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This paper proposes a three-step method for detecting multiple structural breaks for piecewise stationary vector autoregressive processes. The number of structural breaks can be large and unknown with the locations of the breaks being different among different components. The proposed method is established via a link between a structural break problem and a high-dimensional regression problem. By means of this connection, a group orthogonal greedy algorithm, originated from the high-dimensional variable selection context, is developed for efficiently screening out potential break-points in the first step. A high-dimensional information criterion is proposed for consistent structural breaks estimation in the second step. In the third step, the information criterion further determines the specific components in which structural breaks occur. Monte Carlo experiments are conducted to demonstrate the finite sample performance, and applications to stock data are provided to illustrate the proposed method.

This paper establishes asymptotic theory for optimal estimation of change points in general time series models under α-mixing conditions. We show that the Bayes-type estimator is asymptotically minimax for change-point estimation under squared error loss. Two bootstrap procedures are developed to construct confidence intervals for the change points. An approximate limiting distribution of the change-point estimator under small change is also derived. Simulations and real data applications are presented to investigate the finite sample performance of the Bayes-type estimator and the bootstrap procedures.

A new time‐frequency model and a method to classify time series data are proposed in this article. By viewing the observed signals as realizations of locally dyadic stationary (LDS) processes, a LDS model can be used to provide a time‐frequency decomposition of the signals, under which the evolutionary Walsh spectrum and related statistics can be defined and estimated. The classification procedure is as follows. First choose a training data set that comprises two groups of time series with a known group. Then compute the time frequency feature (the energy) using the training data set, and use a best tree method to maximize the discrepancy of this feature between the two groups. Finally, choose the testing data set with the unknown group as validation data, and use a discriminant statistic to classify the validation data to one of the groups. The classification method is illustrated via an electroencephalographic dataset and the Ericsson B transaction time dataset. The proposed classification method performs better for integer‐valued time series in terms of classification error rates in both simulations and real‐life applications.

ENTHIS LINK GOES TO A ENGLISH SECTIONFRTHIS LINK GOES TO A FRENCH SECTION Bartlett correction constitutes one of the attractive features of empirical likelihood because it enables the construction of confidence regions for parameters with improved coverage probabilities. We study the Bartlett correction of spatial frequency domain empirical likelihood (SFDEL) based on general spectral estimating functions for regularly spaced spatial data. This general formulation can be applied to testing and estimation problems in spatial analysis, for example testing covariance isotropy, testing covariance separability as well as estimating the parameters of spatial covariance models. We show that the SFDEL is Bartlett correctable. In particular, the improvement in coverage accuracies of the Bartlett‐corrected confidence regions depends on the underlying spatial structures. The Canadian Journal of Statistics 47: 455–472; 2019 © 2019 Statistical Society of Canada

Customized personal rate offering is of growing importance in the insurance industry. To achieve this, an important step is to identify subgroups of insureds from the corresponding heterogeneous claim frequency data. In this paper, a penalized Poisson regression approach for subgroup analysis in claim frequency data is proposed. Subjects are assumed to follow a zero-inflated Poisson regression model with group-specific intercepts, which capture group characteristics of claim frequency. A penalized likelihood function is derived and optimized to identify the group-specific intercepts and effects of individual covariates. To handle the challenges arising from the optimization of the penalized likelihood function, an alternating direction method of multipliers algorithm is developed and its convergence is established. Simulation studies and real applications are provided for illustrations.

Interaction of capital market participants is a complicated dynamic process. A stochastic model is proposed to describe the dynamics to predict short-term stock price behaviors. Independent compound Poisson processes are introduced to describe the occurrences of market orders, limit orders and cancellations of limit orders, respectively. Based on high-frequency observations of the limit order book, the maximum empirical likelihood estimator (MELE) is applied to estimate the parameters of the compound Poisson processes. Moreover, an analytical formula is derived to compute the probability distribution of the first-passage time of a compound Poisson process. Based on this formula, the conditional probability of a price increase and the conditional distribution of the duration until the first change in mid-price are obtained. A novel approach of short-term stock price prediction is proposed and this methodology works reasonably well in the data analysis. Copyright © 2016 John Wiley & Sons, Ltd.

An effective model for time-varying quantiles of a time series is of considerable practical importance across various disciplines. In particular, in financial risk management, computation of Value-at-risk (VaR), one of the most popular risk measures, involves knowledge of quantiles of portfolio returns. This paper examines the random walk behavior of VaRs constructed under two most common approaches, viz. historical simulation and the parametric approach using GARCH models. We find that sequences of historical VaRs appear to follow a unit root model, which can be an artifact under some settings, whereas its counterpart constructed via the parametric approach does not follow a random walk model by default.

Analysis of high-dimensional time series data is of increasing interest among different fields. This article studies high-dimensional time series from a dimension reduction perspective using factor modelling. Statistical inference is conducted using eigen-analysis of a certain non-negative definite matrix related to autocovariance matrices of the time series, which is applicable to fixed or increasing dimension. When the dimension goes to infinity, the rate of convergence and limiting distributions of estimated factors are established. Using the limiting distributions of estimated factors, a high-dimensional final prediction error criterion is proposed to select the number of factors. Asymptotic properties of the criterion are illustrated by simulation studies and real applications.

Interest in online auctions has been growing in recent years. There is an extensive literature on this topic, whereas modeling online auction price process constitutes one of the most active research areas. Most of the research, however, only focuses on modeling price curves, ignoring the bidding process. In this paper, a semiparametric regression model is proposed to model the online auction process. This model captures two main features of online auction data: changing arrival rates of bidding processes and changing dynamics of prices. A new inference procedure using B-splines is also established for parameter estimation. The proposed model is used to forecast the price of an online auction. The advantage of this proposed approach is that the price can be forecast dynamically and the prediction can be updated according to newly arriving information. The model is applied to Xbox data with satisfactory forecasting properties. Copyright © 2016 John Wiley & Sons, Ltd.

Bartlett correction, which improves the coverage accuracies of confidence regions, is one of the desirable features of empirical likelihood. For empirical likelihood with dependent data, previous studies on the Bartlett correction are mainly concerned with Gaussian processes. By establishing the validity of Edgeworth expansion for the signed root empirical log-likelihood ratio statistics, we show that the Bartlett correction is applicable to empirical likelihood for short-memory time series with possibly non-Gaussian innovations. The Bartlett correction is established under the assumptions that the variance of the innovation is known and the mean of the underlying process is zero for a single parameter model. In particular, the order of the coverage errors of Bartlett-corrected confidence regions can be reduced from O(n−1) to O(n−2).

This article investigates likelihood inferences for high-dimensional factor analysis of time series data. We develop a matrix decomposition technique to obtain expressions of the likelihood functions and its derivatives. With such expressions, the traditional delta method that relies heavily on score function and Hessian matrix can be extended to high-dimensional cases. We establish asymptotic theories, including consistency and asymptotic normality. Moreover, fast computational algorithms are developed for estimation. Applications to high-dimensional stock price data and portfolio analysis are discussed. The technical proofs of the asymptotic results and the computer codes are available online.

Modeling online auction prices is a popular research topic among statisticians and marketing analysts. Recent research mainly focuses on two directions: one is the functional data analysis (FDA) approach, in which the price–time relationship is modeled by a smooth curve, and the other is the point process approach, which directly models the arrival process of bidders and bids. In this paper, a novel model for the bid arrival process using a self-exciting point process (SEPP) is proposed and applied to forecast auction prices. The FDA and point process approaches are linked together by using functional data analysis technique to describe the intensity of the bid arrival point process. Using the SEPP to model the bid arrival process, many stylized facts in online auction data can be captured. We also develop a simulation-based forecasting procedure using the estimated SEPP intensity and historical bidding increment. In particular, prediction interval for the terminal price of merchandise can be constructed. Applications to eBay auction data of Harry Potter books and Microsoft Xbox show that the SEPP model provides more accurate and more informative forecasting results than traditional methods. Copyright © 2014 John Wiley & Sons, Ltd.

Consider a structural break autoregressive (SBAR) process where j = 1, …, m + 1, {t1, …, tm} are change-points, 1 = t0 < t1 < ⋅⋅⋅ < tm + 1 = n + 1, σ( · ) is a measurable function on , and {ϵt} are white noise with unit variance. In practice, the number of change-points m is usually assumed to be known and small, because a large m would involve a huge amount of computational burden for parameters estimation. By reformulating the problem in a variable selection context, the group least absolute shrinkage and selection operator (LASSO) is proposed to estimate an SBAR model when m is unknown. It is shown that both m and the locations of the change-points {t1, …, tm} can be consistently estimated from the data, and the computation can be efficiently performed. An improved practical version that incorporates group LASSO and the stepwise regression variable selection technique are discussed. Simulation studies are conducted to assess the finite sample performance. Supplementary materials for this article are av...

Let X"t=@?"j"="0^~c"j@e"t"-"j be a moving average process with GARCH (1, 1) innovations {@e"t}. In this paper, the asymptotic behavior of the quadratic form Q"n=@?"j"="1^n@?"s"="1^nb(t-s)X"tX"s is derived when the innovation {@e"t} is a long-memory and heavy-tailed process with tail index @a, where {b(i)} is a sequence of constants. In particular, it is shown that when 1 =4, Q"n has an asymptotic normal distribution. These results not only shed light on the singular behavior of the quadratic forms when both long-memory and heavy-tailed properties are present, but also have applications in the inference for general linear processes driven by heavy-tailed GARCH innovations.

Consider an AR(p) process , where {ɛt} is a sequence of i.i.d. random variables lying in the domain of attraction of a stable law with index 0

This paper investigates the asymptotic theory of the least squares estimators (LSE) for a long-memory nearly unstable model when the innovation sequences are functionals of moving averages. It is shown that the limit distribution of the LSE is a functional of the Hermite Ornstein–Uhlenbeck process. This result not only generalizes the result of Buchmann and Chan [Ann. Statist. 35 (2007), 2001–2017], but also that of Wu [Economet. Theory 22 (2006), 1–14].

The maximum likelihood estimate (MLE) of the autoregressive coefficient of a near-unit root autoregressive process Yt = ?nYt-1 + ?t with a-stable noise {?t} is studied in this paper. Herein ?n = 1 - ?/n, ? = 0 is a constant, Y0 is a fixed random variable and et is an a-stable random variable with characteristic function f(t,?) for some parameter ?. It is shown that when 0 1 and E?1 = 0, the limit distribution of the MLE of ?n and ? are mixtures of a stable process and Gaussian processes. On the other hand, when a > 1 and E?1 ? 0, the limit distribution of the MLE of ?n and ? are normal. A Monte Carlo simulation reveals that the MLE performs better than the usual least squares procedures, particularly for the case when the tail index a is less than 1.

It is known that the tail index of a GARCH model is determined by a moment equation, which involves the underlying unknown parameters of the model. A tail index estimator can therefore be constructed by solving the sample moment equation with the unknown parameters being replaced by its quasi-maximum likelihood estimates (QMLE). To construct a confidence interval for the tail index, one needs to estimate the non-trivial asymptotic variance of the QMLE. In this paper, an empirical likelihood method is proposed for interval estimation of the tail index. One advantage of the proposed method is that interval estimation can still be achieved without having to estimate the complicated asymptotic variance. A simulation study confirms the advantage of the proposed method.

Consider the nonparametric heteroscedastic regression model Y = m ( X ) + σ ( X ) ɛ , where m ( · ) is an unknown conditional mean function and σ ( · ) is an unknown conditional scale function. In this paper, the limit distribution of the quantile estimate for the scale function σ ( X ) is derived. Since the limit distribution depends on the unknown density of the errors, an empirical likelihood ratio statistic based on quantile estimator is proposed. This statistics is used to construct confidence intervals for the variance function. Under certain regularity conditions, it is shown that the quantile estimate of the scale function converges to a Brownian motion and the empirical likelihood ratio statistic converges to a chi-squared random variable. Simulation results demonstrate the superiority of the proposed method over the least squares procedure when the underlying errors have heavy tails.

Levy processes have been receiving increasing attention in financial modeling. One distinctive feature of such models is that their characteristic functions are readily available. Inference based on characteristic functions is very useful for studying Levy processes. By incorporating the recent advances in nonparametric approaches, empirical likelihood methods based on characteristic functions are developed in this paper for parameter estimation, testing a particular parametric class including the presence of a jump component in the Levy process and testing for symmetry of a distribution. Simulation and case studies confirm the effectiveness of the proposed method.

By virtue of long-memory time series, it is illustrated in this paper that white noise calculus can be used to handle subtle issues of stochastic integral convergence that often arise in the asymptotic theory of time series. A main difficulty of such an issue is that the limiting stochastic integral cannot be defined path-wise in general. As a result, continuous mapping theorem cannot be directly applied to deduce the convergence of stochastic integrals $\int^{1}_{0}H_{n}(s)\,dZ_{n}(s)$ to $\int^{1}_{0}H(s)\,dZ(s)$ based on the convergence of $(H_{n},Z_{n})$ to $(H,Z)$ in distribution. The white noise calculus, in particular the technique of $\mathcal{S}$-transform, allows one to establish the asymptotic results directly.

We introduce a long-memory dynamic Tobit model, defining it as a censored version of a fractionally integrated Gaussian ARMA model, which may include seasonal components and/or additional regression variables. Parameter estimation for such a model using standard techniques is typically infeasible, since the model is not Markovian, cannot be expressed in a finite-dimensional state-space form, and includes censored observations. Furthermore, the long-memory property renders a standard Gibbs sampling scheme impractical. Therefore we introduce a new Markov chain Monte Carlo sampling scheme, which is orders of magnitude more efficient than the standard Gibbs sampler. The method is inherently capable of handling missing observations. In case studies, the model is fit to two time series: one consisting of volumes of requests to a hard disk over time, and the other consisting of hourly rainfall measurements in Edinburgh over a 2-year period. The resulting posterior distributions for the fractional differencing parameter demonstrate, for these two time series, the importance of the long-memory structure in the models. Copyright © 2006 John Wiley & Sons, Ltd.

This paper studies asymptotic properties of the exact maximum likelihood estimates (MLE) for a general class of Gaussian seasonal long-range-dependent processes. This class includes the commonly used Gegenbauer and seasonal autoregressive fractionally integrated moving average processes. By means of an approximation of the spectral density, the exact MLE of this class are shown to be consistent, asymptotically normal and efficient. Finite sample performance of these estimates is examined by Monte Carlo simulations and it is shown that the estimates behave very well even for moderate sample sizes. The estimation methodology is illustrated by a real-life Internet traffic example.

Summary The development of time series models for traffic volume data constitutes an important step in constructing automated tools for the management of computing infrastructure resources. We analyse two traffic volume time series: one is the volume of hard disc activity, aggregated into half-hour periods, measured on a workstation, and the other is the volume of Internet requests made to a workstation. Both of these time series exhibit features that are typical of network traffic data, namely strong seasonal components and highly non-Gaussian distributions. For these time series, a particular class of non-linear state space models is proposed, and practical techniques for model fitting and forecasting are demonstrated.

Bartlett correction is one of the desirable features of empirical likelihood (EL) since it allows constructions of confidence regions with improved coverage probabilities. Previous studies demonstrated the Bartlett correction of EL for independent observations and for short-memory time series. By establishing the validity of Edgeworth expansion for the signed root empirical log-likelihood ratio, the validity of Bartlett correction of EL for Gaussian long-memory time series is established. In particular, orders of the coverage error of confidence regions can be reduced from $\log^{6}n/n$ to $\log^{3}n/n$, which is different from the classical rate of reduction from $n^{-1}$ to $n^{-2}$.

This paper addresses the issues of maximum likelihood estimation and forecasting of a long-memory time series with missing values. A state-space representation of the underlying long-memory process is proposed. By incorporating this representation with the Kalman filter, the proposed method allows not only for an efficient estimation of an ARFIMA model but also for the estimation of future values under the presence of missing data. This procedure is illustrated through an analysis of a foreign exchange data set. An investment scheme is developed which demonstrates the usefulness of the proposed approach. © 1997 John Wiley & Sons, Ltd.

We consider nonparametric maximum-likelihood estimation of a log-concave density in case of interval-censored, right-censored and binned data. We allow for the possibility of a subprobability density with an additional mass at $+\infty$, which is estimated simultaneously. The existence of the estimator is proved under mild conditions and various theoretical aspects are given, such as certain shape and consistency properties. An EM algorithm is proposed for the approximate computation of the estimator and its performance is illustrated in two examples.

A moment bound for the normalized conditional-sum-of-squares (CSS) estimate of a general autoregressive fractionally integrated moving average (ARFIMA) model with an arbitrary unknown memory parameter is derived in this paper. To achieve this goal, a uniform moment bound for the inverse of the normalized objective function is established. An important application of these results is to establish asymptotic expressions for the one-step and multi-step mean squared prediction errors (MSPE) of the CSS predictor. These asymptotic expressions not only explicitly demonstrate how the multi-step MSPE of the CSS predictor manifests with the model complexity and the dependent structure, but also offer means to compare the performance of the CSS predictor with the least squares (LS) predictor for integrated autoregressive models. It turns out that the CSS predictor can gain substantial advantage over the LS predictor when the integration order is high. Numerical findings are also conducted to illustrate the theoretical results., Published in at http://dx.doi.org/10.1214/13-AOS1110 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In this paper, a uniform (over some parameter space) moment bound for the inverse of Fisher's information matrix is established. This result is then applied to develop moment bounds for the normalized least squares estimate in (nonlinear) stochastic regression models. The usefulness of these results is illustrated using time series models. In particular, an asymptotic expression for the mean squared prediction error of the least squares predictor in autoregressive moving average models is obtained. This asymptotic expression provides a solid theoretical foundation for some model selection criteria., Published in at http://dx.doi.org/10.1214/10-AOS861 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Remark 3.1. This corollary reflects the effects of the slower convergence rate of the estimated parameter a$n. This fact serves as a reminiscence of the classical Kolmogorov-Smirnov statistics problem when the underlying parame? ters are estimated; see Durbin (1976). When cxq is known, the test statistic (1.5) is still valid, however. As pointed out by the reviewer, when F ? F(x,0) involves an unknown parameter ?, one should consider Kn with F(x) being replaced by F(x,0n). When H < 1/2, it can be shown that the limit distribution of the statistic exists by means of the result of Wu (2003). The closed form of such a limit dis? tribution is rather complicated and does not possess a simple expression, however, and is not presented here.

A nonparametric test statistic based on the distance between the joint and marginal densities is developed to test for the serial dependence for a given sequence of time series data. The key idea lies in observing that, under the null hypothesis of independence, the joint density of the observations is equal to the product of their individual marginals. Histograms are used in constructing such a statistic which is nonparametric and consistent. It possesses high power in capturing subtle or diffuse dependence structure. A bilinear time series model is used to illustrate its performance with the classical correlation approach.

Consider $Z^f_t(u)=\int_0^{tu}f(N_s) ds$, $t>0$, $u\in[0,1]$, where $N=(N_t)_{t\in\mathbb{R}}$ is a normal process and $f$ is a measurable real-valued function satisfying $Ef(N_0)^23/4$, respectively, whereas our result covers $H=3/4$., Comment: Published in at http://dx.doi.org/10.1214/08-AAP531 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

This paper studies the quasi-maximum likelihood estimator (QMLE) of non-stationary GARCH(p,q) models. By expressing GARCH models in matrix form, the log-likelihood function is written in terms of the product of random matrices. Oseledec’s multiplicative ergodic theorem is then used to establish the asymptotic properties of the log-likelihood function and thereby, showing the weak consistency and the asymptotic normality of the QMLE for non-stationary GARCH(p,q) models.

This paper studies the residual empirical process of long- and short-memory time series regression models and establishes its uniform expansion under a general framework. The results are applied to the stochastic regression models and unstable autoregressive models. For the long-memory noise, it is shown that the limit distribution of the Kolmogorov-Smirnov test statistic studied in Ho and Hsing [Ann. Statist. 24 (1996) 992-1024] does not hold when the stochastic regression model includes an unknown intercept or when the characteristic polynomial of the unstable autoregressive model has a unit root. To this end, two new statistics are proposed to test for the distribution of the long-memory noises of stochastic regression models and unstable autoregressive models. (With Correction.), Comment: Published in at http://dx.doi.org/10.1214/07-AOS543 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

An autoregressive time series is said to be near-integrated (nearly nonstationary) if some of its characteristic roots are close to the unit circle. Statistical inference for the least squares estimators of near-integrated AR(1) models has been under rigorous study recently both in the statistics and econometric literatures. Although classical asymptotics are no longer available, through the study of weak convergence of stochastic processes, one can establish the asymptotic theories in terms of simple diffusion processes or Brownian motions. Such results rely heavily on the finiteness of the variance of the noise. When this finite variance condition fails, whereas many physical and economic phenomena are believed to be generated by an infinite variance noise sequence, the aforementioned asymptotics are not applicable. In this article, a unified theory concerning near-integrated autoregressive time series with infinite variance is developed. In particular, when the noise sequence {e t } belongs to...

This paper considers the effect of least squares procedures for nearly unstable linear time series with strongly dependent innovations. Under a general framework and appropriate scaling, it is shown that ordinary least squares procedures converge to functionals of fractional Ornstein--Uhlenbeck processes. We use fractional integrated noise as an example to illustrate the important ideas. In this case, the functionals bear only formal analogy to those in the classical framework with uncorrelated innovations, with Wiener processes being replaced by fractional Brownian motions. It is also shown that limit theorems for the functionals involve nonstandard scaling and nonstandard limiting distributions. Results of this paper shed light on the asymptotic behavior of nearly unstable long-memory processes., Published in at http://dx.doi.org/10.1214/009053607000000136 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

A mean square error criterion is proposed in this paper to provide a systematic approach to approximate a long-memory time series by a short-memory ARMA(1, 1) process. Analytic expressions are derived to assess the effect of such an approximation. These results are established not only for the pure fractional noise case, but also for a general autoregressive fractional moving average long-memory time series. Performances of the ARMA(1,1) approximation as compared to using an ARFIMA model are illustrated by both computations and an application to the Nile river series. Results derived in this paper shed light on the forecasting issue of a long-memory process. Copyright © 2001 John Wiley & Sons, Ltd.

This paper develops a state space modeling for long-range dependent data. Although a long-range dependent process has an infinite-dimensional state space representation, it is shown that by using the Kalman filter, the exact likelihood function can be computed recursively in a finite number of steps. Furthermore, an approximation to the likelihood function based on the truncated state space equation is considered. Asymptotic properties of these approximate maximum likelihood estimates are established for a class of long-range dependent models, namely, the fractional autoregressive moving average models. Simulation studies show rapid converging properties of the approximate maximum likelihood approach.