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.

scholarly journals Parameter Estimation for Stochastic Differential Equations Driven by Mixed Fractional Brownian Motion

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Na Song ◽  
Zaiming Liu
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Fractional Brownian Motion ◽  
Asymptotic Properties ◽  
Regularity Conditions ◽  
Minimum Distance Estimator ◽  
Mixed Fractional Brownian Motion ◽  
Distance Estimator ◽  
Drift Parameter

We study the asymptotic properties of minimum distance estimator of drift parameter for a class of nonlinear scalar stochastic differential equations driven by mixed fractional Brownian motion. The consistency and limit distribution of this estimator are established as the diffusion coefficient tends to zero under some regularity conditions.

scholarly journals On Hybridizations of Fourth Order Kernel of the Beta Polynomial Family.

2019 ◽  
pp. 819-829
Author(s):  
Israel Uzuazor SILOKO ◽  
Osayomore IKPOTOKIN ◽  
Edith Akpevwe SILOKO
Keyword(s):  
Fourth Order ◽  
Convergence Rates ◽  
Higher Order ◽  
Kernel Functions ◽  
Mean Integrated Squared Error ◽  
Squared Error ◽  
Integrated Squared Error ◽  
Nonparametric Kernel ◽  
Order Kernel ◽  
Polynomial Family

The usual second order nonparametric kernel estimators are of wide uses in data analysis and visualization but constrained with slow convergence rate. Higher order kernels provide a faster convergence rates and are known to be bias reducing kernels. In this paper, we propose a hybrid of the fourth order kernel which is a merger of two successive fourth order kernels and the statistical properties of these hybrid kernels were study. The results of our simulation reveals that the proposed higher order hybrid kernels outperformed their corresponding parent’s kernel functions using the asymptotic mean integrated squared error.

scholarly journals Convergence Rates of First- and Higher-Order Dynamics for Solving Linear Ill-Posed Problems

2021 ◽  
Author(s):  
Radu Boţ ◽  
Guozhi Dong ◽  
Peter Elbau ◽  
Otmar Scherzer
Keyword(s):  
Linear Operator ◽  
Operator Equation ◽  
Convex Analysis ◽  
Convergence Rates ◽  
Linear Operator Equation ◽  
Optimal Convergence Rates ◽  
Ill Posed ◽  
Thresholding Algorithm ◽  
The Given ◽  
Theoretical Results

AbstractRecently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov’s algorithm and the fast iterative shrinkage-thresholding algorithm, respectively. In this paper, we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residual of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. However, in the restricted situation of this paper they can often be significantly improved. Moreover, since we show that the proposed flows for minimising the norm of the residual of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions, the given rates can be considered the benchmarks for further studies in convex analysis.

scholarly journals Transmetric Density Estimation

2016 ◽  
Vol 5 (2) ◽  
pp. 35
Author(s):  
Sigve Hovda
Keyword(s):  
Density Estimation ◽  
Kernel Density Estimation ◽  
Kernel Density ◽  
Convergence Order ◽  
Graphical Display ◽  
Mean Integrated Squared Error ◽  
Squared Error ◽  
Integrated Squared Error ◽  
The Mean ◽  
Convergence Orders

<div>A transmetric is a generalization of a metric that is tailored to properties needed in kernel density estimation.  Using transmetrics in kernel density estimation is an intuitive way to make assumptions on the kernel of the distribution to improve convergence orders and to reduce the number of dimensions in the graphical display.  This framework is required for discussing the estimators that are suggested by Hovda (2014).</div><div> </div><div>Asymptotic arguments for the bias and the mean integrated squared error is difficult in the general case, but some results are given when the transmetric is of the type defined in Hovda (2014).  An important contribution of this paper is that the convergence order can be as high as $4/5$, regardless of the number of dimensions.</div>

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