scholarly journals Asymptotic Behaviour of the Empirical Distance Covariance for Dependent Data

2021 ◽  
Author(s):  
Marius Kroll
Keyword(s):  
Asymptotic Distribution ◽  
Metric Spaces ◽  
General Theorem ◽  
Almost Sure Convergence ◽  
Dependent Data ◽  
Strong Mixing ◽  
Mixing Condition ◽  
Asymptotic Results ◽  
Absolute Regularity ◽  
Distance Covariance

AbstractWe give two asymptotic results for the empirical distance covariance on separable metric spaces without any iid assumption on the samples. In particular, we show the almost sure convergence of the empirical distance covariance for any measure with finite first moments, provided that the samples form a strictly stationary and ergodic process. We further give a result concerning the asymptotic distribution of the empirical distance covariance under the assumption of absolute regularity of the samples and extend these results to certain types of pseudometric spaces. In the process, we derive a general theorem concerning the asymptotic distribution of degenerate V-statistics of order 2 under a strong mixing condition.

Asymptotic for LS estimators in the EV regression model for dependent errors

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4845-4856
Author(s):  
Konrad Furmańczyk
Keyword(s):  
Regression Model ◽  
Functional Dependence ◽  
Dependent Data ◽  
Errors In Variables ◽  
Mixing Conditions ◽  
Asymptotic Results ◽  
Dependence Measure ◽  
Wide Range ◽  
Ev Regression Model ◽  
Linear And Nonlinear

We study consistency and asymptotic normality of LS estimators in the EV (errors in variables) regression model under weak dependent errors that involve a wide range of linear and nonlinear time series. In our investigations we use a functional dependence measure of Wu [16]. Our results without mixing conditions complete the known asymptotic results for independent and dependent data obtained by Miao et al. [7]-[10].

INDEFINITE KERNEL NETWORK WITH DEPENDENT SAMPLING

2013 ◽  
Vol 11 (05) ◽  
pp. 1350020 ◽  
Author(s):  
HONGWEI SUN ◽  
QIANG WU
Keyword(s):  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Function Class ◽  
Positive Definiteness ◽  
Reproducing Kernel Hilbert Spaces ◽  
Strong Mixing ◽  
Mixing Condition ◽  
Indefinite Kernel ◽  
Learning Rates ◽  
Kernel Hilbert Spaces

We study the asymptotical properties of indefinite kernel network with coefficient regularization and dependent sampling. The framework under investigation is different from classical kernel learning. Positive definiteness is not required by the kernel function and the samples are allowed to be weakly dependent with the dependence measured by a strong mixing condition. By a new kernel decomposition technique introduced in [27], two reproducing kernel Hilbert spaces and their associated kernel integral operators are used to characterize the properties and learnability of the hypothesis function class. Capacity independent error bounds and learning rates are deduced.

The asymptotic distribution of the Net Benefit estimator in presence of right-censoring

2021 ◽  
pp. 096228022110370
Author(s):  
Brice Ozenne ◽  
Esben Budtz-Jørgensen ◽  
Julien Péron
Keyword(s):  
Asymptotic Distribution ◽  
Nuisance Parameter ◽  
Real Data ◽  
R Package ◽  
Right Censoring ◽  
Drop Out ◽  
Net Benefit ◽  
Asymptotic Results ◽  
Finite Samples ◽  
Benefit Risk Assessment

The benefit–risk balance is a critical information when evaluating a new treatment. The Net Benefit has been proposed as a metric for the benefit–risk assessment, and applied in oncology to simultaneously consider gains in survival and possible side effects of chemotherapies. With complete data, one can construct a U-statistic estimator for the Net Benefit and obtain its asymptotic distribution using standard results of the U-statistic theory. However, real data is often subject to right-censoring, e.g. patient drop-out in clinical trials. It is then possible to estimate the Net Benefit using a modified U-statistic, which involves the survival time. The latter can be seen as a nuisance parameter affecting the asymptotic distribution of the Net Benefit estimator. We present here how existing asymptotic results on U-statistics can be applied to estimate the distribution of the net benefit estimator, and assess their validity in finite samples. The methodology generalizes to other statistics obtained using generalized pairwise comparisons, such as the win ratio. It is implemented in the R package BuyseTest (version 2.3.0 and later) available on Comprehensive R Archive Network.

Almost sure relative stability of the maximum of a stationary sequence

2003 ◽  
Vol 35 (03) ◽  
pp. 721-736
Author(s):  
Philippe Naveau
Keyword(s):  
Relative Stability ◽  
Almost Sure Convergence ◽  
Stationary Sequence ◽  
Asymptotic Independence ◽  
Necessary Condition ◽  
Mixing Condition ◽  
Tail Distribution

The almost sure convergence of the maximum of a strictly stationary sequence is studied. We show that, if a particular mixing condition, related to the asymptotic independence of maxima defined by O'Brien, is satisfied, then the maximum behaves asymptotically like the quantile F ←(1-1/n) whenever the marginal tail distribution decreases quickly enough. A necessary condition for the almost sure convergence of the maximum is derived. Applications are also discussed.

scholarly journals On a ‘Replicating Character String’ Model

2014 ◽  
Vol 51 (2) ◽  
pp. 512-527
Author(s):  
Richard C. Bradley
Keyword(s):  
Dna Sequences ◽  
Input Sequence ◽  
String Model ◽  
Strong Mixing ◽  
Mixing Rate ◽  
Output Sequence ◽  
Absolute Regularity ◽  
Set Up ◽  
Strict Stationarity ◽  
Further Development

In Chaudhuri and Dasgupta's 2006 paper a certain stochastic model for ‘replicating character strings’ (such as in DNA sequences) was studied. In their model, a random ‘input’ sequence was subjected to random mutations, insertions, and deletions, resulting in a random ‘output’ sequence. In this paper their model will be set up in a slightly different way, in an effort to facilitate further development of the theory for their model. In their 2006 paper, Chaudhuri and Dasgupta showed that, under certain conditions, strict stationarity of the ‘input’ sequence would be preserved by the ‘output’ sequence, and they proved a similar ‘preservation’ result for the property of strong mixing with exponential mixing rate. In our setup, we will in spirit slightly extend their ‘preservation of stationarity’ result, and also prove a ‘preservation’ result for the property of absolute regularity with summable mixing rate.

On the Stochastic Dynamical Behaviors of a Nonlinear Oscillator Under Combined Real Noise and Harmonic Excitations

2016 ◽  
Vol 12 (3) ◽  
Author(s):  
Chen Kong ◽  
Zhen Chen ◽  
Xian-Bin Liu
Keyword(s):  
Nonlinear Oscillator ◽  
First Passage Time ◽  
Adjoint Operator ◽  
Strong Mixing ◽  
Linear Filter ◽  
Mixing Condition ◽  
Filter System ◽  
Real Noise ◽  
Band Noise ◽  
Harmonic Excitations

The exit problem and global stability of a nonlinear oscillator excited by an ergodic real noise and harmonic excitations are examined. The real noise is assumed to be a scalar stochastic function of an n-dimensional Ornstein–Uhlenbeck vector process which is the output of a linear filter system. Due to the existence of t-dependent excitation, two two-dimensional Fokker–Planck–Kolmogorov (FPK) equations governing the van der Pol variables process and the amplitude-phase process, respectively, are obtained and discussed through a perturbation method and the spectrum representations of the FPK operator and its adjoint operator of the linear filter system, while the detailed balance condition and the strong mixing condition are removed. Based on these FPK equations, the global properties of one-dimensional nonlinear oscillators with external or (and) internal periodic excitations under external or (and) internal real noises can be examined. Finally, a Duffing oscillator excited by a parametric real noise and parametric harmonic excitations is presented as an example, and the mean first-passage time (MFPT) about the oscillator's exit behavior between limit cycles is obtained under both wide-band noise and narrow-band noise excitations. The analytical result is verified by digital simulation.

scholarly journals Second-Order Least Squares Estimation in Nonlinear Time Series Models with ARCH Errors

Econometrics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 41
Author(s):  
Mustafa Salamh ◽  
Liqun Wang
Keyword(s):  
Time Series ◽  
Real Data ◽  
Error Distribution ◽  
Likelihood Method ◽  
Strong Mixing ◽  
Efficiency Gain ◽  
Economic Time Series ◽  
Mixing Condition ◽  
Conditional Moments ◽  
Dynamic Regression Model

Many financial and economic time series exhibit nonlinear patterns or relationships. However, most statistical methods for time series analysis are developed for mean-stationary processes that require transformation, such as differencing of the data. In this paper, we study a dynamic regression model with nonlinear, time-varying mean function, and autoregressive conditionally heteroscedastic errors. We propose an estimation approach based on the first two conditional moments of the response variable, which does not require specification of error distribution. Strong consistency and asymptotic normality of the proposed estimator is established under strong-mixing condition, so that the results apply to both stationary and mean-nonstationary processes. Moreover, the proposed approach is shown to be superior to the commonly used quasi-likelihood approach and the efficiency gain is significant when the (conditional) error distribution is asymmetric. We demonstrate through a real data example that the proposed method can identify a more accurate model than the quasi-likelihood method.

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