Variance bounds for estimators in autoregressive models with constraints

We consider nonlinear and heteroscedastic autoregressive models whose residuals are martingale increments with conditional distributions that fulfil certain constraints. We treat two classes of constraints: residuals depending on the past through some function of the past observations only, and residuals that are invariant under some finite group of transformations. We determine the efficient influence function for estimators of the autoregressive parameter in such models, calculate variance bounds, discuss information gains, and suggest how to construct efficient estimators. Without constraints, efficient estimators can be given by weighted least squares estimators. With the constraints considered here, efficient estimators are obtained differently, as one-step improvements of some initial estimator, similarly as in autoregressive models with independent increments.     

Reliability estimation in stress–strength models: an MCMC approach

In this paper, we consider the estimation of the stress–strength parameter R=P(Y<X) when X and Y are independent and both are modified Weibull distributions with the common two shape parameters but different scale parameters. The Markov Chain Monte Carlo sampling method is used for posterior inference of the reliability of the stress–strength model. The maximum-likelihood estimator of R and its asymptotic distribution are obtained. Based on the asymptotic distribution, the confidence interval of R can be obtained using the delta method. We also propose a bootstrap confidence interval of R. The Bayesian estimators with balanced loss function, using informative and non-informative priors, are derived. Different methods and the corresponding confidence intervals are compared using Monte Carlo simulations.     

Kullback–Leibler information of a censored variable and its applications

In this paper, we study the Kullback–Leibler (KL) information of a censored variable, which we will simply call it censored KL information. The censored KL information is shown to have the necessary monotonicity property in addition to inherent properties of nonnegativity and characterization. We also present a representation of the censored KL information in terms of the relative risk and study its relation with the Fisher information in censored data. Finally, we evaluate the estimated censored KL information as a goodness-of-fit test statistic.     

Non-linear models 1

This is the first application of a new method for testing stationary random point processes. Consider the class of all stationary ergodic point processes on the real line with arbitrary dependences among the inter–point distances (spacing).The hypothesis is :The observed process φ is a homogeneous Poisson process or more (resp.less) regular than a Poisson process.The sample is the vector of the first n points t₁, …,t_n.There is a close relation between our method for testing and queueing theory: For finding an appropriate test statistic, we observe the behaviour of a single server queue with the input φ.A table of critical values is given.     

Exact Likelihood Equations for Autoregression Models with Multivariate Elliptically Contoured Distributions

Abstract

The multivariate elliptically contoured distributions provide a viable framework for modeling time-series data. It includes the multivariate normal, power exponential, t, and Cauchy distributions as special cases. For multivariate elliptically contoured autoregressive models, we derive the exact likelihood equations for the model parameters. They are closely related to the Yule-Walker equations and involve simple function of the data. The maximum likelihood estimators are obtained by alternately solving two linear systems and illustrated using the simulation data.     

Adaptive Test for Periodicity in Self-Exciting Threshold Autoregressive Models

This article is devoted to the study of the periodicity testing problem in a self-exciting threshold autoregressive (SETAR) model. The local asymptotic normality (LAN) property is shown via the adapted sufficient conditions due to Swensen (1985 Swensen , A. R. ( 1985 ). The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend . Journal of Multivariate Analysis 16 : 54 – 70 .[Crossref], [Web of Science ®] , [Google Scholar]). Moreover, the LAN of the central sequence is established. First, we consider the case where the innovation density is specified and we obtain a parametric local asymptotic test. Second, we construct an adaptive test in the case where this density is unspecified but symmetric. The performances of these established tests are shown via simulation studies.     

Modifications of the Empirical Likelihood Interval Estimation with Improved Coverage Probabilities

The empirical likelihood (EL) technique has been well addressed in both the theoretical and applied literature in the context of powerful nonparametric statistical methods for testing and interval estimations. A nonparametric version of Wilks theorem (Wilks, 1938 Wilks , S. S. ( 1938 ). The large-sample distribution of the likelihood ratio for testing composite hypotheses . Annals of Mathematical Statistics 9 : 60 – 62 .[Crossref] , [Google Scholar]) can usually provide an asymptotic evaluation of the Type I error of EL ratio-type tests. In this article, we examine the performance of this asymptotic result when the EL is based on finite samples that are from various distributions. In the context of the Type I error control, we show that the classical EL procedure and the Student's t-test have asymptotically a similar structure. Thus, we conclude that modifications of t-type tests can be adopted to improve the EL ratio test. We propose the application of the Chen (1995 Chen , L. ( 1995 ). Testing the mean of skewed distributions . Journal of the American Statistical Association 90 : 767 – 772 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) t-test modification to the EL ratio test. We display that the Chen approach leads to a location change of observed data whereas the classical Bartlett method is known to be a scale correction of the data distribution. Finally, we modify the EL ratio test via both the Chen and Bartlett corrections. We support our argument with theoretical proofs as well as a Monte Carlo study. A real data example studies the proposed approach in practice.     

The Effect of Pseudo-Exogenous Instrumental Variables on Hausman Test

Hausman test is popularly used to examine the endogeneity of explanatory variables in a regression model. To derive a well-defined asymptotic distribution of Hausman test, the correlation between the instrumental variables and the error term needs to converge to zero. However, it is possible that there remains considerable correlation in finite samples between the instruments and the error, even though their correlation eventually converges to zero. This article investigates the potential problem that such “pseudo-exogenous” instruments may create. We show that the performance of Hausman test is deteriorated when the instruments are asymptotically exogenous but endogenous in finite samples, through Monte Carlo simulations.     

Testing Equality of Mean Vectors in Two Sample Problem with Missing Data

Finding the influence of traffic accident on the road is helpful to analyze the characteristics of traffic flow, and take reasonable and effective control measures. Here, the detrended fluctuation analysis method is applied to investigate the complexity of time series in mixed traffic flow with a blockage induced by an accident. As a parameter to depict the long-term evolutionary behavior of the time series in traffic flow, the scaling exponent is analyzed. According to the scaling exponent, it is shown that the traffic flow time series can display long-range correlation characteristics, short-range correlation characteristics, and non-power-law relation in the long-range correlation characteristics, which is strongly dependent on the entering probability of vehicle, the ratio of slow vehicle and the blockage duration time.     

Dirichlet Processes in Nonlinear Mixed Effects Models

In this article, we use two efficient approaches to deal with the difficulty in computing the intractable integrals when implementing Gibbs sampling in the nonlinear mixed effects model (NLMM) based on Dirichlet processes (DP). In the first approach, we compute the Laplace's approximation to the integral for its high accuracy, low cost, and ease of implementation. The second approach uses the no-gaps algorithm of MacEachern and Müller (1998 MacEachern , S. , Müller , P. ( 1998 ). Estimating mixtures of Dirichlet process models . Journal of Computational and Graphical Statistics 7 : 223 – 238 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) to perform Gibbs sampling without evaluating the difficult integral. We apply both approaches to real problems and simulations. Results show that both approaches perform well in density estimation and prediction and are superior to the parametric analysis in that they can detect important model features, such as skewness, long tails, and multimodality, whereas the parametric analysis cannot.