www.ebook3000.com STATISTICAL PORTFOLIO ESTIMATION www.ebook3000.com STATISTICAL PORTFOLIO ESTIMATION Masanobu Taniguchi, Hiroshi Shiraishi, Junichi Hirukawa, Hiroko Kato Solvang, and Takashi Yamashita MATLAB• is a trademark of The MathWorks, Inc and is used with permission The MathWorks does not warrant the accuracy of the text or exercises in this book This book’s use or discussion of MATLAB • software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB • software CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2018 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Printed on acid-free paper Version Date: 20170720 International Standard Book Number-13: 978-1-4665-0560-5 (Hardback) This book contains information obtained from authentic and 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please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com www.ebook3000.com Contents Preface ix Introduction Preliminaries 2.1 Stochastic Processes and Limit Theorems 5 Portfolio Theory for Dependent Return Processes 3.1 Introduction to Portfolio Theory 3.1.1 Mean-Variance Portfolio 3.1.2 Capital Asset Pricing Model 3.1.3 Arbitrage Pricing Theory 3.1.4 Expected Utility Theory 3.1.5 Alternative Risk Measures 3.1.6 Copulas and Dependence 3.1.7 Bibliographic Notes 3.1.8 Appendix 3.2 Statistical Estimation for Portfolios 3.2.1 Traditional Mean-Variance Portfolio Estimators 3.2.2 Pessimistic Portfolio 3.2.3 Shrinkage Estimators 3.2.4 Bayesian Estimation 3.2.5 Factor Models 3.2.5.1 Static factor models 3.2.5.2 Dynamic factor models 3.2.6 High-Dimensional Problems 3.2.6.1 The case of m/n → y ∈ (0, 1) 3.2.6.2 The case of m/n → y ∈ (1, ∞) 3.2.7 Bibliographic Notes 3.2.8 Appendix 3.3 Simulation Results 3.3.1 Quasi-Maximum Likelihood Estimator 3.3.2 Efficient Frontier 3.3.3 Difference between the True Point and the Estimated Point 3.3.4 Inference of µP 3.3.5 Inference of Coefficient 3.3.6 Bibliographic Notes 19 19 20 23 27 28 33 36 38 38 41 42 44 46 48 52 52 55 58 59 62 63 63 65 66 68 69 70 72 73 Multiperiod Problem for Portfolio Theory 4.1 Discrete Time Problem 4.1.1 Optimal Portfolio Weights 4.1.2 Consumption Investment 75 76 76 78 v vi CONTENTS 4.2 4.3 4.1.3 Simulation Approach for VAR(1) model 4.1.4 Bibliographic Notes Continuous Time Problem 4.2.1 Optimal Consumption and Portfolio Weights 4.2.2 Estimation 4.2.2.1 Generalized method of moments (GMM) 4.2.2.2 Threshold estimation method 4.2.3 Bibliographic Notes Universal Portfolio 4.3.1 µ-Weighted Universal Portfolios 4.3.2 Universal Portfolios with Side Information 4.3.3 Successive Constant Rebalanced Portfolios 4.3.4 Universal Portfolios with Transaction Costs 4.3.5 Bibliographic Notes 4.3.6 Appendix Portfolio Estimation Based on Rank Statistics 5.1 Introduction to Rank-Based Statisticsrank 5.1.1 History of Ranks 5.1.1.1 Wilcoxon’s signed rank and rank sum tests 5.1.1.2 Hodges–Lehmann and Chernoff–Savage 5.1.2 Maximal Invariantsmaximal invariant 5.1.2.1 Invariance of sample space, parameter space and tests 5.1.2.2 Most powerful invariant testmost powerful invariant test 5.1.3 Efficiencyefficiency of Rank-Based Statistics 5.1.3.1 Least favourableleast favourable density and most powerfulmost powerful test 5.1.3.2 Asymptotically most powerful rank test 5.1.4 U-Statistics for Stationary Processes 5.2 Semiparametrically Efficient Estimation in Time Series 5.2.1 Introduction to Rank-Based Theory in Time Series 5.2.1.1 Testing for randomness against ARMA alternatives 5.2.1.2 Testing an ARMA model against other ARMA alternatives 5.2.2 Tangent Spacetangent space 5.2.3 Introduction to Semiparametric Asymptotic Optimal Theory 5.2.4 Semiparametrically Efficient Estimation in Time Series, and Multivariate Cases 5.2.4.1 Rank-based optimal influence functions (univariate case) 5.2.4.2 Rank-based optimal estimation for elliptical residuals 5.3 Asymptotic Theory of Rank Order Statistics for ARCH Residual Empirical Processes 5.4 Independent Component Analysis 5.4.1 Introduction to Independent Component Analysis 5.4.1.1 The foregoing model for financial time series 5.4.1.2 ICA modeling for financial time series 5.4.1.3 ICA modeling in frequency domain for time series 5.5 Rank-Based Optimal Portfolio Estimation 5.5.1 Portfolio Estimation Based on Ranks for Independent Components 5.5.2 Portfolio Estimation Based on Ranks for Elliptical Residualselliptical residuals www.ebook3000.com 80 84 84 85 90 91 95 98 98 98 101 104 106 108 109 113 113 113 113 117 124 124 125 126 126 129 137 139 139 139 146 149 155 159 159 164 170 180 180 180 187 191 202 202 204 CONTENTS Portfolio Estimation Influenced by Non-Gaussian Innovations and Exogenous Variables 6.1 Robust Portfolio Estimation under Skew-Normal Return Processes 6.2 Portfolio Estimators Depending on Higher-Order Cumulant Spectra 6.3 Portfolio Estimation under the Utility Function Depending on Exogenous Variables 6.4 Multi-Step Ahead Portfolio Estimation 6.5 Causality Analysis 6.6 Classificationclassification by Quantile Regressionquantile regression 6.7 Portfolio Estimation under Causal Variables Numerical Examples 7.1 Real Data Analysis for Portfolio Estimation 7.1.1 Introduction 7.1.2 Data 7.1.3 Method 7.1.3.1 Model selection 7.1.3.2 Confidence region 7.1.3.3 Locally stationary estimation 7.1.4 Results and Discussion 7.1.5 Conclusions 7.1.6 Bibliographic Notes 7.2 Application for Pension Investment 7.2.1 Introduction 7.2.2 Data 7.2.3 Method 7.2.4 Results and Discussion 7.2.5 Conclusions 7.3 Microarray Analysis Using Rank Order Statistics for ARCH Residual 7.3.1 Introduction 7.3.2 Data 7.3.3 Method 7.3.3.1 The rank order statistic for the ARCH residual empirical process 7.3.3.2 Two-group comparison for microarray data 7.3.3.3 GO analysis 7.3.3.4 Pathway analysis 7.3.4 Simulation Study 7.3.5 Results and Discussion 7.3.5.1 Simulation data 7.3.5.2 Affy947 expression dataset 7.3.6 Conclusions 7.4 Portfolio Estimation for Spectral Density of Categorical Time Series Data 7.4.1 Introduction 7.4.2 Method 7.4.2.1 Spectral Envelope 7.4.2.2 Diversification analysis 7.4.2.3 An extension of SpecEnv to the mean-diversification efficient frontier 7.4.3 Data 7.4.3.1 Simulation data 7.4.3.2 DNA sequence data 7.4.4 Results and Discussion 7.4.4.1 Simulation study vii 207 207 211 215 221 224 227 229 235 235 235 235 237 237 238 239 240 242 243 243 244 244 244 248 248 248 249 250 252 252 252 254 254 254 255 255 255 257 259 259 259 259 260 260 261 261 261 262 262 viii CONTENTS 7.5 7.4.4.2 DNA sequence for the BNRF1 genes 7.4.5 Conclusions Application to Real-Value Time Series Data for Corticomuscular Functional Coupling for SpecEnv and the Portfolio Study 7.5.1 Introduction 7.5.2 Method 7.5.3 Results and Discussion 263 268 270 270 274 274 Theoretical Foundations and Technicalities 8.1 Limit Theorems for Stochastic Processes 8.2 Statistical Asymptotic Theory 8.3 Statistical Optimal Theory 8.4 Statistical Model Selection 8.5 Efficient Estimation for Portfolios 8.5.1 Traditional mean variance portfolio estimators 8.5.2 Efficient mean variance portfolio estimators 8.5.3 Appendix 8.6 Shrinkage Estimation 8.7 Shrinkage Interpolation for Stationary Processes 283 283 287 293 300 312 313 315 324 331 342 Bibliography 349 Author Index 365 Subject Index 371 www.ebook3000.com Preface The field of financial engineering has developed as a huge integration of economics, probability theory, statistics, etc., for some decades The composition of porfolios is one of the most fundamental and important methods of financial engineering to control the risk of investments This book provides a comprehensive development of statistical inference for portfolios and its applications Historically, Markowitz (1952) contributed to the advancement of modern portfolio theory by laying the foundation for the diversification of investment portfolios His approach is called the meanvariance portfolio, which maximizes the mean of portfolio return with reducing its variance (risk of portfolio) Actually, the mean-variance portfolio coefficients are expressed as a function of the mean and variance matrix of the return process Optimal portfolio coefficients based on the mean and variance matrix of return have been derived by various criteria Assuming that the return process is i.i.d Gaussian, Jobson and Korkie (1980) proposed a portfolio coefficient estimator of the optimal portfolio by making the sample version of the mean-variance portfolio However, empirical studies show that observed stretches of financial return are often are non-Gaussian dependent In this situation, it is shown that portfolio estimators of the mean-variance type are not asymptotically efficient generally even if the return process is Gaussian, which gives a strong warning for use of the usual portfolio estimators We also provide a necessary and sufficient condition for the estimators to be asymptotically efficient in terms of the spectral density matrix of the return This motivates the fundamental important issue of the book Hence we will provide modern statistical techniques for the problems of portfolio estimation, grasping them as optimal statistical inference for various return processes We will introduce a variety of stochastic processes, e.g., non-Gaussian stationary processes, non-linear processes, non-stationary processes, etc For them we will develop a modern statistical inference by use of local asymptotic normality (LAN), which is due to LeCam The approach is a unified and very general one Based on this we address a lot of important problems for portfolios It is well known that a Markowitz portfolio is to optimize the mean and variance of portfolio return However, recent years have seen increasing development of new risk measures instead of the variance such as Value at Risk (VaR), Expected Shortfall (ES), Conditional Value at Risk (CVaR) and Tail Conditional Expectation (TCE) We show how to construct optimal portfolio estimators based on these risk measures Thanks to the advancement of computer technology, a variety of financial transactions have became possible in a very short time in recent years From this point of view, multiperiod problems are one of the most important issues in portfolio theory We discuss this problems from three directions of perspective In the investment of the optimal portfolio strategy, multivariate time series models are required Moreover, the assumption that the innovation densities underlying those models are known seems quite unrealistic If those densities remain unspecified, the model becomes a semiparametric one, and rank-based inference methods naturally come into the picture Rank-based inference methods under very general conditions are known to achieve semiparametric efficiency bounds However, defining ranks in the context of multivariate time series models is not obvious Two distinct definitions can be proposed The first one relies on the assumption that the innovation process is described by some unspecified independent component analysis model The second one relies on the assumption that the innovation density is some unspecified elliptical density Applications to portfolio management problems, rank statistics and mean-diversification efficient frontier are discussed These examples give readers a practical hint for applying the introduced portfolio theories This book contains applications ranging widely, from ix 364 BIBLIOGRAPHY Yingying Xu, Zhuwu Wu, Long Jiang, and Xuefeng Song A maximum entropy method for a robust portfolio problem Entropy, 16(6):3401–3415, 2014 Menahem E Yaari The dual theory of choice under risk Econometrica, 55(1):95–115, January 1987 Yee Hwa Yang, Yuanyuan Xiao, and Mark R Segal Identifying differentially expressed genes from microarray experiments via statistic synthesis Bioinformatics, 21(7):1084–1093, 2005 Nakahiro Yoshida Estimation for diffusion processes from discrete observation J Multivariate Anal., 41(2):220–242, 1992 Ken-ichi Yoshihara Limiting behavior of {U}-statistics for stationary, absolutely regular processes Z Wahrscheinlichkeitstheorie und Verw Gebiete, 35(3):237–252, 1976 Xi Zhao, Einar Andreas Rødland, Therese Sørlie, Bjørn Naume, Anita Langerød, Arnoldo Frigessi, Vessela N Kristensen, Anne-Lise Børresen-Dale, and Ole Christian Lingjærde Combining gene signatures improves prediction of breast cancer survival PLoS One, 6(3):e17845, 2011 Xi Zhao, Einar Andreas Rødland, Therese Sørlie, Hans Kristian Moen Vollan, Hege G Russnes, Vessela N Kristensen, Ole Christian Lingjærde, and Anne-Lise Børresen-Dale Systematic assessment of prognostic gene signatures for breast cancer shows distinct influence of time and er status BMC Cancer, 14(1):1, 2014 Rongxi Zhou, Ru Cai, and Guanqun Tong Applications of entropy in finance: A review Entropy, 15(11):4909–4931, 2013 www.ebook3000.com Author Index ˇ ak, Z., 170 Sid´ Box, G.E.P., 274 Brammer, M.J., 224 Brandt, M.W., 76, 80, 81, 83, 84 Breeden, D.T., 26 Brettschneider, J., 251 Brillinger, D.R., 8, 64, 193, 211, 212, 230, 232, 284, 285 Brockwell, P.J., 8, 44, 55, 214, 302, 319 Brown, B.M., 16 Brown, P.O., 254 Brown, S.J., 38 Băuhlmann, P., 244 Børresen-Dale, A.L., 249, 251, 280 Absil, P., 257 Acerbi, C., 20, 35 Adcock, C.J., 207 Ait-Sahalia, Y., 41, 50, 63, 78, 80, 84, 89, 90, 96, 98, 207 Akaike, H., 238, 252, 300, 302, 309 Akslen, L.A., 251 Alexander, G.J., 27 Allais, M., 31 Allen, M., 238 Amano, T., 84, 244, 245 Amaro, E., 224 An, Z., 174 Anderson, T.W., 8, 230, 333 Andrews, F.C., 117 Artzner, P., 20, 34, 35 Aue, A., 26 Cacho-Diaz, J., 84, 89, 90, 98 Cai, R., 259 Caldas, C., 257 Campain, A., 249, 250 Campbell, J.Y., 41 Cardoso, E.F., 224 Chamberlain, G., 26, 56 Chandra, S.A., 174, 176, 178–180, 250 Chapados, N., 84 Chen, C.W.S., 229 Chen, M., 174 Chernoff, H., 120, 121, 123, 170 Cherry, J.M., 254 Chew, S.H., 31 Chin, K., 251 Chochola, O., 26 Choquet, G., 32 Choy, K., 298 Chu, G., 249 Claeskens, G., 300 Claffey, K.P., 250 Clarke, R.B., 251 Collin, F., 251 Connor, G., 41 Cope, L., 251 Cover, T.M., 2, 75, 98–103, 108–111 Cvitanic, J., 78 Baba, K., 270 Bagasheva, B.S., 49, 63 Bai, Z., 58, 60, 61 Bancroft, T.A., 300, 307 Baptista, A.M., 27 Barkai, N., 250 Basak, G., 1, 41 Basawa, I.V., 299 Bassett, G., 20, 32, 35, 41, 45, 46, 227 Bellman, R., 76, 314 Benjamini, Y., 249, 252 Beran, J., 244 Bergh, J., 251 Bickel, P.J., 149, 151–154 Binkley, G., 254 Birney, E., 254 Black, F., 50 Blum, A., 98, 106–108, 111 Bodnar, T., 58, 62, 63 Bollerslev, T., 13, 180 Bolstad, B.M., 251 Bos, P.D., 251 Bose, A., 238 Botstein, D., 254 d’Assignies, M.S., 251 D’Eustachio, P., 254 365 366 AUTHOR INDEX Dahlhaus, R., 239, 312 Datta, S., 238 Davis, R.A., 8, 44, 55, 214, 302, 319 de Bono, B., 254 De Luca, G., 207 Delbaen, F., 20, 34, 35 Dellaportas, P., 182, 183 Delorenzi, M., 251 Dempster, A.P., 198–200 Dempster, M.A.H., 106, 108 Der, Z.A., 196, 201 Desmedt, C., 251 DeVries, S., 251 DiCiccio, T., 209, 210 Diehn, M., 254 Domrachev, M., 250 Drost, F C., 203 Dunsmuir, W., 285 Eber, J.M., 20, 34, 35 Eden, E., 254 Edgar, R., 250 Efron, B., 238 Eisen, M.B., 251 EL Karoui, N., 58, 60, 61 Ellis, P., 251 Elton, E.J., 38 Embrechts, P., 38 Eng, J.K., 250 Engle, R.F., 6, 180, 181, 250 Fabozzi, F.J., 49, 63 Fama, E.F., 53, 75, 78 Fan, J., 184, 186 Fang, H., 27 Fishburn, P.C., 33 Florens-Zmirou, D., 96 Foekens, J.A., 251 Forni, M., 11, 12, 56–58, 63 Frahm, G., 61 Franke, J., 238 French, K.R., 53 Frey, R., 38 Fridlyand, J., 251 Friedman, J., 63 Frigessi, A., 249 Frost, P.A., 47 Fujita, A., 224 Gaivoronski, A.A., 98, 104–106, 108 Garat, P., 191, 192 Garel, B., 294 Gasser, T., 317 Gastwirth, J.L., 124 Genton, M.G., 207 George, J., 251 Gerald, W.L., 251 Geweke, J., 224 Giacometti, R., 20, 33 Gillespie, M., 254 Gillet, C., 251 Giraitis, L., 13 Giri, D.D., 251 Girvan, M., 254, 257 Glass, K., 254, 257 Glombek, K, 60 Glombek, K., 58–61, 63 Goetzmann, W.N., 38 Goldberger, A.S., 50 Gonc¸alves, S., 244 Gopinah, G.R., 254 Gourieroux, C., 63, 79, 84, 207 Goyal, A., 76, 80, 81, 83, 84 Granger, C.W.J., 224 Grenander, U., 221, 342 Gruber, M.J., 38 Gupta, G.P., 251 Hăardle, W., 14, 224, 274, 317 Hafner, C.M., 317 Haggan, V., 274 Haibe-Kains, B., 251 Hajek, J., 126, 128–130, 132–136, 141, 144, 163, 297 Hall, P., 15, 243, 251 Hallett, M., 270 Hallin, M., 11, 12, 56–58, 63, 139, 141–143, 145, 146, 148, 149, 155–163, 165–168, 170, 202, 203, 205, 206, 294, 298 Hamada, K., 213, 217, 223, 346 Han, D.K., 250 Hannan, E.J., 8–10, 15, 65, 174, 222, 225, 226, 238, 285, 288, 300, 309, 314, 332, 339, 342 Hansen, L., 41, 50, 63, 78, 84, 98 Harris, A., 251 Harvey, C.R., 27 Hashimoto, N., 270 Hastie, T., 63 Heath, D., 20, 34, 35 Hernandez-Boussard, T., 254 Hewitt, E., 10 Hey, Y., 251 www.ebook3000.com AUTHOR INDEX Heyde, C.C., 15, 93, 98 Hirukawa, J., 202, 203, 227, 261, 299, 305, 308, 312, 321, 322, 324, 326, 327, 329, 330, 332 Hjort, N.L., 300 Hobbs, B., 251 Hochberg, Y., 249, 252 Hodges, J.L., 117, 119 Honda, M., 14, 270, 271, 273, 274, 291, 298, 304, 317, 319 Horlings, H.M., 250 Horv´ath, L., 26, 170–172, 177 Hosoya, Y., 63, 65, 224, 284, 285, 288, 315, 316, 344 Hotta, L.K., 36 Howell, A., 251 Hsu, J.S.J., 49, 63 Hsu, Y.T., 229 Hurd, T.R., 84, 89, 90, 98 Huskova, M., 26 Hwang, S., 27 Hăormann, S., 26 H´ajek, J., 170 Ibragimov, I.A., 16, 137 Iino, M., 250 Ikeda, A., 270 Ingenbleek, J.F., 139, 141–143, 145, 161, 294 Inoue, Y., 270 Irizarry, R.A., 251 Jacod, J., 291 Jagannathan, R., 41 James, W., 47, 331, 342 Jasiak, J., 63, 79, 84, 207 Jassal, B., 254 Jeffrey, S.S., 251 Jeffreys, H., 50 Jeganathan, P., 296 Jeong, K., 224 Jiang, R., 259 Jiang, Z., 12, 14 Jin, H., 254 Jobson, J., 1, 41, 47 Joe, H., 36–38 John, G., 55 Johnsen, H., 251 Jorion, P., 41, 47 Joshi-Tope, G., 254 Journ´ee, M., 257 Kahneman, D., 32 367 Kakizawa, Y., 8, 9, 227, 287, 289, 295, 304, 313, 314, 316, 317 Kakouris, I., 36, 38 Kalai, A., 98, 106–108, 111 Kase, T., 209, 210 Kataoka, S., 20, 33 Kato, H., 14, 248, 270, 271, 273, 274, 291, 298, 304, 317, 319 Keenan, D.M., 213 Kelly, J.L., 2, 75, 78 Kessler, M., 96 Kilian, L., 244 Klaar, S., 251 Klaassen, C.A.J., 149, 151–154, 203 Knight, K., 228 Kobayashi, I., 229, 233, 234 Koenker, R., 20, 32, 35, 38, 41, 45, 46, 63, 227, 229 Kokoszka, P., 13, 170–172, 177 Kondo, M., 226, 227 Konno, H., 33 Kordas, G., 20, 32, 35, 41, 45, 46 Koren, A., 250 Korkie, B., 1, 41, 47 Korkie, R.M., 41, 47 Koul, H.L., 300, 307 Kraus, A., 26 Kreiss, J.P., 238, 293, 294 Kristensen, V., 249, 251, 280 Kroner, K.F., 181 Kruskal, W.H., 117 Kunieda, T., 270 Kuo, W.L., 251 Lahiri, S.N., 238, 243 Lai, T.Y., 27 Laird, N.M., 198–200 Lallemand, F., 251 Langerød, A., 249 Lapuk, A., 251 Lash, A.E., 250 Lauprete, G.J., 41 Lazrak, A., 78 LeCam, L., 1, 271, 293 Ledoit, O., 41, 47 Lee, S., 250, 252, 298 Lehmann, E.L., 117, 119, 124–126, 128 Leippold, M., 78 Leipus, R., 13 Lewis, S., 254 Li, D., 78 Lim, K.G., 27 368 AUTHOR INDEX Lin, W.L., 180–182 Lingjærde, O.C., 249, 251, 280 Lintner, J., 24 Lippi, M., 11, 12, 56–58, 63 Lipson, D., 254 Litterman, R., 50 Litzenberger, R.H., 26 Liu, E.T., 251 Liu, H., 58, 60, 61 Liu, Y., 343 Ljung, G.M., 274 Lo, A.W., 41 Loi, S., 251 Loperfido, N., 207 Lozza, S., 20, 33 Lu, Z., 12, 14 Luenberger, D.G., 24, 26–28, 30, 38, 76 Lukepohl, H., 224 Lundgren, D.H., 250 MacCrimmmon, K.R., 31 Mackinlay, A.C., 41 Magnus, J.R., 226, 332 Mancini, C., 97 Markowitz, H.M., 1, 19, 21, 26, 33, 38 Martin, R.D., 63 Marˇcenko, V.A., 58 Massagu´e, J., 251 Matese, J.C., 254 Matsumoto, R., 270 Matteson, D.S., 187, 189–191 Matthews, L., 254 McDougall, A.J., 235, 259 McNeil, A.J., 38 Merton, R.C., 26, 75, 84, 98 Meucci, A., 259–261 Michailidis, G., 224 Mihara, T., 270 Milhøj, A., 171 Miller, C.J., 251 Miller, L.D., 251 Mima, T., 270 Minn, A.J., 251 Mitra, G., 106, 108 Monti, A.C., 209, 210 Morgan, J.S., 259 Morgenstern, O., 20, 29 Mossin, J., 24, 78 Mykland, P.A., 207 Nagamine, T., 270 Naume, B., 249 Navon, R., 254 Neudecker, H., 226, 332 Neve, R.M., 251 Ng, K.W., 108, 110 Ng, W.L., 78 Noether, G.E., 122 Ogata, H., 84, 244, 245 Ohara, S., 270 Okoniewski, M.J., 251 Olivieri, A., 244 Olshen, A.B., 251 Ordentlich, E., 98, 100–103, 108, 109, 111 Ozaki, T., 250, 274 Paindaveine, D., 165, 166, 168, 170, 205, 206 Palaro, H.P., 36 Parolya, N., 58, 62, 63 Pastur, L.A., 58 Patilea, V., 84, 244, 245 Pawitan, Y., 251 Pepper, S.D., 251 Perou, C.M., 251 Petkovic, A., 209, 210 Pflug, G.C., 36, 106, 108 Pham, D.T., 191, 192 Piette, F., 251 Pitacco, E., 244 Pitman, E.J.G., 117, 118 Pliska, S.R., 76, 79, 84 Ploner, A., 251 Polimenis, V., 27 Politis, D.N., 182, 183 Pollack, J.R., 251 Prakasa Rao, B.L.S., 96, 291, 292, 299 Pratt, J.M., 29 Prigent, J.L., 29–31, 38 Pr´asˇkov´a, Z., 26 Puri, M.L., 139, 141–143, 145, 146, 148, 149, 161, 170, 172, 176, 177, 226, 227, 294 Qian, Z., 251 Quiggin, J., 32 Quinn, B.G., 238, 300 Rachev, S.T., 49, 63 Ratti, V., 41, 47 Rees, C.A., 251, 254 Reichlin, L., 11, 12, 56–58, 63 Reinders, J.T., 250 www.ebook3000.com AUTHOR INDEX Reyal, F., 250 Rezaul, K., 250 Ritov, Y., 149, 151–154 Rockafellar, R.T., 20, 35, 36, 38 Romano, J.P., 124–126, 128 Rosenblatt, M., 221, 342 Ross, D.T., 251 Ross, S.A., 2, 19 Rothschild, M., 56 Roy, A.D., 20, 33 Roydasgupta, R., 251 Rozanov, Y.A., 137 Rubin, D.B., 198–200 Rubinstein, M.E., 26 Rudin, A.M., 259 Ruppert, D., 26–28, 30, 38, 73 Russnes, H.G., 251, 280 Rustem, B., 36, 38 Ryder, T., 251 Ryudiger, F., 38 Rødland, E.A., 249, 251, 280 Saleh, A.K.M.E., 300, 307 Salenius, S., 270 Samarov, A., 41, 336 Samuelson, P.A., 31, 75, 78, 79 Santa-Clara, P., 76, 80, 81, 83, 84 Sargent, T.J., 55 Satchell, S.E., 27 Sato, J.R., 224 Sato, K., 85 Savage, I.R., 170 Savarino, J.E., 47 Savege, I.R., 120, 121, 123 Scherer, B.M., 63 Schmeidler, D., 32 Schmid, W., 58, 62, 63 Schmidt, E., 254 Schwarz, G., 238, 300, 309 Scott, D.J., 299 Segal, M.R., 249 Sen, P.K., 126, 128, 141, 144, 163, 170, 172, 176, 177 Senda, M., 339 Sepulchre, R., 257 Serroukh, A., 298 Shannon, C.E., 2, 75 Sharpe, W., 2, 19, 23 Sherlock, G., 254 Shibasaki, H., 270 Shibata, R., 300, 304, 306 Shimizu, Y., 96–98 369 Shiohama, T., 45, 46 Shiraishi, H., 41, 80, 84, 239, 242, 244, 245, 279, 312 Shiryaev, A.N., 8, 291 Shojaie, A., 224 Shreve, S.E., 87 Shu, W., 251 Shumway, R.H., 196, 201, 227 Shutes, K., 207 Sidak, Z., 126, 128, 141, 144, 163, 297 Siddique, A., 27 Siegel, P.M., 251 Simpson, K., 251 Sims, A.H., 251 Sims, C.A., 55, 224 Smeds, J., 251 Smethurst, G.J., 251 Solev, V.N., 137 Song, S., 224 Song, X., 259 Speed, T.P., 251 Spellman, P.T., 251 Spitzer, F., 306 Stein, C., 46, 47, 331, 339 Stein, L., 254 Steinebach, J.G., 26 Steinfeld, I., 254 Steinskog, D.J., 12 Stella, F., 98, 104–106, 108 Stoffer, D.S., 235, 250, 259, 261, 262, 266, 267, 274, 275 Stout, W.F., 15 Strasser, H., 297 Stromberg, K., 10 Stroud, J.R., 76, 80, 81, 83, 84 Stulajter, F., 36 Sun, G., 41 Suto, Y., 342, 343 Swensen, A.R., 293, 294, 324 Sørensen, M., 13, 96, 286, 291 Sørlie, T., 249, 251, 280 Taki, W., 270 Tamaki, K., 227, 261 Tang, M.L., 108, 110 Taniai, H., 45, 46, 202, 203 Taniguchi, M., 8, 9, 14, 41, 63, 65, 84, 174, 176, 178–180, 202, 203, 209, 210, 213, 217, 223, 226, 227, 229, 233, 234, 239, 242, 244, 245, 248, 250, 252, 261, 271, 273, 274, 279, 284, 285, 287–289, 291, 295, 298–300, 370 AUTHOR INDEX 304, 305, 308, 312, 315–317, 319, 321, 322, 324, 326, 327, 329, 330, 332, 339, 342–344, 346 Taqqu, M.S., 336, 338 Tasche, D., 20, 35 Telser, L., 20, 33 Teschendorff, A.E., 257 Teyssiere, G., 170–172, 177 Theil, H., 50 Thomas, J.A., 98, 99, 108–110 Thomaz, C.E., 224 Thumar, J.K., 250 Tian, G.L., 108, 110 Tiao, G.C., 274 Tibshirani, R., 63, 249 Tirosh, I., 250 Tjøstheim, D., 12, 172, 176, 177 Tong, G., 259 Tong, H., Trojani, F., 78 Tsay, R.S., 63, 73, 187, 189–191, 274 Tsybakov, A., 14, 274 Tusher, V.G., 249 Tutt, A.M., 251 Tversky, A., 32 Tyler, D.E., 235, 250, 259, 261, 262, 266, 267, 274, 275 Uryasev, S.P., 20, 35, 36, 38 van de Rijn, M., 251 van de Vijver, M.J., 250 van der Weide, R., 183, 184 van Vliet, M.H., 250 Vanini, P., 78 Vastrik, I., 254 Vega, V.B., 251 Veredas, D., 84, 244, 245 Vergara, L., 251 Viale, A., 251 Viale, G., 251 Vollan, H.K.M., 251, 280 von Neumann, J., 20, 29 Vrontos, I.D., 182, 183 Wendt, D.A., 250, 259, 261, 262, 266, 267, 274, 275 Werker, B.J.M., 155–163, 167, 203 Wessels, L.F.A., 250 Wilcoxon, F., 113, 114, 116, 117 Wilson, L., 250 Wofl, M., 41, 47 Wolff, S.S., 124 Wong, W.K., 58, 60, 61 Wooldridge, J.M., 180 Wu, C.F.J., 244 Wu, G.R., 254 Wu, Z., 259 Xia, Y., 33, 38 Xiao, Y., 249 Xiao, Z., 227, 229 Xu, Y., 259 Yaari, M.E., 32 Yakhini, Z., 254 Yamashita, T., 229, 233, 234 Yamazaki, H., 33 Yang, L., 14, 274 Yang, Y.H., 249, 250 Yao, Q., 12, 184, 186 Ye, D.W., 217 Yoshida, N., 96–98 Yoshihara, K., 137–139, 142, 143 Zhang, Y., 251 Zhao, X., 249, 251, 280 Zhou, R., 259 Wallis, W.A., 117 Wang, M., 184, 186 Wang, S., 33, 38 Wang, T., 78 Wang, Y., 251 Wellner, J.A., 149, 151–154 Welsch, R.E., 41 www.ebook3000.com Subject Index D-distance, 186 F statistic, 118 H test, 117 K-factor GARCH, 181 K-factor GARCH model, 180 K-sample problem, 117 P-local characteristics, 291 U-statistic, 137 U-statistics, 142, 143, 161 α-CVaR, 36 α-risk, 20, 34 δ-method, 44, 98, 316 µ-weighted universal portfolio, 100 µ-weighted universal portfolio with side information, 102 φ-mixing, 137 c1 -test, 121 h-step ahead prediction, 221 p-dependent, 142 (n, m)-asymptotics, 58, 59 asymptotic growth rate, 100, 101, 104–106 asymptotic normality, 61, 121 asymptotic power, 134 asymptotic relative efficiency, 117–123, 145, 177 asymptotically centering, 296 asymptotically efficient, 297 asymptotically equivalent, 142 asymptotically most powerful, 130 asymptotically most powerful test, 148, 149 asymptotically orthogonal, 156 asymptotically unbiased estimator, 302 asymptotically unbiased of level α, 297 asymptotically uniformly most powerful, 134 autocovariance function, autoregressive conditional heteroscedastic model (ARCH(q)), 2, 6, 170, 250 autoregressive moving average process of order (p, q) (ARMA(p, q)), 146 autoregressive moving average process of order (p, q)(ARMA(p, q)), autoregressive process of order p (AR(p)), absolute rank, 116 absolute risk aversion (ARA), 30 absolutely continuous, 9, 46, 131, 293 absolutely regular, 137 absolutely regular process, 142 adapted, 12 adapted stochastic process, 293 adaptive varying-coefficient spatiotemporal model, 12 adaptiveness, 158, 159 admissible, 309 Affy947 expression dataset, 250 Akaike’s information criterion (AIC), 238, 300 approximate factor model, 56 AR bootstrap (ARB), 80, 81, 84, 235, 238 arbitrage opportunity, 28 arbitrage pricing theory (APT), 19, 27 ARCH(∞), 13 ARCH(∞)-SM model, 2, 298 Arrow–Pratt measure, 29 asymptotic efficiency, 97 asymptotic filter, 96 bandwidth, 239 BARRA approach, 53 Bayesian estimation, 41 Bayesian information criterion (BIC), 238, 300 Bayesian optimal portfolio estimator, 50, 51 Bellman equation, 75, 77, 81, 88 best constant rebalanced portfolio (BCRP), 104–106 best constant rebalanced portfolio with transaction costs, 107, 108 best constant rebalanced portfolio(BCRP), 99–101 best critical region, 134 best linear predictor, 221 best linear unbiased estimator (BLUE), 197 best state constant rebalanced portfolio (BSCRP), 102, 104 beta, 25 Black–Litterman model, 50 371 372 SUBJECT INDEX blind separation, 192 BLUE, 314 bootstrapped reserve fund, 247 Brownian motion process, 7, 84 budget constraint, 76, 79 cancer cells, 257 cancer-signaling pathways, 257 canonical correlation, 230 canonical correlation analysis (CCA), 2, 230 canonical representation, 291 canonical variate, 230 capital asset pricing model (CAPM), 2, 19, 23 capital market line (CML), 24 Cauchy–Schwarz inequality, 124 causal (nonanticipating) portfolio, 99 causal (nonanticipating) portfolio with transaction costs, 107 causal variable, central sequence, 155, 157–159, 162, 164, 166, 167, 210 certainty equivalent, 29 cesaro sum, 331 chain rule, 151 characteristic equation, 141 characteristic function, 85 Choquet distortion measure, 32 Choquet expected utility, 32 classification, 227 classification rule, 227 coherence, 20 coherent risk measure, 34 common (dynamic) eigenvalue, 57 compensator, 13, 95, 291 compound Poisson process, 84, 95 comutation matrix, 166 conditional heteroscedastic autoregressive nonlinear (CHARN), 2, 14, 235 conditional least squares estimator, 83 conditional value at risk (CVaR), 20 conditionally uncorrelated components (CUCs), 184, 186 confidence region, 241 conjugate prior, 50 consistency, 41, 60, 309 constant absolute risk aversion (CARA), 30 constant conditional correlation (CCC) model, 65 constant rebalanced portfolio (CRP), 99 constant rebalanced portfolio with transaction cost, 107 constant relative risk aversion (CRRA), 30 consumption process, 79, 86 consumption-based CAPM (CCAPM), 26, 79 contiguity, 134 contiguous, 131, 227 continuity, 29 continuous time stochastic process, continuously differentiable, 44 contrast function, 96 convex risk measure, 34 copula, 20, 36 copula density, 37 corticomuscular functional coupling (CMC), 270 counting process, Cramer–Wold device, 144 critical function, 127 critical region, 134 critical value, 121 cross information, 169 CRRA utility, 75, 77, 80, 81, 83, 89, 95 cumulant, 43 cumulant spectra, 207 cyclic trend, 341 cylinder set, 14 decreasing absolute risk aversion (DARA), 30 diagonal multivariate GARCH, 181 differentially expressed (DE), 249 diffusion, diffusion process, diffusion process with jump, 84, 87, 90, 96 Dirac measure, 95 Dirichlet distribution, 101 discount rate, 86 discrete Fourier transform (DFT), 192, 197 discrete time stochastic process, discriminant analysis, 227 disparity, 227 dispersion measure, 20, 33 distance correlation, 145 distance covariance, 145 distribution-CAPM, 27 DNA sequence, DNA sequence for Epstein–Barr virus (ELBV), 261 Doob’s martingale convergence theorem, 15 drift, dynamic factor model, 11, 55 www.ebook3000.com SUBJECT INDEX dynamic orthogonal components (DOCs), 187, 191, 202 dynamic portfolio, 78 dynamic programming, 75 Edgeworth expansion, 214 efficiency, 126 efficient central sequence, 162, 168 efficient central sequences, 167 efficient frontier, 21, 235 efficient influence function, 150, 152, 153, 157 efficient portfolio, 21 efficient score, 157 efficient score function, 154, 164 Einstein’s summation conversion, 216 elastic-net, 53 electomyograms (EMGs), 270 electroencephalogram (EEG), 270 ellipical density, 164 elliptical random variable, 164 elliptical residuals, 204 EM algorithm, 199, 200 empirical distribution function, 302 end-of-period target, 244 envelope power function, 126 envelope ralation, 79 ergodic, 14 estimated efficient frontier, 69 estimating equation, 92, 192 Euler condition, 79 exogenous variable, 207 expected excess rate of return, 25 expected shortfall (ES), 20 expected utility, 20, 30 exponential AR (ExpAR) models, 274 factor, 52 factor analysis, 41 factor loading, 28, 52 factor model, 19, 27 false discovery rate (FDR), 249 Fama–French approach, 53 FARIMA model, 298 Fatou’s lemma, 335 feasible region, 21 feasible set, 20 filtration, 12 final prediction error (FPE), 237 finite Fourier transform, 211 first-order condition (FOC), 75, 77, 80, 89 first-order stochastic dominance, 31 373 Fisher information, 130, 150, 155, 163, 166 Fisher information matrix, 155, 158, 159, 305 focused information criterion (FIC), 300 four moments CAPM, 27 Fourier transform, 196 Fr´echet differentiable, 150 full-factor GARCH, 182 functional central limit theorem (FCLT), 138 fundamental factor model, 53 Gaussian copula, 37 Gaussian process, gene expression omnibus (GEO), 250 gene ontology (GO) analysis, 254 generalized AIC (GAIC), 2, 300 generalized ARMA process, 147 generalized autoregressive conditional heteroscedastic model (GARCH(p, q)), 13 generalized consumption investment problem, 79 generalized dynamic factor model, 56 generalized EM (GEM) algorithm, 200 generalized inverse matrix, 62 generalized least squares (GLS), 53 generalized method of moments (GMM), 75, 91, 190 generalized orthogonal GARCH (GO-GARCH) model, 183 global minimum variance portfolio, 22, 58, 59, 69, 239, 242 Granger causality, 224 Grenander’s condition, 220, 338 group, 124 group invariance, 167 Hamilton–Jacobi–Bellman (HJB) equation, 75 Hannan Quinn criterion (HQ), 238, 300 harmonic absolute risk aversion (HARA), 30 hedging demand, 78 Hermite polynomial, 214 Hermitian nonnegative, 338 high-dimensional problem, 41 homogeneous Poisson process, 6, 84 homogeneous Poisson random measure, 95 Huber’s function, 190 hyper-parameter, 49 idiosyncratic (dynamic) eigenvalue, 57 idiosyncratic risk, 26 374 SUBJECT INDEX independence, 29 independent component analysis (ICA), 187 inference function, 162 infinite time horizon problem, 88 influence function, 157 influence measure criterion (IMC), 210 information bound, 150, 153 information matrix, 153, 166 information set process, 76 information theory, 98 informative prior, 50 innovation process, 140 intensity parameter, 84 intercept of factor model, 27 interest rate, 244 interpolation, 342 intertemporal CAPM (ICAPM), 26 intrinsic annotation, 251 invariance, 156 invariant, 15, 124, 129 invariant principle, 168 invariant test, 125 inverse Wishart distribution, 50 invertibility condition, 42 Itˆo’s lemma, 86, 87, 90 James–Stein estimator, 331 Japanese Government Pension Investment Fund (GPIF), 84, 235, 244 Jeffreys prior, 50 Jensen’s inequality, 124 kernel function, 239 Kroneker product, 166 L´evy measure, 85 L´evy process, 84 LASSO, 53 least favourable, 126, 128, 129, 153, 156, 157, 161, 162 least favourable density, 126, 127 least favourable direction, 156 least favourable distribution, 128 least squares estimator, 339 Lebesgue bounded convergence theorem, 118 LeCam’s first lemma, 141, 158 LeCam’s third lemma, 144, 297 limit theorem, 283 Lindeberg condition, 16, 134 linear interpolation, 125 linear process, linear projection, 224 linear rank statistics, 114 linear serial rank statistic, 139, 145 linear serial rank statistics, 144, 148 Lipschitz condition, 140 Lipschitz continuity, 96 local alternative, 120 local asymptotic linearity, 158 local asymptotic normality (LAN), 1, 210, 271, 293 local martingale, 15 local optimality, 169 locally asymptotic mixed normal (LAMN), 299 locally asymptotically maximin efficient, 170 locally asymptotically normal (LAN), 155, 156, 158, 159, 162, 164 locally asymptotically optimal, 298 locally stationary process, 2, 235, 239, 299 logarithmic utility, 75, 78 long memory process, loss function, 296 lottery, 28 m-vector process, macroeconomic factor model, 52 Markov property, 88 Markov time, 12 martingale, martingale central limit theorem, 92, 160 martingale difference, 16 maximal invariant, 124, 125, 160, 167 maximum likelihood estimator (MLE), 3, 291, 292 mean absolute deviation model, 33 mean absolute error (MAE), 215 mean semivariance model, 33 mean squared error (MSE), 48, 196, 339 mean squared interpolation error (MSIE), 342 mean target model, 33 mean variance efficient portfolio, 21 mean-CVaR portfolio, 1, 35, 36 mean-ES portfolio, 35 mean-TCE portfolio, 35 mean-VaR portfolio, 1, 34 mean-variance portfolio, 1, 208 mean-variance-skewness (MVS) model, 31 measure of disparity, 300 measure preserving, 15 microarray data, 2, 235, 248, 249 midrank, 114 www.ebook3000.com SUBJECT INDEX minimum Pitman efficiency, 117 minimum-variance portfolio, 22 minimum-variance set, 21 misspecification, 346 misspecified prediction error, 222 mixing, 286 mixing transformation, 187 mixture successive constant rebalanced portfolio (MSCRP), 106 modern portfolio theory, 19 molecular subtypes of breast tumours, 251 monthly log-return, 235 Moore Penrose inverse, 62 Moore Penrose pseudo inverse, 63 most efficient statistic, 145 most powerful, 126–129 most powerful invariant test, 125 most powerful test, 126, 127 moving average process of order q(MA(q)), moving block bootstrap (MBB), 84 multiperiod problem, multiplicatively modulated exponential autoregressive model (mmExpAR), 271 multiplicatively modulated nonlinear autoregressive model (mmNAR), 271 multivariate Student t-distribution, 50 myopic portfolio, 75, 78 naive portfolio, 58, 59 Neyman–Pearson lemma, 128, 134 non-Gaussian linear process, 41, 211 non-Gaussian stationary process, noncentral chi square distribution, 118 nonlinear autoregressive model, nonparametric estimator, 239 nonparametric portfolio estimator, 235 nonparametric test statistic, 120 nonsystematic risk, 26 nuisance parameter, 129 optimal consumption, 89 optimal estimating function, 93 optimal portfolio, 1, 211 optimal portfolio estimator, 41 optimal portfolio weight, 42 optimal shrinkage factor, 47 order statistic, 114 order statistics, 125 ordinary least squares (OLS), 52, 53, 237 375 orthogonal GARCH, 183 orthogonal increment process, parametric experiments, 155 parametric submodels, 155 pathway analysis, 254 pathwise differentiable, 152 pension investment, 243 periodogram, 283 permutation, 125 pessimistic, 35 pessimistic portfolio, 1, 32, 35, 44 Pitman efficiency, 121 pointwise ergodic theorem, 15 portfolio weight process, 76 posterior density, 49 power function, 128 power functions, 122 predictable process, 15 predictive return density, 49 price relative, 98 principal component analysis (PCA), 54, 188 prior distribution, 49 probability simplex, 98 probability space, projection, 157 projection operator, 151, 152 prospect theory, 32 pseudo-Mahalanobis signs and ranks, 205 quadratic characteristic, 15 quadratic variation process, 286 quadratically integrable derivative, 129, 132 quantile, 35 quantile regression, 41, 45, 227 quasi-Gaussian maximum likelihood estimator, 66, 67, 287 quasi-maximum likelihood, 191 radial basis function (RBF), 273 Radon–Nikodym density, 293 random coefficient regression, 45 rank, 113, 125, 165, 167, 205 rank autocorrelation coefficient, 148 rank dependent expected utility (RDEU) theory, 30, 32 rank order statistics, 173 rank order statistics for ARCH residual, 250 rank statistic, 114, 131 ranks, 129 rapidly increasing experimental design, 96 rational, 28 376 SUBJECT INDEX regression spectral measure, 219 regular, 299 regular parametric model, 150 regular parametric submodel, 152 regular point of the parametrization, 150 relative risk aversion (RRA), 30 required reserve fund, 244 resampled efficent frontier, 239 resampled return, 239 reserve fund, 244, 245 residual analysis for fitting time series modeling, 274 residual Fisher information, 156 return process, 2, 42, 207 ridge regression, 53 risk aversion, 29 risk aversion coefficient, 214 risk diversification, 259 risk measure, 20 risk premium, 25, 29 risk-free asset, 76, 216 risky asset, 76 robust, 114, 210 run statistic, 139 safety first, 33 safety measure, 20, 33 sampling scheme, 95 Scheffe’s theorem, 330 score function, 139, 150, 160, 169 score generating function, 145 score generation function, 140 second order stochastic dominance, 31 second-order stationary, 9, 42 second-order stationary process, 63 security market line (SML), 25 self-exciting threshold autoregressive model (SETAR(k; p, , p)), self-financing, 76 semi-parametrically efficient inference, 162 semimartingale, 13 semiparametric efficiency, 156, 157 semiparametric inference, 151 semiparametric model, 153, 162 semiparametric models, 155 semiparametrically efficient, 157 semiparametrically efficient central sequence, 156 separating matrix, 187 sequence of experiments, 295 sequencing data analysis, 235 shape efficient central sequence, 168 Sharpe ratio, 245 shrinkage estimation, 3, 41 shrinkage estimator, 47, 331 shrinkage factor, 47 shrinkage interpolator, 342 shrinkage sample mean, 47 shrinkage target, 47 side information, 98, 101 sieve bootstrap (SB), 244 sign, 165, 167, 205 sign function, 116 sign test, 128 signed rank statistic, 116 skew-GARCH, 207 skew-normal distribution, 2, 207 Sklar’s theorem, 36 Sobolev space, 166 Spearman’s rank correlation coefficient, 140 specific risk, 26 spectral density matrix, 9, 42, 57 spectral density of exponential type, 310 spectral distribution matrix, spectrum envelope (SpecEnv), 235, 259 square summable, 56 stable, 286 state constant rebalanced portfolio (SCRP), 102 state space, stationary process, 41 statistical factor model, 54 statistical model selection, 300 stochastic basis, 12 stochastic dominance, 31 stochastic integral, stochastic process, stopping time, 13 strictly stationary, 8, 137 strong mixing, 191 Student t-distribution, 214 Student t-test, 119 successive constant rebalanced portfolio (SCRP), 104, 105 surface, 151, 152 symmetric restriction, 124 systematic risk, 26 tail conditional expectation (TCE), 20 tangency portfolio, 23, 58, 59 tangent space, 149, 151, 152, 157, 167 terminal condition, 77, 79–81 terminal wealth, 76, 80 threshold estimation method, 75, 97 www.ebook3000.com SUBJECT INDEX time series multifactor model, 27 time-varying spectral density, 320 trading time, 76 training sample, 229 transaction cost, 98, 107 transformation, 124 turning point statistic, 139 two-sample location problem, 113 unbiased martingale estimating function, 92 uncorrelated, 52 uniform local asymptotic normality (ULAN), 165, 166, 202, 203 uniformly most powerful (UMP), 125, 127, 128 uninformative prior, 50 unit root, 220 universal portfolio (UP), 2, 75, 98, 99 universal portfolio with transaction cost, 108 universal portfolio with transaction costs, 107 utility, 29, 76, 215 value at risk (VaR), 20, 34 value function, 75, 76, 79, 87 VAR(p), 11, 237, 238 VaR-CAPM, 27 variational distance, 296 VARMA(p, q), 11 vector GARCH equation, 66 vector linear process, 11, 42 vector-valued ARMA, 42 Volterra series expansion, 204 von Mises’s differentiable statistical functional, 138 wealth process, 86 wealth relative, 98 weight process, 76 weighted successive constant rebalanced portfolio (WSCRP), 105, 106 weighted utility theory, 31 white noise, 52 Whittle estimator, 222 Whittle’s approximate likelihood, 246 Wiener process, Wilcoxon signed rank statistic, 116 Wilcoxon statistic, 114 Wilcoxon test, 114, 119, 120 Wilcoxon’s rank sum test, 113 Wilcoxon’s signed rank test, 113 wild bootstrap (WB), 244 377 www.ebook3000.com .. .STATISTICAL PORTFOLIO ESTIMATION www.ebook3000.com STATISTICAL PORTFOLIO ESTIMATION Masanobu Taniguchi, Hiroshi Shiraishi, Junichi Hirukawa,... 3.1.8 Appendix 3.2 Statistical Estimation for Portfolios 3.2.1 Traditional Mean-Variance Portfolio Estimators 3.2.2 Pessimistic Portfolio 3.2.3 Shrinkage Estimators 3.2.4 Bayesian Estimation 3.2.5... domain for time series 5.5 Rank-Based Optimal Portfolio Estimation 5.5.1 Portfolio Estimation Based on Ranks for Independent Components 5.5.2 Portfolio Estimation Based on Ranks for Elliptical Residualselliptical