×

Matrices – compensating the loss of Anschauung. (English) Zbl 1508.00023

Friedman, Michael (ed.) et al., Model and mathematics: from the 19th to the 21st century. Cham: Birkhäuser. Trends Hist. Sci., 363-377 (2022).
For the entire collection see [Zbl 1495.00074].

MSC:

00A71 General theory of mathematical modeling

References:

[1] Aalen, O., Andersen, P.K., Borgan, Ø., Gill, R., Keiding, N.: History of applications of martingales in survival analysis. Electronic Journal for History of Probability and Statistics 5(1) (2009). 28 pp. · Zbl 1170.01364
[2] Anderson, T.W.: The Statistical Analysis of Time Series. Wiley (1971) · Zbl 0225.62108
[3] Anderson, TW; Taylor, JB, Some experimental results on the statistical properties of least squares estimates in control problems, Econometrica, 44, 6, 1289-1302 (1976) · doi:10.2307/1914261
[4] Andrieu, C.; Doucet, A.; Holenstein, R., Particle Markov Chain Monte Carlo methods, Journal of the Royal Statistical Society: Series B, 72, 3, 269-342 (2010) · Zbl 1411.65020 · doi:10.1111/j.1467-9868.2009.00736.x
[5] Bahadur, RR, A note on the fundamental identity of sequential analysis, Ann. Math. Statist., 29, 534-543 (1958) · Zbl 0099.13404 · doi:10.1214/aoms/1177706628
[6] Bartroff, J.; Lai, TL; Narasimhan, B., A new approach to designing phase I-II cancer trials for cytotoxic chemotherapies, Stat. Med., 33, 2718-2735 (2014) · doi:10.1002/sim.6124
[7] Bartroff, J.; Lai, TL; Shih, MC, Sequential Experimentation in Clinical Trials: Design and Analysis (2013), New York: Springer, New York · Zbl 1281.62001 · doi:10.1007/978-1-4614-6114-2
[8] Baumgartner, R., Chen, J., Lai, T.L.: Real World Data and Evidence: Applications in Precision Medicine and Healthcare. Chapman & Hall/CRC (2021). Forthcoming
[9] Bellman, R.E.: Dynamic Programming. Princeton University Press (1957) · Zbl 0077.13605
[10] Blackwell, D.; Girshick, MA, On functions of sequences of independent chance vectors with applications to the problem of the “random walk” in \(k\) dimensions, Ann. Math. Statist., 17, 310-317 (1946) · Zbl 0060.29007 · doi:10.1214/aoms/1177730943
[11] Bloomfield, P.: Fourier Analysis of Time Series: An Introduction. Wiley (1976) · Zbl 0353.62051
[12] Blum, JR, Multidimensional stochastic approximation methods, Ann. Math. Statist., 25, 737-744 (1954) · Zbl 0056.38305 · doi:10.1214/aoms/1177728659
[13] Box, G.E.P., Jenkins, G.M.: Time Series Analysis: Forecasting and Control. Holden-Day (1971) · Zbl 0249.62009
[14] Box, GEP; Wilson, KB, On the experimental attainment of optimum conditions, J. Roy. Statist. Soc. Ser. B, 13, 1, 1-38 (1951) · Zbl 0043.34402
[15] Brockett, R.W.: Nonlinear systems and nonlinear estimation theory. In: J. Hazewinkel, J.C. Willems (eds.) Stochastic Systems: The Mathematics of Filtering and Identification, and Applications, pp. 441-477. Springer (1981) · Zbl 0505.93064
[16] Brockett, R.W., Clark, J.M.C.: The geometry of the conditional density functions. In: O.L.R. Jacobs et al. (ed.) Analysis and Optimization of Stochastic Systems, pp. 299-309. Academic Press (1980) · Zbl 0496.93049
[17] Chan, HP; Lai, TL, A general theory of particle filters in hidden Markov models and some applications, Ann. Statist., 41, 2877-2904 (2013) · Zbl 1293.60071 · doi:10.1214/13-AOS1172
[18] Chen, J., Heyse, J., Lai, T.L.: Medical Product Safety Evaluation: Biological Models and Statistical Methods. Chapman & Hall/CRC (2018)
[19] Chopin, N.; Jacob, PE; Papaspiliopoulos, O., SMC2: an efficient algorithm for sequential analysis of state space models, Journal of the Royal Statistical Society: Series B, 75, 3, 397-426 (2013) · Zbl 1411.62242 · doi:10.1111/j.1467-9868.2012.01046.x
[20] Chow, YS; Robbins, H.; Teicher, H., Moments of randomly stopped sums, Annals of Mathematical Statistics, 36, 3, 789-799 (1965) · Zbl 0134.35601 · doi:10.1214/aoms/1177700053
[21] Darling, DA; Robbins, H., Iterated logarithm inequalities, Proc. Nat. Acad. Sci. U.S.A., 57, 1188-1192 (1967) · Zbl 0167.46902 · doi:10.1073/pnas.57.5.1188
[22] Doob, JL, Stochastic Processes (1953), New York: Wiley, New York · Zbl 0053.26802
[23] Dvoretzky, A., On stochastic approximation, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, I, 39-55, 1954-1955 (1956) · Zbl 0072.34701
[24] Farrell, RH, Asymptotic behavior of expected sample size in certain one sided tests, Ann. Math. Statist., 35, 36-72 (1964) · Zbl 0156.39306 · doi:10.1214/aoms/1177703731
[25] Ghosh, B.K.: Sequential Tests of Statistical Hypotheses. Addison-Wesley (1970) · Zbl 0223.62097
[26] Gill, R., Censoring and Stochastic Integrals, Mathematical Centre Tracts 124 (1980), Amsterdam: Mathematisch Centrum, Amsterdam · Zbl 0456.62003
[27] Gittins, JC, Bandit processes and dynamic allocation indices, J. Roy. Statist. Soc. Ser. B, 41, 148-177 (1979) · Zbl 0411.62055
[28] Gittins, J.C., M., J.D.: A dynamic allocation index for the sequential design of experiments. Tech. rep., University of Cambridge, Department of Engineering (1974) · Zbl 0303.62064
[29] Gladyšev, E.G.: On stochastic approximation. Teor. Verojatnost. i Primenen. 10 (1965) · Zbl 0147.18002
[30] Gordon, NJ; Salmond, DJ; Smith, AFM, Novel approach to nonlinear/non-Gaussian Bayesian state estimation, IEE Proc. F (Radar and Signal Processing), 140, 107-113 (1993) · doi:10.1049/ip-f-2.1993.0015
[31] Guo, X., Lai, T., H., S., Wong, S.P.: Quantitative Trading: Algorithms, Analytics, Data, Models, Optimization. Chapman & Hall/CRC (2017)
[32] Hoeffding, W.; Robbins, H., The central limit theorem for dependent random variables, Duke Mathematical Journal, 15, 3, 773-780 (1948) · Zbl 0031.36701 · doi:10.1215/S0012-7094-48-01568-3
[33] Hotelling, H., A general mathematical theory of depreciation, Journal of the American Statistical Association, 20, 151, 340-353 (1925) · doi:10.1080/01621459.1925.10503499
[34] Hotelling, H., The general welfare in relation to problems of taxation and of railway and utility rates, Econometrica, 6, 3, 242-269 (1938) · doi:10.2307/1907054
[35] Hotelling, H., Tubes and spheres in \(n\)-spaces, and a class of statistical problems, American Journal of Mathematics, 61, 2, 440-460 (1939) · JFM 65.0795.02 · doi:10.2307/2371512
[36] Hotelling, H.: Experimental determination of the maximum of a function. Ann. Math, Statist. 12, 20-45 (1941) · Zbl 0024.43104
[37] Johnstone, I.; Siegmund, D., On Hotelling’s formula for the volume of tubes and Naiman’s inequality, Ann. Statist., 17, 184-194 (1989) · Zbl 0678.62066 · doi:10.1214/aos/1176347010
[38] Kiefer, J.; Wolfowitz, J., Stochastic estimation of the maximum of a regression function, Ann. Math. Statist., 23, 462-466 (1952) · Zbl 0049.36601 · doi:10.1214/aoms/1177729392
[39] Kim, D.W., Lai, T.L., Xu, H.: Multi-armed bandits with covariates: Theory and applications. Statistica Sinica 31, in press. doi:10.5705/ss.202,020.0454 (2021) · Zbl 1524.62374
[40] Lai, T.; Robbins, H., A class of dependent random variables and their maxima, Z. Wahrsch. verw. Gebiete, 42, 2, 89-111 (1978) · Zbl 0377.60022 · doi:10.1007/BF00536046
[41] Lai, TL, Space-time processes, parabolic functions and one-dimensional diffusions, Trans. Amer. Math. Soc., 175, 409-438 (1973) · Zbl 0262.60056 · doi:10.1090/S0002-9947-1973-0334337-X
[42] Lai, TL, On Chernoff-Savage statistics and sequential rank tests, Ann. Statist., 3, 825-845 (1975) · Zbl 0331.62059 · doi:10.1214/aos/1176343185
[43] Lai, TL, On confidence sequences, Ann. Statist., 4, 2, 265-280 (1976) · Zbl 0346.62035 · doi:10.1214/aos/1176343406
[44] Lai, TL, Power-one tests based on sample sums, Ann. Statist., 5, 866-880 (1977) · Zbl 0368.62011 · doi:10.1214/aos/1176343943
[45] Lai, TL, Adaptive treatment allocation and the multi-armed bandit problem, Ann. Statist., 15, 1091-1114 (1987) · Zbl 0643.62054 · doi:10.1214/aos/1176350495
[46] Lai, TL, Likelihood ratio identities and their applications to sequential analysis, Sequential Anal., 23, 467-497 (2004) · Zbl 1075.62070 · doi:10.1081/SQA-200038994
[47] Lai, T.L.: Martingales in sequential analysis and time series, 1945-1985. Electronic Journal for History of Probability and Statistics 5(1) (2009). 31 pp. · Zbl 1170.01372
[48] Lai, T.L.: Recursive particle filters for joint state and parameter estimation in hidden markov models with multifaceted applications (2021). Proceedings of International Congress of Chinese Mathematicians, Beijing, 2019, in press. International Press of Boston
[49] Lai, TL; Choi, A.; Tsang, KW, Statistical science in information technology and precision medicine, Ann. Math. Sci. & Appl., 4, 413-438 (2019) · Zbl 1432.62011 · doi:10.4310/AMSA.2019.v4.n2.a6
[50] Lai, T.L., Lavori, P.W., Tsang, K.W.: Adaptive design of confirmatory trials: Advances and challenges. Contemp. Clin. Trials 45,10th Anniversary Special Issue, Part A, 93-102 (2015)
[51] Lai, TL; Lavori, PW; Tsang, KW, Adaptive enrichment designs for confirmatory trials, Stat. Med., 38, 613-624 (2019) · doi:10.1002/sim.7946
[52] Lai, TL; Robbins, H., Adaptive design and stochastic approximation, Ann. Statist., 7, 1196-1221 (1979) · Zbl 0426.62059 · doi:10.1214/aos/1176344840
[53] Lai, TL; Robbins, H., Consistency and asymptotic efficiency of slope estimates in stochastic approximation schemes, Z. Wahrsch. verw. Gebiete, 56, 329-360 (1981) · Zbl 0472.62089 · doi:10.1007/BF00536178
[54] Lai, TL; Robbins, H., Asymptotically efficient adaptive allocation rules, Adv. in Appl. Math., 6, 4-22 (1985) · Zbl 0568.62074 · doi:10.1016/0196-8858(85)90002-8
[55] Lai, TL; Sklar, M.; Weissmueller, NT, Novel clinical trial designs and statistical methods in the era of precision medicine, Statistics in Biopharmaceutical Research, 13, 2, 133-146 (2021) · doi:10.1080/19466315.2020.1814403
[56] Lai, T.L., Sklar, M.B., Xu, H.: Bandit and covariate processes with finite or non-denumerable set of arms. Stochastic Processes and Their Applications to appear, to appear (2021) · Zbl 1493.62476
[57] Lai, TL; Wijsman, RA, First exit time of a random walk from the bounds \(f(n) \pm cg(n)\), with applications, Ann. Probab., 7, 672-692 (1979) · Zbl 0413.60043 · doi:10.1214/aop/1176994990
[58] Lai, T.L., Xing, H.: Data Analytics and Risk Management in Finance and Insurance. Chapman & Hall/CRC (2008)
[59] Lai, T.L., Xing, H.: Statistical Models and Methods in Financial Markets. Springer (2008) · Zbl 1149.62086
[60] Lai, TL; Yuan, H., Stochastic approximation: From statistical origin to big-data, multidisciplinary applications, Statistical Science, 36, 2, 291-302 (2021) · Zbl 07368238 · doi:10.1214/20-STS784
[61] Lorden, G., Procedures for reacting to a change in distribution, Ann. Math. Statist., 42, 1897-1908 (1971) · Zbl 0255.62067 · doi:10.1214/aoms/1177693055
[62] Lu, Y.; Small, D.; Ying, Z., A conversation with Tze Leung Lai, Statistical Science, 36, 1, 158-167 (2021) · Zbl 07368225 · doi:10.1214/20-STS775
[63] Mangel, M.; Samaniego, F., Abraham Wald’s work on aircraft survivability, J. Amer. Statist. Assoc., 79, 259-267 (1984) · doi:10.1080/01621459.1984.10478038
[64] Mitter, SK, On the analogy between mathematical problems of non-linear filtering and quantum physics, Ricerche di Automatica, 10, 2, 163-216 (1979)
[65] Morgenstern, O., Abraham Wald, 1902-1950, Econometrica, 19, 4, 361-367 (1951) · Zbl 0043.24510 · doi:10.2307/1907462
[66] Naiman, DQ, Conservative confidence bands in curvilinear regression, Ann. Statist., 14, 896-906 (1986) · Zbl 0607.62077 · doi:10.1214/aos/1176350040
[67] Neyman, J.: Foundations of behavioristic statistics. In: Foundations of Statistical Inference, Proc. Sympos., Univ. Waterloo, Waterloo, Ontario, 1970, pp. 1-19. Holt, Toronto (1971)
[68] Page, W., An interview with Herbert Robbins, College Math. J., 15, 2-24 (1944) · Zbl 0995.01513 · doi:10.2307/3027425
[69] de la Peña, V.; Klass, MJ; Lai, TL, Self-normalized processes: Exponential inequalities, moment bounds and iterated logarithm laws, Annals of Probability, 32, 3, 1902-1933 (2004) · Zbl 1075.60014
[70] de la Peña, V., Lai, T.L.: Theory and applications of decoupling. In: C. Charalambides, M. Koutras, N. Balakrishnan (eds.) Probability and Statistical Models with Applications, pp. 115-145. Chapman & Hall/CRC (2001)
[71] de la Peña, V., Lai, T.L., Shao, Q.M.: Self-Normalized Processes: Limit Theory and Statistical Applications. Springer (2009) · Zbl 1165.62071
[72] Philipp, W.; Stout, W., Almost sure invariance principles for partial sums of weakly dependent random variables (1975), Soc: Amer. Math, Soc · Zbl 0361.60007 · doi:10.1090/memo/0161
[73] Robbins, H., On the measure of a random set, Annals of Mathematical Statistics, 15, 1, 70-74 (1944) · Zbl 0060.29405 · doi:10.1214/aoms/1177731315
[74] Robbins, H., On the measure of a random set II, Annals of Mathematical Statistics, 16, 4, 342-347 (1945) · Zbl 0060.29406 · doi:10.1214/aoms/1177731060
[75] Robbins, H.: Asymptotically subminimax solutions of compound statistical decision problems. Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability pp. 131-148 (1951) · Zbl 0044.14803
[76] Robbins, H., Some aspects of the sequential design of experiments, Bull. Amer. Math. Soc., 58, 5, 527-535 (1952) · Zbl 0049.37009 · doi:10.1090/S0002-9904-1952-09620-8
[77] Robbins, H., Statistical methods related to the law of the iterated logarithm, Ann. Math. Statist., 41, 5, 1397-1409 (1970) · Zbl 0239.62025 · doi:10.1214/aoms/1177696786
[78] Robbins, H.; Monro, S., A stochastic approximation method, Annals of Mathematical Statistics, 22, 3, 400-407 (1951) · Zbl 0054.05901 · doi:10.1214/aoms/1177729586
[79] Robbins, H.; Siegmund, D., Boundary crossing probabilities for the Wiener process and sample sums, Ann. Math. Statist., 41, 1410-1429 (1970) · Zbl 0255.60058 · doi:10.1214/aoms/1177696787
[80] Robbins, H., Siegmund, D.: A convergence theorem for nonnegative almost supermartingales and some applications. In: Optimizing Methods in Statistics (Proc. Sympos., Ohio State Univ.), pp. 233-257. Academic Press (1971) · Zbl 0286.60025
[81] Robbins, H., Siegmund, D.: A class of stopping rules for testing parametric hypotheses. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, 1970/1971 IV: Biology and Health, 37-41 (1972)
[82] Robbins, H.; Siegmund, D., Statistical tests of power one and the integral representation of solutions of certain partial differential equations, Bull. Inst. Math. Acad. Sinica, 1, 93-120 (1973) · Zbl 0289.62055
[83] Robbins, H.; Siegmund, D., The expected sample size of some tests of power one, Ann. Statist., 2, 415-436 (1974) · Zbl 0318.62069 · doi:10.1214/aos/1176342704
[84] Sawyer, S.: A Fatou theorem for the general one-dimensional parabolic equation. Indiana Univ. Math. J. 24, 451-498 (1974/75) · Zbl 0277.60059
[85] Siegmund, D.: Sequential Analysis. Springer (1985) · Zbl 0573.62071
[86] Storvik, G., Particle filters for state-space models with the presence of unknown static parameters, IEEE Transactions on Signal Processing, 50, 2, 281-289 (2002) · doi:10.1109/78.978383
[87] Wald, A., On cumulative sums of random variables, Ann. Math. Statist., 15, 3, 283-296 (1944) · Zbl 0063.08122 · doi:10.1214/aoms/1177731235
[88] Wald, A., Sequential tests of statistical hypothesis, Ann. Math. Statist., 16, 2, 117-186 (1945) · Zbl 0060.30207 · doi:10.1214/aoms/1177731118
[89] Whittle, P.: Multi-armed bandits and the Gittins index. J. Roy. Statist. Soc. Ser. B 42 (1980) · Zbl 0439.90096
[90] Widder, DV, Positive temperatures on an infinite rod, Trans. Amer. Math. Soc., 55, 85-95 (1944) · Zbl 0061.22303 · doi:10.1090/S0002-9947-1944-0009795-2
[91] Widder, DV, Positive temperatures on a semi-infinite rod, Trans. Amer. Math. Soc., 75, 510-525 (1953) · Zbl 0052.32701 · doi:10.1090/S0002-9947-1953-0058104-7
[92] Wijsman, R.: Sequential confidence intervals with beta protection in one-parameter families. In: J. Van Ryzin (ed.) Adaptive Statistical Procedure and Related Topics, pp. 96-107. Institute of Mathematical Statistics (1985) · Zbl 0679.62063
[93] Woodroofe, M.: Nonlinear Renewal Theory in Sequential Analysis. SIAM (1982). CBMS-NSF Regional Conference Series in Applied Mathematics 39 · Zbl 0487.62062
[94] Working, H.; Hotelling, H., Applications of the theory of error to the interpretation of trends, J. Amer. Statist. Assoc., 24, 165, 73-85 (1929) · JFM 55.0928.04 · doi:10.1080/01621459.1929.10506274
[95] Wu, H.T., Lai, T.L., Haddad, G.G., Muotri, A.: Oscillatory biomedical signals: Frontiers in mathematical models and statistical analysis. Frontiers in Applied Mathematics and Statistics 7 (2021). 31 pp.
[96] Yau, S.T., Yau, S.S.T.: Finite-dimensional filters with nonlinear drift. III: Duncan-Mortensen-Zakai equation with arbitrary initial condition for the linear filtering system and the benes filtering system. IEEE Transactions on Aerospace and Electronic Yystems 33(4), 1277-1294 (1997)
[97] Zeng, Y., A partially observed model for micromovement of asset prices with bayes estimation via filtering, Mathematical Finance, 13, 3, 411-444 (2003) · Zbl 1130.91346 · doi:10.1111/1467-9965.t01-1-00022
[98] Zeng, Y., Estimating stochastic volatility via filtering for the micromovement of asset prices, IEEE Transactions on Automatic Control, 49, 3, 338-348 (2004) · Zbl 1366.91166 · doi:10.1109/TAC.2004.824478
[99] Zheng, T., Ying, Z.: Columbia University statistics. In: A. Agresti, X. Meng (eds.) Strength in Numbers: The Rising of Academic Departments in the U.S. Springer (2013)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.