Block Bootstrap Method Research Articles

Model Selection Testing for Diffusion Processes with Applications to Interest Rate and Exchange Rate Models

A model selection test for non-nested misspecified diffusion models is developed by using a criterion based on the Kullback-Leibler information criterion in a new asymptotic framework that accounts for the relative significance of diffusion functions for high frequency data. The test examines the hypothesis that two competing models are equivalent in the criterion. Our approach differentiates the roles of diffusion and drift functions and shows the equivalence of models must be understood differently depending on the sampling frequencies; it is of primary importance for a model to have a diffusion function close to the true diffusion function for superiority when the sampling frequency is high, and we compare drift functions if the models can not be distinguished by the diffusion functions. As the sampling frequencies become higher, the diffusion functions are more important, and the informative signal for ranking the drift functions is weaker. The drift functions are useful only when we sample data for long enough. Our new asymptotics deals with the different rates of information in the diffusion and drift functions by considering both the sampling interval Δ and the sampling span T, and we show the sampling span must increase at a relative speed faster than Δ⁻² (or Δ²T→∞) to ensure sufficient information to be collected for distinguishing two models by their drift functions. The limiting distribution of the test statistic is normal, and we compare different asymptotic approximations to the sampling distribution of the test statistic using the sub-sampling, and the nonparametric block bootstrap methods, as well as the standard normal approximation for the test statistics standardized by the heteroskedasticity auto-correlation consistent variance estimators. We apply our test to the model selection problems for spot interest rate models and exchange rate models. We find that many popular models are observationally equivalent.

Tests of Random Walk: A Comparison of Bootstrap Approaches

This paper compares different versions of the multiple variance ratio test based on bootstrap techniques for the construction of empirical distributions. It also analyzes the crucial issue of selecting optimal block sizes when block bootstrap procedures are used. The comparison of the different approaches using Monte Carlo simulations leads to the conclusion that methodologies using block bootstrap methods present better performance for the construction of empirical distributions of the variance ratio test. Moreover, the results are highly sensitive to methods employed to test the null hypothesis of random walk.

A nonparametric plug-in rule for selecting optimal block lengths for block bootstrap methods

In this paper, we consider the problem of empirical choice of optimal block sizes for block bootstrap estimation of population parameters. We suggest a nonparametric plug-in principle that can be used for estimating ‘mean squared error’-optimal smoothing parameters in general curve estimation problems, and establish its validity for estimating optimal block sizes in various block bootstrap estimation problems. A key feature of the proposed plug-in rule is that it can be applied without explicit analytical expressions for the constants that appear in the leading terms of the optimal block lengths. Furthermore, we also discuss the computational efficacy of the method and explore its finite sample properties through a simulation study.

Uncovering Gene Regulatory Networks from Time-Series Microarray Data with Variational Bayesian Structural Expectation Maximization

We investigate in this paper reverse engineering of gene regulatory networks from time-series microarray data. We apply dynamic Bayesian networks (DBNs) for modeling cell cycle regulations. In developing a network inference algorithm, we focus on soft solutions that can provide a posteriori probability (APP) of network topology. In particular, we propose a variational Bayesian structural expectation maximization algorithm that can learn the posterior distribution of the network model parameters and topology jointly. We also show how the obtained APPs of the network topology can be used in a Bayesian data integration strategy to integrate two different microarray data sets. The proposed VBSEM algorithm has been tested on yeast cell cycle data sets. To evaluate the confidence of the inferred networks, we apply a moving block bootstrap method. The inferred network is validated by comparing it to the KEGG pathway map.

Resampling methods for spatial regression models under a class of stochastic designs

In this paper we consider the problem of bootstrapping a class of spatial regression models when the sampling sites are generated by a (possibly nonuniform) stochastic design and are irregularly spaced. It is shown that the natural extension of the existing block bootstrap methods for grid spatial data does not work for irregularly spaced spatial data under nonuniform stochastic designs. A variant of the blocking mechanism is proposed. It is shown that the proposed block bootstrap method provides a valid approximation to the distribution of a class of M-estimators of the spatial regression parameters. Finite sample properties of the method are investigated through a moderately large simulation study and a real data example is given to illustrate the methodology.

Asymptotic expansions for sums of block-variables under weak dependence

Let {Xi}i=−∞∞ be a sequence of random vectors and $Y_{in}=f_{in}(\mathcal{X}_{i,\ell})$ be zero mean block-variables where $\mathcal{X}_{i,\ell}=(X_{i},\ldots,X_{i+\ell-1})$, i≥1, are overlapping blocks of length ℓ and where fin are Borel measurable functions. This paper establishes valid joint asymptotic expansions of general orders for the joint distribution of the sums ∑i=1nXi and ∑i=1nYin under weak dependence conditions on the sequence {Xi}i=−∞∞ when the block length ℓ grows to infinity. In contrast to the classical Edgeworth expansion results where the terms in the expansions are given by powers of n−1/2, the expansions derived here are mixtures of two series, one in powers of n−1/2 and the other in powers of $[\frac{n}{\ell}]^{-1/2}$. Applications of the main results to (i) expansions for Studentized statistics of time series data and (ii) second order correctness of the blocks of blocks bootstrap method are given.

Bootstrap tests of multiple inequality restrictions on variance ratios

Abstract We develop a block bootstrap method for testing multiple inequality restrictions on variance ratios. The proposed test has reasonable size and power in the presence of strong persistence in conditional variances, making it well suited to applications in financial econometrics.

Field Significance Revisited: Spatial Bias Errors in Forecasts as Applied to the Eta Model

Abstract The spatial structure of bias errors in numerical model output is valuable to both model developers and operational forecasters, especially if the field containing the structure itself has statistical significance in the face of naturally occurring spatial correlation. A semiparametric Monte Carlo method, along with a moving blocks bootstrap method is used to determine the field significance of spatial bias errors within spatially correlated error fields. This process can be completely automated, making it an attractive addition to the verification tools already in use. The process demonstrated here results in statistically significant spatial bias error fields at any arbitrary significance level. To demonstrate the technique, 0000 and 1200 UTC runs of the operational Eta Model and the operational Eta Model using the Kain–Fritsch convective parameterization scheme are examined. The resulting fields for forecast errors for geopotential heights and winds at 850, 700, 500, and 250 hPa over a period of 14 months (26 January 2001–31 March 2002) are examined and compared using the verifying initial analysis. Specific examples are shown, and some plausible causes for the resulting significant bias errors are proposed.

Bootstrap Tests of Multiple Inequality Restrictions on Variance Ratios

We develop a block bootstrap method for testing multiple inequality restrictions on variance ratios. The proposed test has reasonable size and power in the presence of strong persistence in conditional variances, making it well suited to applications in financial econometrics.

Automatic Block-Length Selection for the Dependent Bootstrap

We review the different block bootstrap methods for time series, and present them in a unified framework. We then revisit a recent result of Lahiri [Lahiri, S. N. (1999b). Theoretical comparisons of block bootstrap methods, Ann. Statist. 27:386–404] comparing the different methods and give a corrected bound on their asymptotic relative efficiency; we also introduce a new notion of finite-sample “attainable” relative efficiency. Finally, based on the notion of spectral estimation via the flat-top lag-windows of Politis and Romano [Politis, D. N., Romano, J. P. (1995). Bias-corrected nonparametric spectral estimation. J. Time Series Anal. 16:67–103], we propose practically useful estimators of the optimal block size for the aforementioned block bootstrap methods. Our estimators are characterized by the fastest possible rate of convergence which is adaptive on the strength of the correlation of the time series as measured by the correlogram.

Portfolio performance measurement using APM-free kernel models

A general asset-pricing framework is used to derive a conditional asset-pricing kernel that accounts efficiently for time variation in expected returns and risk, and is suitable to perform (un)conditional evaluations of passive and dynamic investment strategies. The positive abnormal unconditional performance of Canadian equity mutual funds over the period 1989–1999 becomes negative with conditioning, and is robust to the removal of ex post index mimickers. The reversal in the size-based performance results with limited information conditioning is alleviated somewhat with an expansion of the conditioning set. The performance statistics are weakly sensitive to changes in the level of relative risk aversion of the uninformed investor. Unconditional positive performances based on averages of individual fund performances lose their significance when cross-correlations are accounted for using the block-bootstrap method. Estimates of survivorship bias due to the elimination of funds with shorter lives, which range from 36 to 58 basis points per year, are stable across performance models but differ across groupings by fund objective.

Bootstrap Methods for Time Series

SummaryThe bootstrap is a method for estimating the distribution of an estimator or test statistic by resampling one's data or a model estimated from the data. The methods that are available for implementing the bootstrap and the accuracy of bootstrap estimates depend on whether the data are an independent random sample or a time series. This paper is concerned with the application of the bootstrap to time‐series data when one does not have a finite‐dimensional parametric model that reduces the data generation process to independent random sampling. We review the methods that have been proposed for implementing the bootstrap in this situation and discuss the accuracy of these methods relative to that of first‐order asymptotic approximations. We argue that methods for implementing the bootstrap with time‐series data are not as well understood as methods for data that are independent random samples. Although promising bootstrap methods for time series are available, there is a considerable need for further research in the application of the bootstrap to time series. We describe some of the important unsolved problems.RésuméLe bootstrap est une méthode pour estimer la distribution d'un estimateur en rééchantillonnant ses données ou un modéle estiméà partir des données. Les méthodes disponibles pour mettre en oeuvre le bootstrap et la précision des estimateurs de bootstrap dépendent de ce que les données proviennent d'un échantillon aléatoire indépendant ou d'une série temporelle. Cet article concerne l'application du bootstrap aux données des séries temporelles quand on ne dispose pas de modéle paramétrique de dimension finie qui réduise le processus de génération des données à l'échantillonnage aléatoire indépendent. Nous examinons les méthodes qui ont été proposées pour mettre en oeuvre le bootstrap dans cette situation et discutons la precision de ces méthodes comparativement à celle des approximations asymptotiques de premier ordre. Nous montrons que les méthodes pour mettre en oeuvre le bootstrap avec les données des séries temporelles ne sont pas aussi bien comprises que les méthodes pour les données des échantillons aléatoires indépendants. Bien que des méthodes de bootstrap prometteuses pour les séries temporelles soient disponibles, il y a un besoin considérable de recherche supplémental re dans leur application. Nous décrivons quelques problémes importants non résolus.

Accounting for Dependence in a Flexible Multivariate Receptor Model

Formulation and evaluation of environmental policy depends on receptor models that are used to assess the number and nature of pollution sources affecting the air quality for a region of interest. Different approaches have been developed for situations in which no information is available about the number and nature of these sources (e.g., exploratory factor analysis) and the composition of these sources is assumed known (e.g., regression and measurement error models). We propose a flexible approach for fitting the receptor model when only partial pollution source information is available. The use of latent variable modeling allows the direct incorporation of subject matter knowledge into the model, including known physical constraints and partial pollution source information obtained from laboratory measurements or past studies. Because air quality data often exhibit temporal and/or spatial dependence, we consider the importance of accounting for such correlation in estimating model parameters and making valid statistical inferences. We propose an approach for incorporating dependence structure directly into estimation and inference procedures via a new nested block bootstrap method that adjusts for bias in estimating moment matrices. A goodness-of-fit test that is valid in the presence of such dependence is proposed. The application of the approach is facilitated by a new multivariate extension of an existing block size determination algorithm. The proposed approaches are evaluated by simulation and illustrated with an analysis of hourly measurements of volatile organic compounds in the El Paso, Texas/Ciudad Juarez, Mexico area.

Local block bootstrap

For time series that are not stationary, the block bootstrap method is not directly applicable. However, if the underlying stochastic structure is slowly changing with time, one may employ a local block-resampling procedure. We define such a procedure, and give an example of its applicability. To cite this article: E. Paparoditis, D.N. Politis, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 959–962.

THE BOOTSTRAP OF THE MEAN FOR DEPENDENT HETEROGENEOUS ARRAYS

Presently, conditions ensuring the validity of bootstrap methods for the sample mean of (possibly heterogeneous) near epoch dependent (NED) functions of mixing processes are unknown. Here we establish the validity of the bootstrap in this context, extending the applicability of bootstrap methods to a class of processes broadly relevant for applications in economics and finance. Our results apply to two block bootstrap methods: the moving blocks bootstrap of Künsch (1989, Annals of Statistics 17, 1217–1241) and Liu and Singh (1992, in R. LePage & L. Billiard (eds.), Exploring the Limits of the Bootstrap, 224–248) and the stationary bootstrap of Politis and Romano (1994a, Journal of the American Statistical Association 89, 1303–1313). In particular, the consistency of the bootstrap variance estimator for the sample mean is shown to be robust against heteroskedasticity and dependence of unknown form. The first-order asymptotic validity of the bootstrap approximation to the actual distribution of the sample mean is also established in this heterogeneous NED context.

Bootstrapping stationary sequences by the Nadaraya-Watson regression estimator

We propose a new bootstrap method for stationary time series data which uses the Nadaraya-Watson kernel regression estimator. We call this method N-W bootstrap. The N-W bootstrap consists of estimating the conditional autoregressive mean function by the Nadaraya-Watson estimator and bootstrapping the resulting residuals. Indeed we obtain a bootstrapped copy by regenerating a stationary time series data from the Nadaraya-Watson regression estimates and their bootstrapped residuals. A Monte Carlo simulation study is conducted to compare our method to the block bootstrap method (Ku¨nsch, 1989 or Liu and Singh, 1992) in various time series models, which shows the performance of the N-W bootstrap is better or at least comparable to the block bootstrap. Some practical usefulness of the N-W bootstrap is also discussed.

Bootstrap tolerance and confidence limits for two-variable reliability using independent and weakly dependent observations

Tolerance and confidence limits are obtained for two-variable reliability, δ= P( Y< X), using the bootstrap method. X and Y are two random variables with unspecified distributions. Both independent and weakly dependent data cases are considered. Moving Block Bootstrap (MBB) method is used for the later case. Minitab macros are provided for obtaining tolerance and confidence limits. Simulation studies indicate the accuracy of the proposed tolerance and confidence limits.

Theoretical comparisons of block bootstrap methods

In this paper, we compare the asymptotic behavior of some common block bootstrap methods based on nonrandom as well as random block lengths. It is shown that, asymptotically, bootstrap estimators derived using any of the methods considered in the paper have the same amount of bias to the first order. However, the variances of these bootstrap estimators may be different even in the first order. Expansions for the bias, the variance and the mean-squared error of different block bootstrap variance estimators are obtained. It follows from these expansions that using overlapping blocks is to be preferred over nonoverlapping blocks and that using random block lengths typically leads to mean-squared errors larger than those for nonrandom block lengths.

COMPUTING THE DISTRIBUTION OF THE LYAPUNOV EXPONENT FROM TIME SERIES: THE ONE-DIMENSIONAL CASE STUDY

In this paper, a simple and direct statistical method is proposed for estimating the Lyapunov exponent of an unknown dynamic system using its time series of observation data. It is shown that the asymptotic distribution of the estimates obtained from the proposed method is normal. Monte Carlo and block bootstrap methods are used to simulate the estimation for the logistic map, in which they both provide the expectation and variance for the estimates. Computer simulations show that our estimates are very close to the true values of the exponent for the logistic map with different parameters.