Annals Of Statistics Research Articles

Semiparametric smoothing approach to hazard rate estimation

ABSTRACTThis research extends the multiplicative density estimation technique of Naito [(2004), ‘Semiparametric Density Estimation by Local -fitting’, The Annals of Statistics, 32, 1162–1191] to the hazard rate setting. The proposed estimate consists of a parametric estimate of the underlying model times a nonparametric correction factor. The reasoning of this approach is first illustrated by varying the shape parameter involved in the approximation and displaying the benefits of the resulting estimate in an sense for specific example distributions. The sample analogue of this approach is then used as the basis for building an estimator of the true hazard rate function. Establishing its asymptotic properties and specifically its mean square error, reveals that the suggested estimate performs better than its nonparametric counterpart. Detailed instructions are given for calculating the operational characteristics of the estimate, that is, its shape parameter and bandwidth. Finally, its practical performance is illustrated for simulated as well as a real world data example.

Corporate bankruptcy prediction: a high dimensional analysis

Much bankruptcy research has relied on parametric models, such as multiple discriminant analysis and logit, which can only handle a finite number of predictors (Altman in The Journal of Finance 23 (4), 589–609, 1968; Ohlson in Journal of Accounting Research 18 (1), 109–131, 1980). The gradient boosting model is a statistical learning method that overcomes this limitation. The model accommodates very large numbers of predictors which can be rank ordered, from best to worst, based on their overall predictive power (Friedman in The Annals of Statistics 29 (5), 1189–1232, 2001; Hastie et al. 2009). Using a sample of 1115 US bankruptcy filings and 91 predictor variables, the study finds that non-traditional variables, such as ownership structure/concentration and CEO compensation are among the strongest predictors overall. The next best predictors are unscaled market and accounting variables that proxy for size effects. This is followed by market-price measures and financial ratios. The weakest predictors overall included macro-economic variables, analyst recommendations/forecasts and industry variables.

A likelihood ratio like permutation test for one way designs

ABSTRACTWe consider permutation tests based on a likelihood ratio like statistic for the one way or k sample design used in an example in Kolassa and Robinson [(2011), ‘Saddlepoint Approximations for Likelihood Ratio Like Statistics with Applications to Permutation Tests’, Annals of Statistics, 39, 3357–3368]. We give explicitly the region in which the statistic exists, obtaining results which permit calculation of the statistic on the boundary of this region. Numerical examples are given to illustrate improvement in the power of the tests compared to the classical statistics for long-tailed error distributions and no loss of power for normal error distributions.

Special issue in honor of Kathryn Chaloner

We are honored to be guest editors for this special issue of Applied Stochastic Models in Business and Industry, dedicated to Kathryn Chaloner's life and achievements. Contributions from her colleagues and friends in this volume are devoted to Kathryn's early research interest in Bayesian optimal design. She was a pioneer in this field and had a broad impact on the subsequent literature on experimental design. Kathryn was internationally known for her research in Bayesian Statistics, which included design of experiments, outlier detection, prior elicitation, and clinical trials, and for her advancements in the study of HIV/AIDS. She was an accomplished researcher, teacher, and mentor. She actively encouraged women and under-represented minority students to join the field of biostatistics and provided mentorship, support, and encouragement. The special issue contains seven papers. The first three papers present some of the recent work in Bayesian optimal designs for nonlinear models, binary responses, and experiments with adversarial components. This is followed by two articles on Bayesian designs in clinical trials. The last two articles of the special issue involve engineering design problems. The paper by Giovagnoli considers the binary response models of Chaloner and Larntz (1989, Journal of Statistical Planning and Inference, 21:191–208) and develops an adaptive version of A-optimal Bayesian designs that are revised as additional data become available during the experiment. Adaptive Bayesian compound designs is also discussed as another extension. Konstantinou and Dette develop approximate Bayesian D-optimal designs for nonlinear regression models that involve covariates that cannot be observed directly. Applications to specific nonlinear models including the exponential regression model are considered and characterizations of the D-optimal saturated designs are presented. Optimal sample size selection for experiments is considered by De Santis and Gubbiotti in an adversarial setting where the decision makers have different prior opinions while having common utility functions and data. Using a Bayesian decision approach with a quadratic loss function, the authors present results for the one-parameter exponential family and investigate effect of the priors on the optimal sample size. Müller, Xu, and Thall discuss advantages and limitations of Bayesian decision theoretic approach in design of clinical trials. In doing so, the authors present a specific case study and point out the limitations such as the computational difficulties involved in solving sequential problems as well as selection of utility functions capturing different stakeholder interests. The article by Ventz, Parmigiani, and Trippa also consider limitations of the Bayesian framework in design of clinical trials and propose a strategy to alleviate these. The proposed strategy involves combining Bayesian designs with frequentist metrics such as error rates, confidence intervals, etc. that are commonly used by medical community and regulatory agencies. The authors show that this can be achieved by an inclusion of frequentist constraints into the Bayesian formulation. The constrained decision theoretic framework is illustrated via several applications and computational algorithms are presented for implementing the proposed approach. Nakamura, Seepaul, Kadane, and Reeja-Jayan discuss design of experiments in materials science. The authors consider use of a batch-sequential experiment and a regression model to determine optimal levels of input variables for synthesis of ceramic materials. It is illustrated that the optimal settings produce more reliable results than other experiments. The final article by Polson and Soyer considers Bayesian design of accelerated life tests for reliability assessment. An augmented probability simulation (APS) approach using Lindley's (1976, Annals of Statistics, 4:1–10) conjugate utility functions is proposed to compute optimal designs in an efficient manner. The proposed approach is illustrated using single and multiple point designs. Use of particle based methods for APS is also discussed as an alternative to Markov chain Monte Carlo methods.

Probabilistic Opinion Pooling with Imprecise Probabilities

The question of how the probabilistic opinions of different individuals should be aggregated to form a group opinion is controversial. But one assumption seems to be pretty much common ground: for a group of Bayesians, the representation of group opinion should itself be a unique probability distribution (Madansky [44]; Lehrer and Wagner [34]; McConway Journal of the American Statistical Association, 76(374), 410–414, [45]; Bordley Management Science, 28(10), 1137–1148, [5]; Genest et al. The Annals of Statistics, 487–501, [21]; Genest and Zidek Statistical Science, 114–135, [23]; Mongin Journal of Economic Theory, 66(2), 313–351, [46]; Clemen and Winkler Risk Analysis, 19(2), 187–203, [7]; Dietrich and List [14]; Herzberg Theory and Decision, 1–19, [28]). We argue that this assumption is not always in order. We show how to extend the canonical mathematical framework for pooling to cover pooling with imprecise probabilities (IP) by employing set-valued pooling functions and generalizing common pooling axioms accordingly. As a proof of concept, we then show that one IP construction satisfies a number of central pooling axioms that are not jointly satisfied by any of the standard pooling recipes on pain of triviality. Following Levi (Synthese, 62(1), 3–11, [39]), we also argue that IP models admit of a much better philosophical motivation as a model of rational consensus.

NONPARAMETRIC ESTIMATION OF CONDITIONAL VALUE-AT-RISK AND EXPECTED SHORTFALL BASED ON EXTREME VALUE THEORY

We propose nonparametric estimators for conditional value-at-risk (CVaR) and conditional expected shortfall (CES) associated with conditional distributions of a series of returns on a financial asset. The return series and the conditioning covariates, which may include lagged returns and other exogenous variables, are assumed to be strong mixing and follow a nonparametric conditional location-scale model. First stage nonparametric estimators for location and scale are combined with a generalized Pareto approximation for distribution tails proposed by Pickands (1975,Annals of Statistics3, 119–131) to give final estimators for CVaR and CES. We provide consistency and asymptotic normality of the proposed estimators under suitable normalization. We also present the results of a Monte Carlo study that sheds light on their finite sample performance. Empirical viability of the model and estimators is investigated through a backtesting exercise using returns on future contracts for five agricultural commodities.

External and Internal Citation Analyses Can Provide Insight into Serial/Monograph Ratios when Refining Collection Development Strategies in Selected STEM Disciplines

A Review of:
 Kelly, M. (2015). Citation patterns of engineering, statistics, and computer science researchers: An internal and external citation analysis across multiple engineering subfields. College and Research Libraries, 76(7), 859-882. http://doi.org/10.5860/crl.76.7.859
 
 Objective – To determine internal and external citation analysis methods and their potential applicability to the refinement of collection development strategies at both the institutional and cross-institutional levels for selected science, technology, engineering, and mathematics (STEM) subfields.
 
 Design – Multidimensional citation analysis; specifically, analysis of citations from 1) key scholarly journals in selected STEM subfields (external analysis) compared to those from 2) local doctoral dissertations in similar subfields (internal analysis).
 
 Setting – Medium-sized, STEM-dominant public research university in the United States of America. 
 
 Subjects – Two citation datasets: 1) 14,149 external citations from16 journals (i.e., 2 journals per subfield; citations from 2012 volumes) representing bioengineering, civil engineering, computer science (CS), electrical engineering, environmental engineering, operations research, statistics (STAT), and systems engineering; and 2) 8,494 internal citations from 99 doctoral dissertations (18-22 per subfield) published between 2008-–2012 from CS, electrical and computer engineering (ECE), and applied information technology (AIT) and published between 2005-–2012 for systems engineering and operations research (SEOR) and STAT. 
 Methods – Citations, including titles and publication dates, were harvested from source materials and stored in Excel and then manually categorized according to format (book, book chapter, journal, conference proceeding, website, and several others). To analyze citations, percentages of occurrence by subfield were calculated for variables including format, age (years since date cited), journal distribution, and the frequency at which a journal was cited. Top journals for selected subfields were identified based on the percentages of authors citing them in each dataset and, for interdisciplinary journals, according to how often citations for them appeared in subfield groups.
 
 Main Results – For each subfield group, distinct patterns emerged for both internal and external analysis in terms of format, currency, and preferred journals. Regarding format of material cited, journals were dominant for external citations and ranged between 40% of citations (CS) to 94% (bioengineering) of formats cited. Formats were more distributed for internal citations, with ECE, SEOR, and STAT exhibiting journal dominance (61%, 30%, and 59% of citations, respectively) and conference proceedings dominant in CS (43%) and AIT (30%). Regarding currency, almost all cited items (>98% for external citations and 96% for internal citations) were published within the last 50 years, with electrical engineering showing the highest percentage of materials cited within the past five years for external citations (47%). For internal citations, applied information technology illustrated the most use of materials in the five-year timeframe (46%). Top journals for each subfield in which only external data were analyzed include Journal of Biomechanics (bioengineering 54%), Engineering Structures (civil engineering 47%), Water Research (environmental engineering 60%). For CS and AIT, the top journal was Communications of the ACM (external CS citations 29%; internal CS 32%; internal AIT 36%). For electrical engineering, the top journals were Electronics Letters (21% external citations) and Proceedings of the IEEE (50% internal citations). SEOR was broken into three categories (systems engineering, SEOR, and operations research), with Systems Engineering being the top journal according to external citations for the subfield of the same name (48%) and Air Traffic Control Quality as the leading SEOR journal (25% internal citations only). Management Science (77% external citations only) was the top journal for operations research. Top STAT journals were Annals of Statistics (96% internal citations) and Journal of the American Statistical Association (60%). Science was the top interdisciplinary journal for external citations (10%) and IEEE: Transactions on Pattern Analysis and Machine Intelligence for internal citations (13%). 
 
 Conclusion – An approach to citation analysis integrating both internal and external components is useful for institutions aiming to develop balanced STEM collections as well as for collection assessment and budgeting purposes and enables adjustment of serial/monograph ratios to create custom local serial/monograph ratio “blends.” In this institution’s case, internal data suggested a 59:41 serial/monograph ratios versus an external data ratio of 75:25, which indicated that a blended ratio of 67:33 might be appropriate for this institution based on an average of both ratios. In the future, cross-institutional collaboration for external analyses would make it easier for institutions to focus on internal analyses in order to develop appropriate local serial/monograph ratio blends.

Weighted bootstrapped kernel density estimators in two-sample problems

ABSTRACTA weighted bootstrap method is proposed to approximate the distribution of the () norms of two-sample statistics involving kernel density estimators. Using an approximation theorem of Horváth, Kozkoszka and Steineback [(2000) ‘Approximations for Weighted Bootstrap Processes with an Application’, Statistics and Probability Letters, 48, 59–70], that allows one to replace the weighted bootstrap empirical process by a sequence of Gaussian processes, we establish an unconditional bootstrap central limit theorem for such statistics. The proposed method is quite straightforward to implement in practice. Furthermore, through some simulation studies, it will be shown that, depending on the weights chosen, the proposed weighted bootstrap approximation can sometimes outperform both the classical large-sample theory as well as Efron's [(1979) ‘Bootstrap Methods: Another Look at the Jackknife’, Annals of Statistics, 7, 1–26] original bootstrap algorithm.

Bias-corrected maximum likelihood estimation of the parameters of the complex Bingham distribution

In this paper, some bias correction methods are considered for parameter estimation of the complex Bingham distribution. The first method relies on the bias correction formula proposed by Cordeiro and Klein [Statistics & Probability Letters 19 (1994) 169–176]. The second method uses the formulas proposed by Kume and Wood [Statistics & Probability Letters 77 (2007) 832–837] for calculating the derivatives of the log likelihood function. The third method is based on the saddlepoint approximation proposed by Kume and Wood [Biometrika 92 (2005) 465–476]. Bootstrap bias correction methods due to Efron [The Annals of Statistics 7 (1979) 1–26] are also considered. Simulation experiments are used to compare the bias correction methods. In all cases, the analytical and bootstrap bias correction methods have smaller mean square errors. Since the dominant eigenvalue is used to obtain the mean shape, which has practical relevance, it is a key issue for comparing the estimators. The numerical results indicate that the bootstrap methods have a slightly better performance for the dominant eigenvalue.

A Conversation with Arthur Cohen

Arthur Cohen was born in 1933. He received his B.A. in mathematics from Brooklyn College in 1955, and then went to graduate studies in statistics at Columbia University. In 1957, he took leave from Columbia to serve for two years at the Communicable Disease Center, Public Health Services. He returned to Columbia, completed his studies and received his Ph.D. in mathematical statistics in 1963. Art joined the statistics department at Rutgers as an Assistant Professor, and two years later became Associate Professor. From 1968 through 1977, he served as chairman of the department during a critical period in its development. For 52 years, his wisdom has helped guide the department in its rise to excellence. Art served as Editor of the Annals of Statistics for three years, Co-editor of the Journal of Multivariate Analysis for eleven years, as Associate Editor of the Journal of the American Statistical Association and the Journal of Statistical Planning and Inference, each for five years. Art has over 140 publications. In an influential series of fifty-two Annals of Statistics and JASA papers, Art and co-authors developed wide ranging and fundamental results in decision theory, admissibility, Bayes’ procedures, sequential tests, complete class theorems, directional tests, order restricted inference and multiple testing. Art is a Fellow of the Institute of Mathematical Statistics, the American Statistical Association and the International Statistical Institute.

A Conversation with Michael Woodroofe

Michael Woodroofe was born in Corvallis on March 17, 1940, and grew up in a small town called Athena in Oregon. Michael graduated from McEwen High School in 1958 and entered Stanford University, from which he graduated four years later with a major in Mathematics. He earned his masters degree and Ph.D. from the mathematics department at the University of Oregon in 1964 and 1965, respectively. Michael Woodroofe has had a distinguished career and is widely recognized as a preeminent statistician and probabilist. He has broad interests and has made deep and significant contributions in many areas in statistical inference and probability, including biased sampling, shape-restricted inference, sequential analysis, nonlinear renewal theory, modern nonparametric inference, statistics in astronomy and central limit theory for stationary processes. He has published more than 100 research articles, written a SIAM monograph and authored a book. He is a former Editor of the Annals of Statistics, a member of Phi Beta Kappa and a fellow of the Institute of Mathematical Statistics. Michael’s professional positions have included being on the faculty of the Department of Statistics at Carnegie Mellon University and the University of Michigan at Ann Arbor, where he has been on faculty for more than 40 years. He was a founding member of the Department of Statistics at the University Michigan in 1969, retaining a joint appointment with Mathematics, and served as the Chair of the Department of Statistics during 1977–1983. In addition, he has held visiting positions at Columbia University, Massachusetts Institute of Technology and Rutgers University. Michael and his wife, Fran Woodroofe, reside in Ann Arbor. He is the father of one daughter, Caroline, and two sons, Russell and Blake.

Multistage estimation of the difference of locations of two negative exponential populations under a modified Linex loss function: Real data illustrations from cancer studies and reliability analysis

ABSTRACTWe have designed modified two-stage and purely sequential strategies to estimate the difference of location parameters from two independent negative exponential populations having unknown but proportional scale parameters under a modified Linex loss function. This article extends one-sample methodologies of Mukhopadhyay and Bapat (2016, Sequential Analysis). Some preliminary results are established along the lines of Mukhopadhyay and Hamdy (1984, Canadian Journal of Statistics) and Mukhopadhyay and Darmanto (1988, Sequential Analysis). We have resorted to Mukhopadhyay and Duggan (1997, Sankhya, Series A) in developing asymptotic second-order properties for the modified two-stage methodology and to nonlinear renewal theory of Lai and Siegmund (1977, 1979, Annals of Statistics) and Woodroofe (1977, Annals of Statistics) in addressing analogous properties under the purely sequential methodology. Then, we supplement with extensive sets of data analysis via computer simulations validating that both modified two-stage and purely sequential methods perform very well. Both methodologies are also illustrated and implemented using real datasets from cancer studies and reliability analysis.

Bayesian Analysis of Entry Games: A Simulated Likelihood Approach Without Simulation Errors

This paper provides a practical guide to Bayesian estimation of simultaneous entry games of complete information with heterogeneous firms. Bayesian inference requires computation of the likelihood, which is carried out by simulating unobservables. To avoid errors from finite simulations, we apply Andrieu and Roberts’s [2009. The pseudo-marginal approach for efficient Monte Carlo computations, Annals of Statistics, 37(2), pp. 697–725] pseudo-marginal approach. We rely also on adaptive Markov chain Monte Carlo algorithms that improve computational performance.

Multistage point estimation methodologies for a negative exponential location under a modified linex loss function: Illustrations with infant mortality and bone marrow data

ABSTRACTWe have designed Stein-type (Stein, 1945, Annals of Mathematical Statistics) two-stage, modified two-stage (Mukhopadhyay and Duggan, 1997, Sankhya, Series A), and purely sequential strategies (Chow and Robbins, 1965, Annals of Mathematical Statistics) to estimate an unknown location parameter of a negative exponential distribution having an unknown scale parameter under a newly defined and modified Linex loss function. We aim at controlling the associated risk function per unit cost by bounding it from above with a fixed preassigned positive number, ω, and we emphasize both asymptotic first-order and asymptotic second-order properties for the modified two-stage and purely sequential estimation strategies. In developing asymptotic second-order properties for the modified two-stage methodology, we have heavily relied upon basic ideas rooted in Mukhopadhyay and Duggan (1997). In developing asymptotic second-order properties for the purely sequential methodology, however, we have heavily relied upon nonlinear renewal theory (Lai and Siegmund, 1977, 1979, Annals of Statistics; Woodroofe, 1977, Annals of Statistics). Then, we take to extensive data analysis carried out via computer simulations when requisite sample sizes range from small to moderate to large. We find that the Stein-type two-stage estimation methodology oversamples significantly and yet the achieved risk is not close to preset goal ω. On the other hand, both modified two-stage and purely sequential estimation strategies perform remarkably well. We have validated their main theoretical first-order and second-order properties through simulated data. The latter methodologies have been illustrated and implemented using two real data sets from health studies, namely, infant mortality data and bone marrow data.

A New Online Calibration Method for Multidimensional Computerized Adaptive Testing.

Multidimensional-Method A (M-Method A) has been proposed as an efficient and effective online calibration method for multidimensional computerized adaptive testing (MCAT) (Chen & Xin, Paper presented at the 78th Meeting of the Psychometric Society, Arnhem, The Netherlands, 2013). However, a key assumption of M-Method A is that it treats person parameter estimates as their true values, thus this method might yield erroneous item calibration when person parameter estimates contain non-ignorable measurement errors. To improve the performance of M-Method A, this paper proposes a new MCAT online calibration method, namely, the full functional MLE-M-Method A (FFMLE-M-Method A). This new method combines the full functional MLE (Jones & Jin in Psychometrika 59:59-75, 1994; Stefanski & Carroll in Annals of Statistics 13:1335-1351, 1985) with the original M-Method A in an effort to correct for the estimation error of ability vector that might otherwise adversely affect the precision of item calibration. Two correction schemes are also proposed when implementing the new method. A simulation study was conducted to show that the new method generated more accurate item parameter estimation than the original M-Method A in almost all conditions.

Spatial median depth-based robust adjusted empirical likelihood

Empirical likelihood (EL) based inference for parameters defined by general estimating equations of Qin and Lawless [(1994), ‘Empirical Likelihood and General Estimating Equations’,The Annals of Statistics, 22, 300–325] remains an active research topic. However, the performance of the EL method can be hindered by non-robustness and empty set problems. In this paper, we propose a robust adjusted empirical likelihood (RAEL) to address these two problems simultaneously. The resulting RAEL ratio statistic is shown to have inherited the asymptotic properties of both the robust empirical likelihood and the adjusted empirical likelihood. The finite-sample performance of the proposed method is illustrated by simulation and two real-data examples are also presented.

Testing for treatment-biomarker interaction based on local partial-likelihood.

In clinical trials, patients with different biomarker features may respond differently to the new treatments or drugs. In personalized medicine, it is important to study the interaction between treatment and biomarkers in order to clearly identify patients that benefit from the treatment. With the local partial-likelihood estimation (LPLE) method proposed by Fan J, Lin H, Zhou Y. Local partial-likelihood estimation for lifetime data. The Annals of Statistics 2006; 34(1):290Ű325, the treatment effect can be modeled as a flexible function of the biomarker. In this paper, we propose a bootstrap test method for survival outcome data based on the LPLE, for assessing whether the treatment effect is a constant among all patients or varies as a function of the biomarker. The test method is called local partial-likelihood bootstrap (LPLB) and is developed by bootstrapping the martingale residuals. The test statistic measures the amount of change in treatment effects across the entire range of the biomarker and is derived based on asymptotic theories for martingales. The LPLB method is nonparametric and is shown in simulations and data analysis examples to be flexible enough to identify treatment effects in a biomarker-defined subset and more powerful to detect treatment-biomarker interaction of complex forms than the Cox regression model with a simple interaction. We use data from a breast cancer and a prostate cancer clinical trial to illustrate the proposed LPLB test.

Connecting the latent multinomial.

Link et al. (2010, Biometrics 66, 178-185) define a general framework for analyzing capture-recapture data with potential misidentifications. In this framework, the observed vector of counts, y, is considered as a linear function of a vector of latent counts, x, such that y=Ax, with x assumed to follow a multinomial distribution conditional on the model parameters, θ. Bayesian methods are then applied by sampling from the joint posterior distribution of both x and θ. In particular, Link et al. (2010) propose a Metropolis-Hastings algorithm to sample from the full conditional distribution of x, where new proposals are generated by sequentially adding elements from a basis of the null space (kernel) of A. We consider this algorithm and show that using elements from a simple basis for the kernel of A may not produce an irreducible Markov chain. Instead, we require a Markov basis, as defined by Diaconis and Sturmfels (1998, The Annals of Statistics 26, 363-397). We illustrate the importance of Markov bases with three capture-recapture examples. We prove that a specific lattice basis is a Markov basis for a class of models including the original model considered by Link et al. (2010) and confirm that the specific basis used in their example with two sampling occasions is a Markov basis. The constructive nature of our proof provides an immediate method to obtain a Markov basis for any model in this class.

Long range dependency and forecasting of housing price index and mortgage market rate: evidence of subprime crisis

Article history: Received January 20, 2015 Received in revised format 6 February 2015 Accepted 25 March 2015 Available online March 25 2015 In this paper, we examine and forecast the House Price Index (HPI) and mortgage market rate in terms of the description of the subprime crisis. We use a semi-parametric local polynomial Whittle estimator proposed by Shimotsu et al. (2005) [Shimotsu, K., & Phillips, P.C.B. (2005), Exact local Whittle estimation of fractional integration. The Annals of Statistics, 33(4), 18901933.] in a long memory parameter time series. Empirical investigation of HPI and mortgage market rate shows that these variables are more persistent when the d estimates are found on the Shimotsu method than on the one of Kunsch (1987) [Kunsch, H.R. (1987). Statistical aspects of self-similar processes. In Y. Prokhorov and V.V. Sazanov (eds.), Proceedings of the First World Congress of the Bernoulli Society, VNU Science Press, Utrecht, 67-74.]. The estimating forecast values are more realistic and they strongly reflect the present US economy actuality in the two series as indicated by the forecast evaluation topics. Growing Science Ltd. All rights reserved. 5 © 201

ESTIMATION OF CHANGE-POINTS IN LINEAR AND NONLINEAR TIME SERIES MODELS

This paper develops an asymptotic theory for estimated change-points in linear and nonlinear time series models. Based on a measurable objective function, it is shown that the estimated change-point converges weakly to the location of the maxima of a double-sided random walk and other estimated parameters are asymptotically normal. When the magnitude d of changed parameters is small, it is shown that the limiting distribution can be approximated by the known distribution as in Yao (1987, Annals of Statistics 15, 1321–1328). This provides a channel to connect our results with those in Picard (1985, Advances in Applied Probability 17, 841–867) and Bai, Lumsdaine, and Stock (1998, Review of Economic Studies 65, 395–432), where the magnitude of changed parameters depends on the sample size n and tends to zero as n → ∞. The theory is applied for the self-weighted QMLE and the local QMLE of change-points in ARMA-GARCH/IGARCH models. A simulation study is carried out to evaluate the performance of these estimators in the finite sample.