Asymptotic normality of kernel estimators of the conditional mode under strong mixing hypothesis

1999 ◽  
Vol 11 (4) ◽  
pp. 413-442 ◽  
Author(s):  
Djamal Louani ◽  
ELIAS Ould-saïd
Keyword(s):  
Asymptotic Normality ◽  
Strong Mixing ◽  
Kernel Estimators ◽  
Conditional Mode

scholarly journals Asymptotic Distributions of M-Estimates for Parameters of Multivariate Time Series with Strong Mixing Property

2021 ◽  
Vol 5 (1) ◽  
pp. 19
Author(s):  
Alexander Kushnir ◽  
Alexander Varypaev
Keyword(s):  
Time Series ◽  
Asymptotic Normality ◽  
Multivariate Time Series ◽  
Asymptotic Properties ◽  
Asymptotic Distributions ◽  
Stationary Time Series ◽  
Strong Mixing ◽  
Statistical Estimates ◽  
Distribution Parameters ◽  
M Estimates

The publication is devoted to studying asymptotic properties of statistical estimates of the distribution parameters u∈Rq of a multidimensional random stationary time series zt∈Rm, t∈ℤ satisfying the strong mixing conditions. We consider estimates u^nδ(z¯n), z¯n=(z1T,…,znT)T∈Rmn that provide in asymptotic n→∞ the maximum values for some objective functions Qn(z¯n;u), which have properties similar to the well-known property of local asymptotic normality. These estimates are constructed by solving the equations δn(z¯n;u)=0, where δn(z¯n;u) are arbitrary functions for which δn(z¯n;u)−gradhQn(z¯n;u+n−1/2h)→0(n→∞) in Pn,u(z¯n)-probability uniformly on u∈U, were U is compact in Rq. In many cases, the estimates u^nδ(z¯n) have the same asymptotic properties as well-known M-estimates defined by equations u^nQ(z¯n)=arg maxu∈UQn(z¯n;u) but often can be much simpler computationally. We consider an algorithmic method for constructing estimates u^nδ(z¯n), which is similar to the accumulation method first proposed by R. Fischer and rigorously developed by L. Le Cam. The main theoretical result of the article is the proof of the theorem, in which conditions of the asymptotic normality of estimates u^nδ(z¯n) are formulated, and the expression is proposed for their matrix of asymptotic mean-square deviations limn→∞nEn,u{(u^δ(z¯n)−u)(u^δ(z¯n)−u)T}.

Nonparametric Methods for α-Mixing Functional Random Variables

2018 ◽  
Author(s):  
Laurent Delsol
Keyword(s):  
Kernel Smoothing ◽  
Asymptotic Properties ◽  
Strong Mixing ◽  
Kernel Estimators ◽  
Mixing Conditions ◽  
Exponential Inequalities ◽  
Smoothing Methods ◽  
Mixing Coefficients ◽  
Mixing Sequences ◽  
Prediction Problems

This article considers how functional kernel methods can be used to study α-mixing datasets. It first provides an overview of how prediction problems involving dependent functional datasets may arise from the study of time series, focusing on the standard discretized model and modelization that takes into account the functional nature of the evolution of the quantity to be studied over time. It then considers strong mixing conditions, with emphasis on the notion of α-mixing coefficients and α-mixing variables introduced by Rosenblatt (1956). It also describes some conditions for a Markov chain to be α-mixing; some useful tools that provide covariance inequalities, exponential inequalities, and Central Limit Theorem (CLT) for α-mixing sequences; the asymptotic properties of functional kernel estimators; the use of kernel smoothing methods with α-mixing datasets; and various functional kernel estimators corresponding to different prediction methods. Finally, the article highlights some interesting prospects for further research.

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