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Constrained-realization Monte-Carlo method for hypothesis testing. (English) Zbl 0888.62087
Summary: We compare two theoretically distinct approaches to generate artificial (or surrogate) data for testing hypotheses about a given data set. The first and more straightforward approach is to fit a single best model to the original data, and then to generate surrogate data sets that are typical realizations of that model. The second approach concentrates not on the model but directly on the original data; it attempts to constrain the surrogate data sets so that they exactly agree with the original data for a specified set of sample statistics.
Examples of these two approaches are provided for two simple cases; a test for deviations from a Gaussian distribution, and a test for serial dependence in a time series. Additionally, we consider tests for nonlinearity in time series based on a Fourier transform (FT) method and on more conventional autoregressive moving-average (ARMA) fits to the data.
The comparative performance of hypothesis testing schemes based on these two approaches is found to depend on whether or not the discriminating statistic is pivotal. A statistic is pivotal if its distribution is the same for all processes consistent with the null hypothesis. The typical-realization method requires that the discriminating statistic satisfy this property. The constrained-realization approach, on the other hand, does not share this requirement, and can provide an accurate and powerful test without having to sacrifice flexibility in the choice of discriminating statistic.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
bootstrap
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##### References:
 [1] Barnard, G.A., Discussion on: the spectral analysis of point processes (by M.S. bartlett), J. roy. statist. soc. B, 25, 294, (1963) [2] Hope, A.C.A., A simplified Monte-Carlo significance test procedure, J. roy. statist. soc. B, 30, 582-598, (1968) · Zbl 0187.15901 [3] Besag, J.; Diggle, P.J., Simple Monte-Carlo tests for spatial pattern, Appl. statist., 26, 327-333, (1977) [4] Hall, P.; Titterington, D.M., The effect of simulation order on level accuracy and power of Monte-Carlo tests, J. roy. statist. soc. B, 51, 459-467, (1989) · Zbl 0699.62020 [5] Noreen, E.W., Computer intensive methods for testing hypotheses, (1989), Wiley New York [6] Fisher, N.I.; Hall, P., On bootstrap hypothesis testing, Austral. J. statist., 32, 177-190, (1990) [7] Tsay, R.S., Model checking via parametric bootstraps in time series analysis, Appl. statist., 41, 1-15, (1992) · Zbl 0825.62698 [8] Efron, B., Computers and the theory of statistics: thinking the unthinkable, SIAM rev., 21, 460-480, (1979) · Zbl 0417.62001 [9] Efron, B.; Tibshirani, R., Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy, Statist. sci., 1, 54-77, (1986) · Zbl 0587.62082 [10] Hinkley, D.V., Bootstrap methods (with discussion), J. roy. statist. soc. B, 50, 321-337, (1988) · Zbl 0263.62019 [11] Diciccio, T.J.; Romano, J.P., A review of bootstrap confidence intervals (with discussion), J. roy. statist. soc. B, 50, 338-354, (1988) · Zbl 0672.62057 [12] Efron, B.; Tibshirani, R., An introduction to the bootstrap, (1993), Chapman & Hall London · Zbl 0835.62038 [13] Hall, P.; Wilson, S., Two guidelines for bootstrap hypothesis testing, Biometrika, 47, 757-762, (1991) [14] Beran, R., Prepivoting test statistics: A bootstrap view of asymptotic refinements, J. amer. statist. assoc., 83, 687-697, (1988) · Zbl 0662.62024 [15] Farmer, J.D.; Ott, E.; Yorke, J.A., The dimension of chaotic attractors, Physica D, 7, 153-180, (1983) [16] Eckmann, J.P.; Ruelle, D., Ergodic theory of chaos and strange attractors, Rev. modern phys., 57, 617-656, (1985) · Zbl 0989.37516 [17] Theiler, J., Spurious dimension from correlation algorithms applied to limited time series data, Phys. rev. A, 34, 2427-2432, (1986) [18] Daemmig, M.; Mitschke, F., Estimation of Lyapunov exponents from time series: the stochastic case, Phys. lett. A, 178, 385-394, (1993) [19] Mitschke, F.; Daemmig, M., Chaos versus noise in experimental data, Int. J. bifurcation chaos, 3, 693-702, (1993) · Zbl 0875.58023 [20] Marriott, F.H.C., Barnard’s Monte-Carlo tests: how many simulations?, Appl. statist., 28, 75-77, (1978) [21] Freund, J.E.; Williams, F.J., () [22] A.C. Davison et al., Parametric conditional simulation, preprint. [23] Davison, A.C.; Hinckley, D.V.; Schechtman, E., Efficient bootstrap simulation, Biometrika, 73, 555-566, (1986) · Zbl 0613.62018 [24] Fisher, R., The design of experiments, (1951), Hafner New York [25] Scheinkman, J.A.; LeBaron, B., Nonlinear dynamics and stock returns, J. business, 62, 311-338, (1989) [26] () [27] (), 1-505 [28] (), 301-474 [29] () [30] Tong, H., Non-linear time series: A dynamical system approach, (1990), Clarendon Press Oxford [31] Grassberger, P., Do climatic attractors exist?, Nature, 323, 609-612, (1986) [32] Kurths, J.; Herzel, H., An attractor in a solar time series, Physica D, 25, 165-172, (1987) · Zbl 0618.58039 [33] Gantert, C.; Honerkamp, J.; Timmer, J., Analyzing the dynamics of hand tremor time series, Biol. cybernet., 66, 479-484, (1992) · Zbl 0825.92057 [34] Theiler, J.; Eubank, S.; Longtin, A.; Galdrikian, B.; Farmer, J.D., Testing for nonlinearity in time series: the method of surrogate data, Physica D, 58, 77-94, (1992) · Zbl 1194.37144 [35] Theiler, J.; Linsay, P.S.; Rubin, D.M., Detecting nonlinearity in data with long coherence times, (), 429-455 [36] Prichard, D.; Theiler, J., Generating surrogate data for time series with several simultaneously measured variables, Phys. rev. lett., 73, 951-954, (1994) [37] Caputo, J.G., Practical remarks on the estimation of dimension and entropy from experimental data, (), 99-110 [38] Theiler, J., Estimating fractal dimension, J. opt. soc. amer. A, 7, 1055-1073, (1990) [39] Cutler, C.D., Some results on the behavior and estimation of the fractal dimensions of distributions on attractors, J. statist. phys., 62, 651-708, (1991) · Zbl 0738.58029 [40] Smith, R.L., Optimal estimation of fractal dimension, (), 115-136 [41] Smith, R.L., Estimating dimension in noisy chaotic time series, J. roy. statist. soc. B, 54, 329-352, (1992) · Zbl 0775.62246 [42] Wolf, A.; Swift, J.B.; Swinney, H.L.; Vastano, J.A., Determining Lyapunov exponents from a time series, Physica D, 16, 285, (1985) · Zbl 0585.58037 [43] Sano, M.; Sawada, Y., Measurement of the Lyapunov spectrum from chaotic time series, Phys. rev. lett., 55, 1082, (1985) [44] Eckmann, J.P.; Kamphorst, S.O.; Ruelle, D.; Ciliberto, S., Liapunov exponents from a time series, Phys. rev. lett., 55, 1082, (1985) [45] Bryant, P.; Brown, R.; Abarbanel, H.D.I., Computing the Lyapunov spectrum of a dynamical system from observed time series, Phys. rev. lett., 65, 1523-1526, (1990) · Zbl 1050.37520 [46] Zeng, X.; Eykholt, R.; Pielke, R.A., Estimating the Lyapunov-exponent spectrum from short time series of low precision, Phys. rev. lett., 66, 3229-3232, (1991) [47] Ellner, S.; Gallant, A.R.; McCaffrey, D.; Nychka, D., Convergence rates and data requirements for Jacobian-based estimates of Lyapunov exponents from data, Phys. lett. A, 153, 357-363, (1991) [48] Parlitz, U., Identification of true and spurious Lyapunov exponents from time series, Int. J. bifurcation chaos, 2, 155-165, (1992) · Zbl 0878.34047 [49] Farmer, J.D.; Sidorowich, J.J., Predicting chaotic time series, Phys. rev. lett., 59, 845-848, (1987) [50] Casdagli, M., Nonlinear prediction of chaotic time series, Physica D, 35, 335-356, (1989) · Zbl 0671.62099 [51] Sugihara, G.; May, R., Nonlinear forecasting as a way of distinguishing chaos from measurement error in forecasting, Nature, 344, 734-741, (1990) [52] Casdagli, M., Chaos and deterministic versus stochastic nonlinear modeling, J. roy. statist. soc. B, 54, 303-328, (1992) [53] Brock, W.A.; Dechert, W.D.; Scheinkman, J., A test for independence based on the correlation dimension, () · Zbl 0893.62034 [54] Savit, R.; Green, M., Time series and dependent variables, Physica D, 50, 951-1116, (1991) [55] Kaplan, D.T.; Glass, L., Direct test for determinism, Phys. rev. lett., 68, 427-430, (1992) [56] Kaplan, D.T., Evaluating deterministic structure in maps deduced from discrete-time measurement, Int. J. bifurcation chaos, 3, 617-623, (1993) · Zbl 0870.58042 [57] Kaplan, D.T., Exceptional events as evidence for determinism, Physica D, 73, 38-48, (1994) · Zbl 0809.60048 [58] Heńon, M., A two-dimensional mapping with a strange attractor, Comm. math. phys., 50, 69-77, (1976) · Zbl 0576.58018 [59] Akaike, H., Information theory and an extension of the maximum likelihood principle, (), 267-281 · Zbl 0283.62006 [60] Schwarz, G., Estímating the dimension of a model, Ann. statist., 6, 461-464, (1978) · Zbl 0379.62005
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