Uniform Convergence Rates for Nonparametric Estimators of a Density Function and its Derivatives when the Density has a Known Pole

2021 ◽  
Author(s):  
Sorawoot Srisuma
Keyword(s):  
Uniform Convergence ◽  
Density Function ◽  
Convergence Rates ◽  
Nonparametric Estimators

UNIFORM CONVERGENCE RATES OF KERNEL-BASED NONPARAMETRIC ESTIMATORS FOR CONTINUOUS TIME DIFFUSION PROCESSES: A DAMPING FUNCTION APPROACH

2016 ◽  
Vol 33 (4) ◽  
pp. 874-914 ◽  
Author(s):  
Shin Kanaya
Keyword(s):  
Uniform Convergence ◽  
Continuous Time ◽  
Convergence Rates ◽  
Diffusion Processes ◽  
Kernel Functions ◽  
Damping Function ◽  
Nonparametric Estimators ◽  
Empirical Process Theory ◽  
Convergence Results ◽  
The Moment

In this paper, we derive uniform convergence rates of nonparametric estimators for continuous time diffusion processes. In particular, we consider kernel-based estimators of the Nadaraya–Watson type, introducing a new technical device called adamping function. This device allows us to derive sharp uniform rates over an infinite interval with minimal requirements on the processes: The existence of the moment of any order is not required and the boundedness of relevant functions can be significantly relaxed. Restrictions on kernel functions are also minimal: We allow for kernels with discontinuity, unbounded support, and slowly decaying tails. Our proofs proceed by using the covering-number technique from empirical process theory and exploiting the mixing and martingale properties of the processes. We also present new results on the path-continuity property of Brownian motions and diffusion processes over an infinite time horizon. These path-continuity results, which should also be of some independent interest, are used to control discretization biases of the nonparametric estimators. The obtained convergence results are useful for non/semiparametric estimation and testing problems of diffusion processes.

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).

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