Abstract
We compare two estimates of the cumulant generating function of a stationary linear process. The first estimate is based on the empirical moment generating function. The second estimate uses the linear representation of the process and the empirical moment generating function of the innovations. Asymptotic expressions for the mean square errors are derived under short- and long-range dependence. For long-memory processes, the estimate based on the linear representation turns out to have a better rate of convergence. Thus, exploiting the linear structure of the process leads to an infinite gain in asymptotic efficiency.
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References
Beran J. (1991). M-estimators of location for data with slowly decaying serial correlations. Journal of the Americal Statistical Association 86:704–708
Beran J. (1994). Statistics for long-memory processes. Chapman and Hall, London
Brockwell P.J., Davis R.A. (1987). Time series: Theory and methods. Springer, Berlin Heidelberg New York
Csörgő S. (1981). Limit behaviour of the empirical characteristic function. The Annals of Probability 9:130–144
Csörgő S. (1982). The empirical moment generating function. In: Gnedenko B.V., Puri M.L., Vincze I. (eds). Nonparametric statistical inference. North-Holland, London, pp 139–150
Csörgő, S. (1986). Testing for normality in arbitrary dimension. The Annals of Statistics 14:708–723
Csörgő S., Mielniczuk J. (1995). Density estimation under long-range dependence. The Annals of Statistics 23: 990–999
Dehling H., Taqqu M.S. (1989). The empirical process of some long-range dependent sequences with an application to U-statistics. The Annals of Statistics 17:1767–1783
Engel J., Herrmann E., Gasser T. (1994). An iterative bandwidth selector for kernel estimation of densities and their derivatives. Journal of Nonparametric Statistics 4:21–34
Feuerverger A., Mureika R.A. (1977). The empirical characteristic function and its applications. The Annals of Statistics 5: 88–97
Ghosh S. (1996). A new graphical tool to detect non-normality. Journal of the Royal Statistical Society B 58: 691–702
Ghosh, S. (1999). T3-plot. Encyclopedia for statistical sciences. Update Vol. 3 (pp 739–744). New York: Wiley
Ghosh S. (2003). Estimating the moment generating function of a linear process. Student 4(3):211–218
Ghosh S., Beran J. (2000). Two sample T3-plot: A graphical comparison of two distributions. Journal of Computational and Graphical Statistics 9(1):167–179
Ghosh S., Ruymgaart F. (1992). Applications of empirical characteristic functions in some multivariate problems. The Canadian Journal of Statistics 20:429–440
Giraitis L., Surgailis D. (1985). Central limit theorems and other limit theorems for functionals of Gaussian processes. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 70:191–212
Giraitis L., Surgailis D. (1990). A central limit theorem for quadratic forms in strongly dependent linear variables and applications to asymptotical normality of Whittle’s estimate. Probability Theory and Related Fields 86:87–104
Giraitis L., Surgailis D. (1999). Central limit theorem for the empirical process of a linear sequence with long memory. Journal of Statistical Planning and Inference 80:81–93
Granger C.W.J., Joyeux R. (1980). An introduction to long-range time series models and fractional differencing. Journal of Time Series Analysis 1:15–29
Hall P. (1993). On plug-in rules for local smoothing of density estimators. The Annals of Statistics 21:694–710
Hall P., Hart J. (1990). Convergence rates in density estimation for data from infinite-order moving average processes. Probability Theory and Related Fields 87:253-274
Haslett J., Raftery A.E. (1989). Space-time modelling with long-memory dependence: Assessing Ireland’s wind power resource. Invited paper with discussion. Applied Statistics, 38, 1–50.
Honda T. (2000). Nonparametric density estimation for a long-range dependent linear process. Annals of the Institute of Statistical Mathematics 52:599–611
Hosking J.R.M. (1981). Fractional differencing. Biometrika 68:165–176
Mandelbrot B.B., Wallis J.R. (1969). Computer experiments with fractional Gaussian noises. Water Resources Research 5: 228–267
Murota K., Takeuchi K. (1981). The studentized empirical characteristic function and its application to test for the shape of the distribution. Biometrika 68:55–65
Silverman B.W. (1986). Density estimation for statistics and data analysis. Chapman and Hall, London
Taqqu M.S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 31:287–302
Wu W.B., Mielniczuk (2002). Kernel density estimation for linear processes. The Annals of Statistics 30(5):1441–1459
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Ghosh, S., Beran, J. On Estimating the Cumulant Generating Function of Linear Processes. Ann Inst Stat Math 58, 53–71 (2006). https://doi.org/10.1007/s10463-005-0009-5
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DOI: https://doi.org/10.1007/s10463-005-0009-5