Articles |
Confidence intervals for spectral mean and ratio statistics
Department of Statistics, University of Illinois at Urbana-Champaign, 725 South Wright Street, Champaign, Illinois 61820, U.S.A. xshao{at}uiuc.edu
Received for publication 1 January 2008. Revision received 1 June 2008.
We propose a new method, to construct confidence intervals for spectral mean and related ratio statistics of a stationary process, that avoids direct estimation of their asymptotic variances. By introducing a bandwidth, a self-normalization procedure is adopted and the distribution of the new statistic is asymptotically nuisance-parameter free. The bandwidth is chosen using information criteria and a moving average sieve approximation. Through a simulation study, we demonstrate good finite sample performance of our method when the sample size is moderate, while a comparison with an empirical likelihood-based method for ratio statistics is made, confirming a wider applicability of our method.
Key Words: Autocorrelation • Cumulant • Ratio statistic • Spectral density • Spectral distribution function
References
-
Berk K.N. Consistent autoregressive spectral estimates. Ann. Statist. (1974) 2:489–502.[CrossRef]
Billingsley P. Convergence of Probability Measures (1968) New York: Wiley.
Brillinger D. R. Time Series: Data Analysis and Theory (1975) San Francisco, CA: Holden Day.
Brockwell P. J., Davis R. A. Time Series: Theory and Methods (1991) 2nd ed. New York: Springer.
Bühlmann P. Sieve bootstrap for time series. Bernoulli (1997) 3:123–48.[CrossRef][Web of Science]
Chiu S. T. Weighted least squares estimators on the frequency domain for the parameters of a time series. Ann. Statist. (1988) 16:1315–26.[CrossRef]
Dahlhaus R. Asymptotic normality of spectral estimates. J. Mult. Anal. (1985) 16:412–31.[CrossRef]
Dahlhaus R., Janas D. A frequency domain bootstrap for ratio statistics in time series analysis. Ann. Statist. (1996) 24:1934–63.[CrossRef]
Franke J., Härdle W. On bootstrapping kernel spectral estimates. Ann. Statist. (1992) 20:121–45.[CrossRef]
Hannan E. J. Central limit theorem for time series regression. Z. Wahr. verw. Geb. (1973) 26:157–70.[CrossRef]
Keenan D. M. Limiting behavior of functionals of higher-order sample cumulant spectra. Ann. Statist. (1987) 15:134–51.[CrossRef]
Kreiss J. P., Paparoditis E. Autoregressive-aided periodogram bootstrap for time series. Ann. Statist. (2003) 31:1923–55.[CrossRef]
Lobato I. N. Testing that a dependent process is uncorrelated. J. Am. Statist. Assoc. (2001) 96:1066–76.[CrossRef][Web of Science]
Nordman D. J., Lahiri S. N. A frequency domain empirical likelihood for short- and long-range dependence. Ann. Statist. (2006) 34:3019–50.[CrossRef]
Owen A. B. Empirical Likelihood (2001) London: Chapman & Hall.
Politis D. N., Romano J. P., Wolf M. Subsampling (1999) New York: Springer.
Romano J. L., Thombs L. A. Inference for autocorrelations under weak assumptions. J. Am. Statist. Assoc. (1996) 91:590–600.[CrossRef][Web of Science]
Rosenblatt M. Stationary Sequences and Random Fields (1985) Boston, MA: Birkhäuser.
Shao X., Wu W. B. Asymptotic spectral theory for nonlinear time series. Ann. Statist. (2007) 35:1773–1801.[CrossRef]
Taniguchi M. On estimation of the integrals of the 4th order cumulant spectral density. Biometrika (1982) 69:117–22.
Wu W. B. Nonlinear system theory: another look at dependence. Proc. Nat. Acad. Sci. USA (2005) 102:14150–4.
Wu W. B., Min W. On linear processes with dependent innovations. Stoch. Proces. Appl. (2005) 115:939–58.[CrossRef]
Wu W. B., Shao X. Limit theorems for iterated random functions. J. Appl. Prob. (2004) 41:425–36.[CrossRef][Web of Science]
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