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1.4. Notes and References.
Sacks, J., and Ylvisaker, D. (1966), Designs for regression with correlated errors, Ann. Math. Statist. 37, 68–89.
Sacks, J., and Ylvisaker, D. (1968), Designs for regression problems with correlated errors; many parameters, Ann. Math. Statist. 39, 49–69.
Sacks, J., and Ylvisaker, D. (1970a), Design for regression problems with correlated errors III, Ann. Math. Statist. 41, 2057–2074.
Sacks, J., and Ylvisaker, D. (1970b), Statistical design and integral approximation, in: Proc. 12th Bienn. Semin. Can. Math. Congr., R. Pyke, ed., pp. 115–136, Can. Math. Soc., Montreal.
Benhenni, K., and Cambanis, S. (1992a), Sampling designs for estimating integrals of stochastic processes, Ann. Statist. 20, 161–194.
Benhenni, K., and Cambanis, S. (1992b), Sampling designs for estimating integrals of stochastic processes using quadratic mean derivatives, in: Approximation theory, G. A. Anastassiou, ed., pp. 93–123, Dekker, New York.
Su, Y., and Cambanis, S. (1993), Sampling designs for estimation of a random process, Stochastic Processes Appl. 46, 47–89.
Istas, J., and Laredo, C. (1994), Estimation d'intégrales de processus aléatoires à partir d'observations discrétisées, C. R. Acad. Sci. 319, 85–88.
Istas, J., and Laredo, C. (1997), Estimating functionals of a stochastic process, Adv. Appl. Prob. 29, 249–270.
Müller-Gronbach, T. (1996), Optimal designs for approximating the path of a stochastic process, J. Statist. Planning Inf. 49, 371–385.
Novak, E., Ritter, K., and Woźniakowski, H. (1995), Average case optimality of a hybrid secant-bisection method, Math. Comp. 64, 1517–1539.
Ritter, K. (1996a), Asymptotic optimality of regular sequence designs, Ann. Statist. 24, 2081–2096.
Müller-Gronbach, T. (1996), Optimal designs for approximating the path of a stochastic process, J. Statist. Planning Inf. 49, 371–385.
Novak, E., Ritter, K., and Woźniakowski, H. (1995), Average case optimality of a hybrid secant-bisection method, Math. Comp. 64, 1517–1539.
Ritter, K. (1996a), Asymptotic optimality of regular sequence designs, Ann. Statist. 24, 2081–2096.
Novak, E., Ritter, K., and Woźniakowski, H. (1995), Average case optimality of a hybrid secant-bisection method, Math. Comp. 64, 1517–1539.
2.4. Notes and References
Barrow, D. L., and Smith, P. W. (1979), Asymptotic properties of optimal quadrature formulas, in: Numerische Integration, G. Hämmerlin, ed., ISNM 45, pp. 54–66, Birkhäuser Verlag, Basel.
Eubank, R. L., Smith, P. L., and Smith, P. W. (1981), Uniqueness and eventual uniqueness of optimal designs in some time series models, Ann. Statist. 9, 486–493.
Eubank, R. L., Smith, P. L., and Smith, P. W. (1982), A note on optimal and asymptotically optimal designs for certain time series models, Ann. Statist. 10, 1295–1301.
Sacks, J., and Ylvisaker, D. (1966), Designs for regression with correlated errors, Ann. Math. Statist. 37, 68–89.
Sacks, J., and Ylvisaker, D. (1970a), Design for regression problems with correlated errors III, Ann. Math. Statist. 41, 2057–2074.
Benhenni, K., and Cambanis, S. (1992a), Sampling designs for estimating integrals of stochastic processes, Ann. Statist. 20, 161–194.
Ritter, K. (1996a), Asymptotic optimality of regular sequence designs, Ann. Statist. 24, 2081–2096.
Sacks, J., and Ylvisaker, D. (1966), Designs for regression with correlated errors, Ann. Math. Statist. 37, 68–89.
Sacks, J., and Ylvisaker, D. (1970b), Statistical design and integral approximation, in: Proc. 12th Bienn. Semin. Can. Math. Congr., R. Pyke, ed., pp. 115–136, Can. Math. Soc., Montreal.
Barrow, D. L., and Smith, P. W. (1979), Asymptotic properties of optimal quadrature formulas, in: Numerische Integration, G. Hämmerlin, ed., ISNM 45, pp. 54–66, Birkhäuser Verlag, Basel.
Eubank, R. L., Smith, P. L., and Smith, P. W. (1982), A note on optimal and asymptotically optimal designs for certain time series models, Ann. Statist. 10, 1295–1301.
Benhenni, K., and Cambanis, S. (1992a), Sampling designs for estimating integrals of stochastic processes, Ann. Statist. 20, 161–194.
Wahba, G. (1971), On the regression design problem of Sacks and Ylvisaker, Ann. Math. Statist. 42, 1035–1043.
Wahba, G. (1974), Regression design for some equivalence classes of kernels, Ann. Statist. 2, 925–934.
Hájek, J., and Kimeldorf, G. S. (1974), Regression designs in autoregressive stochastic processes, Ann. Statist. 2, 520–527.
Ritter, K. (1996a), Asymptotic optimality of regular sequence designs, Ann. Statist. 24, 2081–2096.
Temirgaliev, N. (1988), On an application of infinitely divisible distributions to quadrature problems, Anal. Math. 14, 253–258.
Voronin, S. M., and Skalyga, V. I. (1984), On quadrature formulas, Soviet Math. Dokl. 29, 616–619.
Lee, D., and Wasilkowski, G. W. (1986), Approximation of linear functionals on a Banach space with a Gaussian measure, J. Complexity 2, 12–43.
Gao, F. (1993), On the role of computable error estimates in the analysis of numerical approximation algorithms, in: Proceedings of the Smalefest, M. W. Hirsch, J. E. Marsden, and M. Shub, eds., pp. 387–394, Springer-Verlag, New York.
3.6. Notes and References
Speckman, P. (1979), L p approximation of autoregressive Gaussian processes, Tech. Rep., Dept. of Statistics, Univ. of Oregon, Eugene.
Su, Y., and Cambanis, S. (1993), Sampling designs for estimation of a random process, Stochastic Processes Appl. 46, 47–89.
Müller-Gronbach, T. (1996), Optimal designs for approximating the path of a stochastic process, J. Statist. Planning Inf. 49, 371–385.
Ritter, K. (1996c), Average case analysis of numerical problems, Habilitations-schrift, Mathematisches Institut, Univ. Erlangen-Nürnberg.
Papageorgiou, A., and Wasilkowski, G. W. (1990), On the average complexity of multivariate problems, J. Complexity 6, 1–23.
Maiorov, V. E. (1992), Widths of spaces endowed with a Gaussian measure, Russian Acad. Sci. Dokl. Math. 45, 305–309.
Sun, Yong-Sheng, and Wang, Chengyong (1994), μ-average n-widths on the Wiener space, J. Complexity 10, 428–436.
Sun, Yong-Sheng, and Wang, Chengyong (1994), μ-average n-widths on the Wiener space, J. Complexity 10, 428–436.
Wasilkowski, G. W. (1992), On average complexity of global optimization problems, Math. Programming 57, 313–324.
Wasilkowski, G. W. (1996), Average case complexity of multivariate integration and function approximation, an overview, J. Complexity 12, 257–272.
5.4. Notes and References
Buslaev, A. P., and Seleznjev, O. V. (1999), On certain extremal problems in the theory of approximation of random processes, East J. Approx. Theory 4, 467–481.
Cambanis, S., and Masry, E. (1994), Wavelet approximation of deterministic and random signals: convergence properties and rates, IEEE Trans. Inform. Theory IT-40, 1013–1029.
Seleznjev, O. V. (2000), Spline approximation of random processes and design problems, J. Statist. Planning Inf. 84, 249–262.
Stein, M. L. (1995c), Predicting integrals of stochastic processes, Ann. Appl. Prob. 5, 158–170.
Benhenni, K. (1997), Approximation d'intégrales de processus stochastiques, C. R. Acad. Sci. 325, 659–663.
Benhenni, K. (1998), Approximating integrals of stochastic processes: extensions, J. Appl. Probab. 35, 843–855.
Stein, M. L. (1995c), Predicting integrals of stochastic processes, Ann. Appl. Prob. 5, 158–170.
Sun, Yong-Sheng, and Wang, Chengyong (1995), Average error bounds of best approximation of continuous functions on the Wiener space, J. Complexity 11, 74–104.
Stein, M. L. (1995c), Predicting integrals of stochastic processes, Ann. Appl. Prob. 5, 158–170.
Istas, J., and Laredo, C. (1994), Estimation d'intégrales de processus aléatoires à partir d'observations discrétisées, C. R. Acad. Sci. 319, 85–88.
Istas, J., and Laredo, C. (1997), Estimating functionals of a stochastic process, Adv. Appl. Prob. 29, 249–270.
Seleznjev, O. V. (1991), Limit theorems for maxima and crossings of a sequence of Gaussian processes and approximation of random processes, J. Appl. Probab. 28, 17–32.
Seleznjev, O. V. (1996), Large deviations in the piecewise linear approximation of Gaussian processes with stationary increments, Adv. Appl. Prob. 28, 481–499.
Hüsler, J. (1999), Extremes of Gaussian processes, on results of Piterbarg and Seleznjev, Stat. Prob. Letters 44, 251–258.
Istas, J., and Laredo, C. (1997), Estimating functionals of a stochastic process, Adv. Appl. Prob. 29, 249–270.
Stein, M. L. (1995c), Predicting integrals of stochastic processes, Ann. Appl. Prob. 5, 158–170.
Benhenni, K., and Istas, J. (1998), Minimax results for estimating integrals of analytic processes, ESAIM, Probab. Stat. 2, 109–121.
Weba, M. (1991a), Quadrature of smooth stochastic processes, Probab. Theory Relat. Fields 87, 333–347.
Weba, M. (1991b), Interpolation of random functions, Numer. Math. 59, 739–746.
Weba, M. (1992), Simulation and approximation of stochastic processes by spline functions, SIAM J. Sci. Stat. Comput. 13, 1085–1096.
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Ritter, K. (2000). Integration and approximation of univariate functions. In: Ritter, K. (eds) Average-Case Analysis of Numerical Problems. Lecture Notes in Mathematics, vol 1733. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0103938
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