Skip to main content Accessibility help
×
Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2025-01-02T01:07:59.239Z Has data issue: false hasContentIssue false

References

Published online by Cambridge University Press:  19 December 2024

Katsuto Tanaka
Affiliation:
Hitotsubashi University, Tokyo
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2025

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anderson, T. W. (1971). The Statistical Analysis of Time Series. Wiley, New York.Google Scholar
Anderson, T. W. and Darling, D. A. (1952). Asymptotic theory of certain ‘goodness of fit’ criteria based on stochastic processes. Annals of Mathematical Statistics, 23, 193212.CrossRefGoogle Scholar
Biagini, F., Hu, Y., Øksendal, B., and Zhang, T. (2008). Stochastic Calculus for Fractional Brownian Motion and Applications. Springer, London.CrossRefGoogle Scholar
Billingsley, P. (1999). Convergence of Probability Measures (Second Edition). Wiley, New York.CrossRefGoogle Scholar
Bishwal, J. P. N. (2008). Parameter Estimation in Stochastic Differential Equations. Springer, Berlin.CrossRefGoogle Scholar
Bobkoski, M. J. (1983). Hypothesis testing in nonstationary time series, PhD Thesis. University of Wisconsin.Google Scholar
Bowman, F. (2010). An Introduction to Bessel Functions. Dover, New York.Google Scholar
Bronski, J. C. (2003). Asymptotics of Karhunen–Loève eigenvalues and tight constants for probability distributions of passive scalar transport. Communications in Mathematical Physics, 238, 563582.CrossRefGoogle Scholar
Chan, N. H. and Wei, C. Z. (1988). Limiting distributions of least squares estimates of unstable autoregressive processes. Annals of Statistics, 16, 367401.CrossRefGoogle Scholar
Courant, R. and Hilbert, D. (1953). Methods of Mathematical Physics, Vol. I. Wiley, New York.Google Scholar
Darling, D. A. (1955). The Cramér–Smirnov test in the parametric case. Annals of Mathematical Statistics, 26, 120.CrossRefGoogle Scholar
Davydov, Y. (1970). The invariance principle for stationary processes. Theory of Probability and Its Applications, 15, 487498.CrossRefGoogle Scholar
Dickey, D. A. and Fuller, W. A. (1979). Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association, 74, 427431.Google Scholar
Donsker, M. D. (1951). An invariance principle for certain probability limit theorems. Memoires of the American Mathematical Society, 6, 112.Google Scholar
Durbin, J. (1973). Weak convergence of the sample distribution function when parameters are estimated. Annals of Statistics, 1, 279290.CrossRefGoogle Scholar
Engle, R. F. and Granger, C. W. J. (1987). Co-integration and error correction: Representation, estimation, and testing. Econometrica, 55, 251276.CrossRefGoogle Scholar
Evans, G. B. A. and Savin, N. E. (1981). Testing for unit roots: 1. Econometrica, 49, 753779.CrossRefGoogle Scholar
Ferguson, T. S. (1967). Mathematical Statistics: A Decision Theoretic Approach. Academic Press, New York.Google Scholar
Giraitis, L., Koul, H., and Surgailis, D. (2012). Large Sample Inference for Long Memory Processes. Imperial College Press, London.CrossRefGoogle Scholar
Girsanov, I. V. (1960). On transforming a certain class of stochastic processes by absolutely continuous substitution of measures. Theory of Probability and Its Applications, 5, 285301.CrossRefGoogle Scholar
Gradshteyn, I. S. and Ryzhik, I. M. (1965). Table of Integrals, Series, and Products (Fifth Edition). Academic Press, New York.Google Scholar
Gripenberg, G. and Norros, I. (1996). On the prediction of fractional Brownian motion. Journal of Applied Probability, 33, 400410.CrossRefGoogle Scholar
Helland, I. S. (1982). Central limit theorems for martingales with discrete or continuous time. Scandinavian Journal of Statistics, 9, 7994.Google Scholar
Hochstadt, H. (1973). Integral Equations. Wiley, New York.Google Scholar
Hosking, J. R. M. (1981). Fractional differencing. Biometrika, 68, 165176.CrossRefGoogle Scholar
Imhof, J. P. (1961). Computing the distribution of quadratic forms in normal variables. Biometrika, 48, 419426.CrossRefGoogle Scholar
Jazwinski, A. H. (1970). Stochastic Processes and Filtering Theory. Academic Press, New York.Google Scholar
Jin, S. and Li, W. V. (2015). Expectation of the limiting distribution of the LSE of a unit root process. Statistica Sinica, 25, 529536.Google Scholar
Kac, M., Kiefer, J., and Wolfowitz, J. (1955). On tests of normality and other tests of goodness of fit based on distance methods. Annals of Mathematical Statistics, 26, 189211.CrossRefGoogle Scholar
Kariya, T. (1980). Locally robust tests for serial correlation in least squares regression. Annals of Statistics, 8, 10651070.CrossRefGoogle Scholar
Klebaner, F. C. (2012). Introduction to Stochastic Calculus with Applications (Third Edition). Imperial College Press, London.CrossRefGoogle Scholar
Kleptsyna, M. L. and Le Breton, A. (2002). Statistical analysis of the fractional Ornstein–Uhlenbeck type process. Statistical Inference for Stochastic Processes, 5, 229248.CrossRefGoogle Scholar
Kleptsyna, M. L., Le Breton, A., and Roubaud, M. C. (2000). Parameter estimation and optimal filtering for fractional type stochastic systems. Statistical Inference for Stochastic Processes, 3, 173182.CrossRefGoogle Scholar
Kolmogorov, A. N. (1940). Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum. Doklady Akademii Nauk SSSR, 26, 115118.Google Scholar
Kuo, H. -H. (2006). Introduction to Stochastic Integration. Springer, New York.Google Scholar
Liptser, R. S. and Shiryaev, A. N. (2001a). Statistics of Random Processes I: General Theory (Second Edition). Springer-Verlag, New York.Google Scholar
Liptser, R. S. and Shiryaev, A. N. (2001b). Statistics of Random Processes II: Applications (Second Edition). Springer-Verlag, New York.Google Scholar
Loève, M. (1977). Probability Theory I (Fourth Edition). Springer-Verlag, New York.Google Scholar
Loève, M. (1978). Probability Theory II (Fourth Edition). Springer-Verlag, New York.CrossRefGoogle Scholar
MacNeill, I. B. (1974). Tests for change of parameter at unknown times and distributions of some related functionals on Brownian motion. Annals of Statistics, 2, 950962.CrossRefGoogle Scholar
Mandelbrot, B. B. and Van Ness, J. W. (1968). Fractional Brownian motions, fractional Browninan noises and applications. SIAM Review, 10, 422437.CrossRefGoogle Scholar
Marinucci, D. and Robinson, P. M. (1999). Alternative forms of fractional Brownian motion. Journal of Statistical Inference and Planning, 80, 111122.CrossRefGoogle Scholar
Nabeya, S. (1989). Asymptotic distributions of test statistics for the constancy of regression coefficients under a sequence of random walk alternatives. Journal of the Japan Statistical Society, 19, 2333.Google Scholar
Nabeya, S. (1992). Limiting moment generating function of Cramér–von Mises–Smirnov goodness of fit statistics under null and local alternatives. Journal of the Japan Statistical Society, 22, 113122.Google Scholar
Nabeya, S. and Perron, P. (1994). Local asymptotic distributions related to the AR(1) model with dependent errors. Journal of Econometrics, 62, 229264.CrossRefGoogle Scholar
Nabeya, S. and Tanaka, K. (1988). Asymptotic theory of a test for the constancy of regression coefficients against the random walk alternative. Annals of Statistics, 16, 218235.CrossRefGoogle Scholar
Nabeya, S. and Tanaka, K. (1990a). A general approach to the limiting distribution for estimators in time series regression with nonstable autoregressive errors. Econometrica, 58, 145163.CrossRefGoogle Scholar
Nabeya, S. and Tanaka, K. (1990b). Limiting powers of unit-root tests in time-series regression. Journal of Econometrics, 46, 247271.CrossRefGoogle Scholar
Norros, I., Valkeila, E., and Virtamo, J. (1999). An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli, 5, 571587.CrossRefGoogle Scholar
Nourdin, I. (2012). Selected Aspects of Fractional Brownian Motion. Springer, Berlin.CrossRefGoogle Scholar
Nyblom, J. and Mäkeläinen, T. (1983). Comparisons of tests for the presence of random walk coefficients in a simple linear model. Journal of the American Statistical Association, 78, 856864.CrossRefGoogle Scholar
Perron, P. (1991). A continuous-time approximation to the unstable first-order autoregressive model: The case without an intercept. Econometrica, 59, 211236.CrossRefGoogle Scholar
Pettitt, A. N. (1976). A two-sample Anderson–Darling rank statistic. Biometrika, 63, 161168.Google Scholar
Phillips, P. C. B. (1987a). Time series regression with a unit root. Econometrica, 55, 277301.CrossRefGoogle Scholar
Phillips, P. C. B. (1987b). Towards a unified asymptotic theory for autoregression. Biometrika, 74, 535547.CrossRefGoogle Scholar
Phillips, P. C. B. and Magdalinos, T. (2007). Limit theory for moderate deviations from a unit root. Journal of Econometrics, 136, 115130.CrossRefGoogle Scholar
Rao, C. R. (1973). Linear Statistical Inference and Its Applications (Second Edition). Wiley, New York.CrossRefGoogle Scholar
Rutherford, D. E. (1946). Some continuant determinants arising in physics and chemistry. Proceedings of the Royal Society of Edinburgh, A- 62, 229236.Google Scholar
Soong, T. T. (1973). Random Differential Equations in Science and Engineering. Academic Press, New York.Google Scholar
Sowell, F. (1990). The fractional unit root distribution. Econometrica, 58, 495505.CrossRefGoogle Scholar
Sukhatme, S. (1972). Fredholm determinant of a positive definite kernel of a special type and its application. Annals of Mathematical Statistics, 43, 19141926.CrossRefGoogle Scholar
Tanaka, K. (1983). Non-normality of the Lagrange multiplier statistic for testing the constancy of regression coefficients. Econometrica, 51, 15771582.CrossRefGoogle Scholar
Tanaka, K. (1990a). The Fredholm approach to asymptotic inference on nonstationary and noninvertible time series models. Econometric Theory, 6, 411432.CrossRefGoogle Scholar
Tanaka, K. (1990b). Testing for a moving average unit root. Econometric Theory, 6, 433444.CrossRefGoogle Scholar
Tanaka, K. (1996). Time Series Analysis: Nonstationary and Noninvertible Distribution Theory. Wiley, New York.Google Scholar
Tanaka, K. (2013). Distributions of the maximum likelihood and minimum contrast estimators associated with the fractional Ornstein–Uhlenbeck process. Statistical Inference for Stochastic Processes, 16, 173192.CrossRefGoogle Scholar
Tanaka, K. (2014). Distributions of quadratic functionals of the fractional Brownian motion based on a martingale approximation. Econometric Theory, 30, 10781109.CrossRefGoogle Scholar
Tanaka, K. (2015). Maximum likelihood estimation for the non-ergodic fractional Ornstein–Uhlenbeck process. Statistical Inference for Stochastic Processes, 18, 315332.CrossRefGoogle Scholar
Tanaka, K. (2017). Time Series Analysis: Nonstationary and Noninvertible Distribution Theory (Second Edition). Wiley, New York.CrossRefGoogle Scholar
Tanaka, K. (2023). Extensions of Darling’s formula for the Fredholm determinant. Unpublished Manuscript.Google Scholar
Taqqu, M. S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 31, 287302.CrossRefGoogle Scholar
Watson, G. N. (1958). A Treatise on the Theory of Bessel Functions (Second Edition). Cambridge University Press, Cambridge.Google Scholar
Watson, G. S. (1961). Goodness-of-fit tests on a circle. Biometrika, 48, 109114.CrossRefGoogle Scholar
Whittaker, E. T. and Watson, G. N. (1958). A Course of Modern Analysis (Fourth Edition). Cambridge University Press, Cambridge.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • References
  • Katsuto Tanaka, Hitotsubashi University, Tokyo
  • Book: Brownian Motion, the Fredholm Determinant, and Time Series Analysis
  • Online publication: 19 December 2024
  • Chapter DOI: https://doi.org/10.1017/9781009567008.012
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • References
  • Katsuto Tanaka, Hitotsubashi University, Tokyo
  • Book: Brownian Motion, the Fredholm Determinant, and Time Series Analysis
  • Online publication: 19 December 2024
  • Chapter DOI: https://doi.org/10.1017/9781009567008.012
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • References
  • Katsuto Tanaka, Hitotsubashi University, Tokyo
  • Book: Brownian Motion, the Fredholm Determinant, and Time Series Analysis
  • Online publication: 19 December 2024
  • Chapter DOI: https://doi.org/10.1017/9781009567008.012
Available formats
×