel.pdf, 2013.Google Scholar
  • Lai, T. L. and Robbins, H., Iterated least squares in multiperiod control. Advances in Applied Mathematics, 3(1): 50–73, 1982. MR0646499Google Scholar
  • Lai, T. L. and Wei, C. Z., Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems. The Annals of Statistics, 10(1): 154–166, 1982. MR0642726Google Scholar
  • Leray, J. and Schauder, J., Topologie et equations fonctionelles. Annales Scientifiques de l’École Normale Supérieure, 51: 45–78, 1934. MR1509338Google Scholar
  • Loève, M., Probability Theory I. Springer Verlag, New York, Berlin, Heidelberg, 4th edition, 1977a. MR0651017Google Scholar
  • Loève, M., Probability Theory II. Springer Verlag, New York, Berlin, Heidelberg, 4th edition, 1977b. MR0651017Google Scholar
  • McCullagh, P., Quasi-likelihood functions. The Annals of Statistics, 11(1): 59–67, 1983. MR0684863Google Scholar
  • McCullagh, P. and Nelder, J. A., Generalized Linear Models. Chapman & Hall, London, 1983. MR0727836Google Scholar
  • Nelder, J. A. and Wedderburn, R. W. M., Generalized linear models. Journal of the Royal Statistical Society, Series A (General), 135(3): 370–384, 1972.Google Scholar
  • Ortega, J. M. and Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables, volume 30 of SIAM’s Classics in Applied Mathematics. Society for, Industrial and Applied Mathematics, Philadelphia, 2000. MR1744713Google Scholar
  • Pronzato, L., Optimal experimental design and some related control problems. Automatica, 44(2): 303–325, 2008. MR2530779Google Scholar
  • Rusmevichientong, P. and Tsitsiklis, J. N., Linearly parameterized bandits. Mathematics of Operations Research, 35(2): 395–411, 2010. MR2674726LinkGoogle Scholar
  • Small, C. G., Wang, J., and Yang, Z., Eliminating multiple root problems in estimation. Statistical Science, 15(4): 313–332, 2000. MR1819708Google Scholar
  • Spătaru, A., Improved convergence rates for tail probabilities. Bulletin of the Transilvania University of Brasov – Series III: Mathematics, Informatics, Physics, 2(51): 137–142, 2009. MR2642502Google Scholar
  • Stoica, G., Baum-Katz-Nagaev type results for martingales. Journal of Mathematical Analysis and Applications, 336(2): 1489–1492, 2007. MR2353031Google Scholar
  • Tzavelas, G., A note on the uniqueness of the quasi-likelihood estimator. Statistics & Probability Letters, 38(2): 125–130, 1998. MR1627914Google Scholar
  • Wedderburn, R. W. M., Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method. Biometrika, 61(3): 439–447, 1974. MR0375592Google Scholar
  • Yin, C., Zhang, H., and Zhao, L., Rate of strong consistency of maximum quasi-likelihood estimator in multivariate generalized linear models. Communications in Statistics – Theory and Methods, 37(19): 3115–3123, 2008. MR2467755Google Scholar
  • Yue, L. and Chen, X., Rate of strong consistency of quasi maximum likelihood estimate in generalized linear models. Science in China Series A: Mathematics, 47(6): 882–893, 2004. MR2127216Google Scholar
  • Zhang, S. and Liao, Y., On some problems of weak consistency of quasi-maximum likelihood estimates in generalized linear models. Science in China Series A: Mathematics, 51(7): 1287–1296, 2008. MR2417495Google Scholar
  • Zhang, S., Liao, Y., and Ning, W., Asymptotic properties of quasi-maximum likelihood estimates in generalized linear models. Communications in Statistics – Theory and Methods, 40(24): 4417–4430, 2011. MR2864166Google Scholar
  • Zhu, C. and Gao, Q., Asymptotic properties in generalized linear models with natural link function and adaptive designs. Advances in Mathematics (China), 42(1): 121–127, 2013. MR3098890Google Scholar
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