scholarly journals Minimax Risk and Uniform Convergence Rates for Nonparametric Dyadic Regression

2021 ◽  
Author(s):  
Bryan S. Graham ◽  
Fengshi Niu ◽  
James L. Powell
Keyword(s):  
Uniform Convergence ◽  
Convergence Rates ◽  
Minimax Risk

scholarly journals ON THE UNIFORM CONVERGENCE OF DECONVOLUTION ESTIMATORS FROM REPEATED MEASUREMENTS

2021 ◽  
pp. 1-22
Author(s):  
Daisuke Kurisu ◽  
Taisuke Otsu
Keyword(s):  
Multivariate Analysis ◽  
Measurement Error ◽  
Uniform Convergence ◽  
Measurement Errors ◽  
Convergence Rates ◽  
Free Variable ◽  
Error Model ◽  
Repeated Measurements ◽  
Empirical Characteristic Function ◽  
Measurement Error Model

This paper studies the uniform convergence rates of Li and Vuong’s (1998, Journal of Multivariate Analysis 65, 139–165; hereafter LV) nonparametric deconvolution estimator and its regularized version by Comte and Kappus (2015, Journal of Multivariate Analysis 140, 31–46) for the classical measurement error model, where repeated noisy measurements on the error-free variable of interest are available. In contrast to LV, our assumptions allow unbounded supports for the error-free variable and measurement errors. Compared to Bonhomme and Robin (2010, Review of Economic Studies 77, 491–533) specialized to the measurement error model, our assumptions do not require existence of the moment generating functions of the square and product of repeated measurements. Furthermore, by utilizing a maximal inequality for the multivariate normalized empirical characteristic function process, we derive uniform convergence rates that are faster than the ones derived in these papers under such weaker conditions.

scholarly journals UNIFORM CONVERGENCE RATES OVER MAXIMAL DOMAINS IN STRUCTURAL NONPARAMETRIC COINTEGRATING REGRESSION

2017 ◽  
Vol 33 (6) ◽  
pp. 1387-1417 ◽  
Author(s):  
James A. Duffy
Keyword(s):  
Regression Model ◽  
Uniform Convergence ◽  
Lower Order ◽  
Unit Root ◽  
Autoregressive Model ◽  
Convergence Rates ◽  
Kernel Regression ◽  
Infinite Variance ◽  
Standard Unit ◽  
Regression Estimators

This paper presents uniform convergence rates for kernel regression estimators, in the setting of a structural nonlinear cointegrating regression model. We generalise the existing literature in three ways. First, the domain to which these rates apply is much wider than the domains that have been considered in the existing literature, and can be chosen so as to contain as large a fraction of the sample as desired in the limit. Second, our results allow the regression disturbance to be serially correlated, and cross-correlated with the regressor; previous work on this problem (of obtaining uniform rates) having been confined entirely to the setting of an exogenous regressor. Third, we permit the bandwidth to be data-dependent, requiring it to satisfy only certain weak asymptotic shrinkage conditions. Our assumptions on the regressor process are consistent with a very broad range of departures from the standard unit root autoregressive model, allowing the regressor to be fractionally integrated, and to have an infinite variance (and even infinite lower-order moments).

scholarly journals ON RATE OPTIMALITY FOR ILL-POSED INVERSE PROBLEMS IN ECONOMETRICS

2010 ◽  
Vol 27 (3) ◽  
pp. 497-521 ◽  
Author(s):  
Xiaohong Chen ◽  
Markus Reiss
Keyword(s):  
Lower Bounds ◽  
Convergence Rates ◽  
Regularity Conditions ◽  
Minimax Risk ◽  
Squared Error ◽  
Minimum Distance Estimator ◽  
Projection Estimator ◽  
Integrated Squared Error ◽  
Ill Posed ◽  
Distance Estimator

In this paper we clarify the relations between the existing sets of regularity conditions for convergence rates of nonparametric indirect regression (NPIR) and nonparametric instrumental variables (NPIV) regression models. We establish minimax risk lower bounds in mean integrated squared error loss for the NPIR and NPIV models under two basic regularity conditions: the approximation number and the link condition. We show that both a simple projection estimator for the NPIR model and a sieve minimum distance estimator for the NPIV model can achieve the minimax risk lower bounds and are rate optimal uniformly over a large class of structure functions, allowing for mildly ill-posed and severely ill-posed cases.

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