An empirical likelihood goodness-of-fit test for time series

Summary. Standard goodness-of-fit tests for a parametric regression model against a series of nonparametric alternatives are based on residuals arising from a fitted model. When a parametric regression model is compared with a nonparametric model, goodness-of-fit testing can be naturally approached by evaluating the likelihood of the parametric model within a nonparametric framework. We employ the empirical likelihood for an α -mixing process to formulate a test statistic that measures the goodness of fit of a parametric regression model. The technique is based on a comparison with kernel smoothing estimators. The empirical likelihood formulation of the test has two attractive features. One is its automatic consideration of the variation that is associated with the nonparametric fit due to empirical likelihood's ability to Studentize internally. The other is that the asymptotic distribution of the test statistic is free of unknown parameters, avoiding plug-in estimation. We apply the test to a discretized diffusion model which has recently been considered in financial market analysis.     

Uniformly strong consistency and Berry-Esseen bound of frequency polygons for α-mixing samples

In this article, the frequency polygon investigated by Scott is studied as a nonparametric estimator for α-mixing samples. By some known exponent and moment inequalities, we obtain the uniformly strong consistency and Berry-Esseen bound of the estimator. The present results relax the relevant conditions used by Carbon et al. Furthermore, the convergence rate of the uniformly asymptotic normality is derived, which is O(n^{? 1/11}) under the given conditions.     

The almost sure convergence rate of the estimator of optimized certainty equivalent risk measure under α-mixing sequences

Conditional value-at-risk (CVaR) model is a kind of financial risk measure that is extensively supported and accepted by international financial community. Its optimized form can be regarded as an optimized certainty equivalent (OCE) risk measurement. In this paper, we mainly discuss and analyze the strong laws of large numbers and the convergence rate of OCE's estimator under α-mixing sequences. The result shows that the almost sure convergence rate of CVaR estimator is given by the results of OCE estimator. Its convergence rate is inversely proportional to the square root of the sample size under certain conditions. Its effectiveness is verified by simulation experiments for two classical α-mixing sequences.     

Wavelet Estimator in Nonparametric Regression Model with Dependent Error’s Structure

In this article, we consider a nonparametric regression model with replicated observations based on the dependent error’s structure, for exhibiting dependence among the units. The wavelet procedures are developed to estimate the regression function. The moment consistency, the strong consistency, strong convergence rate and asymptotic normality of wavelet estimator are established under suitable conditions. A simulation study is undertaken to assess the finite sample performance of the proposed method.     

On the α-risks for shewhart control charts

The performance of the usual Shewhart control charts for monitoring process means and variation can be greatly affected by nonnormal data or subgroups that are correlated. Define the α_k-risk for a Shewhart chart to be the probability that at least one “out-of-control” subgroup occurs in k subgroups when the control limits are calculated from the k subgroups. Simulation results show that the α_k-risks can be quite large even for a process with normally distributed, independent subgroups. When the data are nonnormal, it is shown that the α_k-risk increases dramatically. A method is also developed for simulating an “in-control” process with correlated subgroups from an autoregressive model. Simulations with this model indicate marked changes in the α_k-risks for the Shewhart charts utilizing this type of correlated process data. Therefore, in practice a process should be investigated thoroughly regarding whether or not it is generating normal, independent data before out-of-control points on the control charts are interpreted to be due to some real assignable cause.     

Statistical inference for α-series process with the inverse Gaussian distribution

Statistical inferences for the geometric process (GP) are derived when the distribution of the first occurrence time is assumed to be inverse Gaussian (IG). An α-series process, as a possible alternative to the GP, is introduced since the GP is sometimes inappropriate to apply some reliability and scheduling problems. In this study, statistical inference problem for the α-series process is considered where the distribution of first occurrence time is IG. The estimators of the parameters α, μ, and σ² are obtained by using the maximum likelihood (ML) method. Asymptotic distributions and consistency properties of the ML estimators are derived. In order to compare the efficiencies of the ML estimators with the widely used nonparametric modified moment (MM) estimators, Monte Carlo simulations are performed. The results showed that the ML estimators are more efficient than the MM estimators. Moreover, two real life datasets are given for application purposes.     

Precise large deviations for φ-mixing and UND random variables with long-tailed distributions

In this paper, we investigate the precise large deviations for sums of φ-mixing and UND random variables with long-tailed distributions. The asymptotic relations for non random sum and random sum of random variables with long-tailed distributions are obtained.     

Semiparametric Regression Smoothing of Non-linear Time Series

In this paper, we consider using a semiparametric regression approach to modelling non-linear autoregressive time series. Based on a finite series approximation to non-parametric components, an adaptive selection procedure for the number of summands in the series approximation is proposed. Meanwhile, a large sample study is detailed and a small sample simulation for the Mackey–Glass system is presented to support the large sample study.     

Normalité asymptotique d'estimateurs convergents du mode conditionnel

We state sufficient conditions for asymptotic normality of convergent estimates of the conditional mode, irrespective of data dependence, and give an application to α-mixing stationary processes.