Asymptotically Unbiased Estimation of the Coefficient of Tail Dependence

Abstract. We introduce and study a class of weighted functional estimators for the coefficient of tail dependence in bivariate extreme value statistics. Asymptotic normality of these estimators is established under a second‐order condition on the joint tail behaviour, some conditions on the weight function and for appropriately chosen sequences of intermediate order statistics. Asymptotically unbiased estimators are constructed by judiciously chosen linear combinations of weighted functional estimators, and variance optimality within this class of asymptotically unbiased estimators is discussed. The finite sample performance of some specific examples from our class of estimators and some alternatives from the recent literature are evaluated with a small simulation experiment.     

Multiresolution anomaly detection method for fractional Gaussian noise

Driven by network intrusion detection, we propose a MultiResolution Anomaly Detection (MRAD) method, which effectively utilizes the multiscale properties of Internet features and network anomalies. In this paper, several theoretical properties of the MRAD method are explored. A major new result is the mathematical formulation of the notion that a two-scaled MRAD method has larger power than the average power of the detection method based on the given two scales. Test threshold is also developed. Comparisons between MRAD method and other classical outlier detectors in time series are reported as well.     

Modeling nonlinear time series with local mixtures of generalized linear models

The authors consider a novel class of nonlinear time series models based on local mixtures of regressions of exponential family models, where the covariates include functions of lags of the dependent variable. They give conditions to guarantee consistency of the maximum likelihood estimator for correctly specified models, with stationary and nonstationary predictors. They show that consistency of the maximum likelihood estimator still holds under model misspecification. They also provide probabilistic results for the proposed model when the vector of predictors contains only lags of transformations of the modeled time series. They illustrate the consistency of the maximum likelihood estimator and the probabilistic properties via Monte Carlo simulations. Finally, they present an application using real data.     

Block Bootstrapping for Kernel Density Estimators under ψ-Weak Dependence

Block bootstrap methods are applied to kernel-type density estimator and its derivatives for ψ-weakly dependent processes. Nonparametric density estimation is discussed via moving block bootstrap (MBB) and disjoint block bootstrap (DBB). Asymptotic validity is proved for MBB and DBB. A Monte-Carlo experiment compares confidence intervals based on MBB and DBB with an existing method based on normal approximation (NA) in terms of serial correlation, dynamic asymmetry, and conditional heteroscedasticity. The experiment shows that, in cases of substantial serial correlation, MBB and DBB perform better than NA and, in the other cases, MBB and DBB perform as good as NA.     

An efficient integrated nonparametric entropy estimator of serial dependence

ABSTRACT

We propose an efficient numerical integration-based nonparametric entropy estimator for serial dependence and show that the new entropy estimator has a smaller asymptotic variance than Hong and White’s (2005 Hong, Y., White, H. (2005). Asymptotic distribution theory for nonparametric entropy measures of serial dependence. Econometrica 73:837–901.[Crossref], [Web of Science ®] , [Google Scholar]) sample average-based estimator. This delivers an asymptotically more efficient test for serial dependence. In particular, the uniform kernel gives the smallest asymptotic variance for the numerical integration-based entropy estimator over a class of positive kernel functions. Moreover, the naive bootstrap can be used to obtain accurate inferences for our test, whereas it is not applicable to Hong and White’s (2005 Hong, Y., White, H. (2005). Asymptotic distribution theory for nonparametric entropy measures of serial dependence. Econometrica 73:837–901.[Crossref], [Web of Science ®] , [Google Scholar]) sample averaging approach. A simulation study confirms the merits of our approach.     

Parametric Inference in Stationary Time Series Models with Dependent Errors

This article is concerned with inference for the parameter vector in stationary time series models based on the frequency domain maximum likelihood estimator. The traditional method consistently estimates the asymptotic covariance matrix of the parameter estimator and usually assumes the independence of the innovation process. For dependent innovations, the asymptotic covariance matrix of the estimator depends on the fourth‐order cumulants of the unobserved innovation process, a consistent estimation of which is a difficult task. In this article, we propose a novel self‐normalization‐based approach to constructing a confidence region for the parameter vector in such models. The proposed procedure involves no smoothing parameter, and is widely applicable to a large class of long/short memory time series models with weakly dependent innovations. In simulation studies, we demonstrate favourable finite sample performance of our method in comparison with the traditional method and a residual block bootstrap approach.     

M‐estimation for general ARMA Processes with Infinite Variance

Abstract. General autoregressive moving average (ARMA) models extend the traditional ARMA models by removing the assumptions of causality and invertibility. The assumptions are not required under a non‐Gaussian setting for the identifiability of the model parameters in contrast to the Gaussian setting. We study M‐estimation for general ARMA processes with infinite variance, where the distribution of innovations is in the domain of attraction of a non‐Gaussian stable law. Following the approach taken by Davis et al. (1992) and Davis (1996) , we derive a functional limit theorem for random processes based on the objective function, and establish asymptotic properties of the M‐estimator. We also consider bootstrapping the M‐estimator and extend the results of Davis & Wu (1997) to the present setting so that statistical inferences are readily implemented. Simulation studies are conducted to evaluate the finite sample performance of the M‐estimation and bootstrap procedures. An empirical example of financial time series is also provided.     

Bootstrap diagnostics and remedies

Bootstrap diagnostics are used to assess the reliability of bootstrap calculations and may suggest useful modified calculations when these are possible. Concern focuses on susceptibility to peculiarities in data, incorrectness of a resampling model, incorrect use of resampling simulation output, and inherent inaccuracy of the bootstrap approach. The last involves issues such as inconsistency of a bootstrap method, the order of correctness of a consistent bootstrap method, and approximate pivotality. The authors review here some of these problems, provide workable diagnostic methods where possible, and discuss fast and simple ways to effect the necessary computations.     

Iterative Bias Correction of the Cross‐Validation Criterion

Abstract. The cross‐validation (CV) criterion is known to be asecond‐order unbiased estimator of the risk function measuring the discrepancy between the candidate model and the true model, as well as the generalized information criterion (GIC) and the extended information criterion (EIC). In the present article, we show that the 2kth‐order unbiased estimator can be obtained using a linear combination from the leave‐one‐out CV criterion to the leave‐k‐out CV criterion. The proposed scheme is unique in that a bias smaller than that of a jackknife method can be obtained without any analytic calculation, that is, it is not necessary to obtain the explicit form of several terms in an asymptotic expansion of the bias. Furthermore, the proposed criterion can be regarded as a finite correction of a bias‐corrected CV criterion by using scalar coefficients in a bias‐corrected EIC obtained by the bootstrap iteration.     

Second‐order bias‐corrected AIC in multivariate normal linear models under non‐normality

This paper deals with a bias correction of Akaike's information criterion (AIC) for selecting variables in multivariate normal linear regression models when the true distribution of observation is an unknown non‐normal distribution. It is well known that the bias of AIC is $O(1)$ , and there are a number of the first‐order bias‐corrected AICs which improve the bias to $O(n^{-1})$ , where $n$ is the sample size. A new information criterion is proposed by slightly adjusting the first‐order bias‐corrected AIC. Although the adjustment is achieved by merely using constant coefficients, the bias of the new criterion is reduced to $O(n^{-2})$ . Then, a variance of the new criterion is also improved. Through numerical experiments, we verify that our criterion is superior to others. The Canadian Journal of Statistics 39: 126–146; 2011 © 2011 Statistical Society of Canada