Asymptotic optimal designs under long-range dependence error structure
Holger Dette, Nikolai Leonenko, Andrey Pepelyshev, and Anatoly Zhigljavsky
Source: Bernoulli Volume 15, Number 4
(2009), 1036-1056.
Abstract
We discuss the optimal design problem in regression models with long-range dependence error structure. Asymptotic optimal designs are derived and it is demonstrated that these designs depend only indirectly on the correlation function. Several examples are investigated to illustrate the theory. Finally, the optimal designs are compared with asymptotic optimal designs which were derived by Bickel and Herzberg [Ann. Statist. 7 (1979) 77–95] for regression models with short-range dependent error.
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.bj/1262962225
Digital Object Identifier: doi:10.3150/09-BEJ185
Mathematical Reviews number (MathSciNet): MR2597582
Zentralblatt MATH identifier: 1200.62084
References
Adenstedt, R.K. (1974). On large-sample estimation of the mean of a stationary random sequence. Ann. Statist. 2 1095–1107.
Mathematical Reviews (MathSciNet): MR368354
Zentralblatt MATH: 0296.62081
Digital Object Identifier: doi:10.1214/aos/1176342867
Project Euclid: euclid.aos/1176342867
Anh, V.V., Knopova, V.P. and Leonenko, N.N. (2004). Continuous-time stochastic processes with cyclical long-range dependence. Aust. N. Z. J. Stat. 46 275–296.
Mathematical Reviews (MathSciNet): MR2076396
Atkinson, A.C. and Donev, A.N. (1992). Optimum Experimental Designs. Oxford: Clarendon Press.
Barndorff-Nielsen, O.E., Jensen, J.L. and Sorensen, M. (1990). Parametric modelling of turbulence. Philosophical Transactions: Physical Sciences and Engineering 332 439–455.
Barndorff-Nielsen, O.E. and Leonenko, N.N. (2005). Spectral properties of superpositions of Ornstein–Uhlenbeck type processes. Methodol. Comput. Appl. Probab. 7 335–352.
Mathematical Reviews (MathSciNet): MR2210585
Zentralblatt MATH: 1089.60014
Digital Object Identifier: doi:10.1007/s11009-005-4521-0
Beran, J. (1992). Statistical methods for data with long-range dependence. Statist. Sci. 7 404–427.
Beran, J. (1994). Statistics for Long-Memory Processes. New York: Chapman and Hall.
Mathematical Reviews (MathSciNet): MR1304490
Zentralblatt MATH: 0869.60045
Beran, J., Sherman, R., Taqqu, M.S. and Willinger, W. (1992). Long-range dependence in variable bit-rate video traffic. IEEE Trans. Commun. 43 1566–1579.
Bickel, P.J. and Herzberg, A.M. (1979). Robustness of design against autocorrelation in time. I. Asymptotic theory, optimality for location and linear regression. Ann. Statist. 7 77–95.
Mathematical Reviews (MathSciNet): MR515685
Zentralblatt MATH: 0403.62051
Digital Object Identifier: doi:10.1214/aos/1176344556
Project Euclid: euclid.aos/1176344556
Bickel, P.J., Herzberg, A.M. and Schilling, M.F. (1981). Robustness of design against autocorrelation in time. II. Optimality, theoretical and numerical results for the first-order autoregressive process. J. Amer. Statist. Assoc. 76 870–877.
Mathematical Reviews (MathSciNet): MR650898
Zentralblatt MATH: 0505.62063
Digital Object Identifier: doi:10.2307/2287582
Dette, H. (1997). Designing of experiments with respect to “standardized” optimality criteria. J. Roy. Statist. Soc. Ser. B 59 97–110.
Mathematical Reviews (MathSciNet): MR1436556
Digital Object Identifier: doi:10.1111/1467-9868.00056
Djrbashian, M.M. (1993). Harmonic Analysis and Boundary Value Problems in the Complex Domain. Boston: Birkhauser.
Mathematical Reviews (MathSciNet): MR1249271
Zentralblatt MATH: 0798.43001
Doukhan, P., Oppenheim, G. and Taqqu, M. (eds.) (2003). Theory and Applications of Long-Range Dependence. Boston: Birkhauser.
Mathematical Reviews (MathSciNet): MR1956041
Fedorov, V.V. (1972). Theory of Optimal Experiments. New York: Academic Press.
Mathematical Reviews (MathSciNet): MR403103
Zentralblatt MATH: 0261.62002
Gneiting, T. (2000). Power-law correlations, related models for long-range dependence and their simulation. J. Appl. Probab. 37 1104–1109.
Mathematical Reviews (MathSciNet): MR1808873
Zentralblatt MATH: 0972.62079
Digital Object Identifier: doi:10.1239/jap/1014843088
Project Euclid: euclid.jap/1014843088
Granger, C.W.J. (1980). Long memory relationships and the aggregation of dynamic models. J. Econometrics 14 227–238.
Mathematical Reviews (MathSciNet): MR597259
Digital Object Identifier: doi:10.1016/0304-4076(80)90092-5
Ivanov, A.V. and Leonenko, N.N. (2004). Asymptotic theory for non-linear regression with long-range dependence. Math. Methods Statist. 13 153–178.
Mathematical Reviews (MathSciNet): MR2090470
Zentralblatt MATH: 1132.62338
Ivanov, A.V. and Leonenko, N.N. (2008). Semiparametric analysis of long-range dependence in nonlinear regression. J. Statist. Plann. Inference 138 1733–1753.
Mathematical Reviews (MathSciNet): MR2400476
Zentralblatt MATH: 1138.62310
Digital Object Identifier: doi:10.1016/j.jspi.2007.06.027
Künsch, H., Beran, J. and Hampel, F. (1993). Contrasts under long-range correlations. Ann. Statist. 21 943–964.
Mandelbrot, B.B. (1973). Le problème de la rétalité des cycle lents et le syndrome de Joseph. Economie Appliquéee 26 349–365.
Metzler, R., Barkai, E. and Klafter, J. (1999). Anomalous diffusion and relaxation close to thermal equilibrium: A fractional Fokker–Planck equation approach. Phys. Rev. Lett. 82 3563–3567.
Metzler, R. and Klafter, J. (2000). The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339 1–77.
Mathematical Reviews (MathSciNet): MR1809268
Digital Object Identifier: doi:10.1016/S0370-1573(00)00070-3
Pázman A. (1986). Foundations of Optimum Experimental Design. Dordrecht: Reidel.
Pázman A. and Müller, C. (1998). Applications of necessary and sufficient conditions for maximum efficient design. Metrika 48 1–19.
Pearson, K. (1902). On the mathematical theory of errors of judgement, with special reference to the personal equation. Philos. Trans. Roy. Soc. Ser. A 198 235–299.
Porter-Hudak, S. (1990). An application of the seasonal fractionally differenced model to the monetary aggregates. J. Amer. Statist. Assoc. 85 338–344.
Pukelsheim, F. (1993). Optimal Design of Experiments. New York: Wiley.
Mathematical Reviews (MathSciNet): MR1211416
Sacks, J. and Ylvisaker, N.D. (1966). Designs for regression problems with correlated errors. Ann. Math. Statist. 37 66–89.
Mathematical Reviews (MathSciNet): MR192601
Digital Object Identifier: doi:10.1214/aoms/1177699599
Project Euclid: euclid.aoms/1177699599
Sacks, J. and Ylvisaker, N.D. (1968). Designs for regression problems with correlated errors; many parameters. Ann. Math. Statist. 39 49–69.
Mathematical Reviews (MathSciNet): MR220424
Zentralblatt MATH: 0165.21505
Digital Object Identifier: doi:10.1214/aoms/1177698504
Project Euclid: euclid.aoms/1177698504
Samarov, A. and Taqqu, M.S. (1988). On the efficiency of the sample mean in long memory noise. J. Time Ser. Anal. 9 191–200.
Mathematical Reviews (MathSciNet): MR943006
Zentralblatt MATH: 0637.62085
Digital Object Identifier: doi:10.1111/j.1467-9892.1988.tb00463.x
Schneider, W.R. (1996). Completely monotone generalized Mittag-Leffler functions. Exposition. Math. 14 3–16.
Mathematical Reviews (MathSciNet): MR1382012
Seneta, E. (1976). Regularly Varying Functions. Lecture Notes in Math. 508. Berlin: Springer.
Mathematical Reviews (MathSciNet): MR453936
Silvey, S.D. (1980). Optimal Design. London: Chapman & Hall.
Mathematical Reviews (MathSciNet): MR606742
Smith, H.F. (1938). An empirical law describing heterogeneity in the yields of agricultural crops. J. Agric. Sci. 28 1–23.
Yajima, Y. (1988). On estimation of a regression model with long-memory stationary errors. Ann. Statist. 16 791–807.
Mathematical Reviews (MathSciNet): MR947579
Zentralblatt MATH: 0661.62090
Digital Object Identifier: doi:10.1214/aos/1176350837
Project Euclid: euclid.aos/1176350837
Yajima, Y. (1991). Asymptotic properties of the LSE in a regression model with long-memory stationary errors. Ann. Statist. 19 158–177.
Mathematical Reviews (MathSciNet): MR1091844
Zentralblatt MATH: 0728.62085
Digital Object Identifier: doi:10.1214/aos/1176347975
Project Euclid: euclid.aos/1176347975
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