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.

UNIFORM CONVERGENCE RATES OF KERNEL ESTIMATORS WITH HETEROGENEOUS DEPENDENT DATA

2009 ◽  
Vol 25 (5) ◽  
pp. 1433-1445 ◽  
Author(s):  
Dennis Kristensen
Keyword(s):  
Uniform Convergence ◽  
Convergence Rates ◽  
Dependent Data ◽  
Regression Estimation ◽  
Time Varying ◽  
Convergence Results ◽  
Inhomogeneous Models ◽  
Estimation Problems ◽  
Continuous Time Processes ◽  
Kernel Regression Estimation

The main uniform convergence results of Hansen (2008,Econometric Theory24, 726–748) are generalized in two directions: Data are allowed to (a) be heterogeneously dependent and (b) depend on a (possibly unbounded) parameter. These results are useful in semiparametric estimation problems involving time-inhomogeneous models and/or sampling of continuous-time processes. The usefulness of these results is demonstrated by two applications: kernel regression estimation of a time-varying AR(1) model and the kernel density estimation of a Markov chain that has not been initialized at its stationary distribution.

scholarly journals Period-Life of a Branching Process with Migration and Continuous Time

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 868
Author(s):  
Khrystyna Prysyazhnyk ◽  
Iryna Bazylevych ◽  
Ludmila Mitkova ◽  
Iryna Ivanochko
Keyword(s):  
Random Process ◽  
Generating Function ◽  
Continuous Time ◽  
Branching Process ◽  
Probability Generating Function ◽  
Time Interval ◽  
The Moment ◽  
First Time ◽  
Critical Branching Process ◽  
Number Of Particles

The homogeneous branching process with migration and continuous time is considered. We investigated the distribution of the period-life τ, i.e., the length of the time interval between the moment when the process is initiated by a positive number of particles and the moment when there are no individuals in the population for the first time. The probability generating function of the random process, which describes the behavior of the process within the period-life, was obtained. The boundary theorem for the period-life of the subcritical or critical branching process with migration was found.

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 Spectral methods for Langevin dynamics and associated error estimates

2018 ◽  
Vol 52 (3) ◽  
pp. 1051-1083 ◽  
Author(s):  
Julien Roussel ◽  
Gabriel Stoltz
Keyword(s):  
Convergence Rates ◽  
Langevin Dynamics ◽  
Galerkin Methods ◽  
Discrete Level ◽  
Rigidity Matrix ◽  
Poisson Equations ◽  
The Poisson Equation ◽  
Convergence Results ◽  
Abstract Setting ◽  
Tensor Basis

We prove the consistency of Galerkin methods to solve Poisson equations where the differential operator under consideration is hypocoercive. We show in particular how the hypocoercive nature of the generator associated with Langevin dynamics can be used at the discrete level to first prove the invertibility of the rigidity matrix, and next provide error bounds on the approximation of the solution of the Poisson equation. We present general convergence results in an abstract setting, as well as explicit convergence rates for a simple example discretized using a tensor basis. Our theoretical findings are illustrated by numerical simulations.

scholarly journals CONVERGENCE RATES OF SUMS OF α-MIXING TRIANGULAR ARRAYS: WITH AN APPLICATION TO NONPARAMETRIC DRIFT FUNCTION ESTIMATION OF CONTINUOUS-TIME PROCESSES

2016 ◽  
Vol 33 (5) ◽  
pp. 1121-1153
Author(s):  
Shin Kanaya
Keyword(s):  
Continuous Time ◽  
Convergence Rates ◽  
Moving Average ◽  
Function Estimation ◽  
Triangular Arrays ◽  
Large Numbers ◽  
Strongly Mixing ◽  
Time Distance ◽  
Continuous Time Processes ◽  
Near Unit Root

The convergence rates of the sums of α-mixing (or strongly mixing) triangular arrays of heterogeneous random variables are derived. We pay particular attention to the case where central limit theorems may fail to hold, due to relatively strong time-series dependence and/or the nonexistence of higher-order moments. Several previous studies have presented various versions of laws of large numbers for sequences/triangular arrays, but their convergence rates were not fully investigated. This study is the first to investigate the convergence rates of the sums of α-mixing triangular arrays whose mixing coefficients are permitted to decay arbitrarily slowly. We consider two kinds of asymptotic assumptions: one is that the time distance between adjacent observations is fixed for any sample size n; and the other, called the infill assumption, is that it shrinks to zero as n tends to infinity. Our convergence theorems indicate that an explicit trade-off exists between the rate of convergence and the degree of dependence. While the results under the infill assumption can be seen as a direct extension of those under the fixed-distance assumption, they are new and particularly useful for deriving sharper convergence rates of discretization biases in estimating continuous-time processes from discretely sampled observations. We also discuss some examples to which our results and techniques are useful and applicable: a moving-average process with long lasting past shocks, a continuous-time diffusion process with weak mean reversion, and a near-unit-root process.

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