×

On the asymptotic distribution of Bartlett’s \(U_ p\)-statistic. (English) Zbl 0602.62081

Suppose X(t), \(t\in {\mathbb{Z}}\) is a real valued stationary time series with mean zero, spectral density f(\(\lambda)\) and spectral distribution F(\(\lambda)\). Suppose that the time series is observed at the time points 0,1,...,T-1. M. S. Bartlett [Publ. Inst. Stat. Univ. Paris 3, 119- 134 (1954; Zbl 0058.357) and ”An introduction to stochastic processes.” 2nd ed. (1966; Zbl 0196.183)] proposed the following test statistic for a goodness-of-fit test: \[ U_ p=F_ T(\lambda_ p)/F_ T(\pi), \] where \(F_ T(\lambda)=(2\pi /T)\sum_{0<2\pi s/T\leq \lambda}I_ T(2\pi s/T)\), \(I_ T(\alpha)=\{2\pi H_ 2^{(T)}(0)\}^{-1}d_ T(\alpha)d_ T(-\alpha)\), \(d_ T(\alpha)=\sum^{T- 1}_{t=0}h(t/T)X(t)e^{-i\alpha T}\), and \(\lambda_ p=2\pi p/T\). Here the author proves that the process \[ \sqrt{T/2}(F_ T(\lambda)/F_ T(\pi)-F(\lambda)/F(\pi)),\quad 0\leq \lambda \leq \pi \] converges weakly in D[0,\(\pi\) ] to a Gaussian process which is a tied down Brownian motion for linear process X(\(\cdot)\). He further obtains the asymptotic distribution of \[ \max_{1\leq p\leq [T/2]}\sqrt{T/2}| U_ p- F(\lambda_ p)/F(\pi)| =\sup_{0\leq \lambda \leq \pi}\sqrt{T/2}| F_ T(\lambda)/F_ T(\pi)-F(\lambda)/F(\pi)| \] and indicates a special case where the limiting distribution is not that of Kolmogorov- Smirnov type. Sufficient conditions for the weak convergence of \(\sqrt{T}(F_ T(\lambda)-F(\lambda))\) in D[0,\(\pi\) ] are given. A goodness-of-fit test for ARMA processes based on the estimated innovation sequence is discussed and it is shown that this test statistic has asymptotically Kolmogorov-Smirnov distribution.
Reviewer: B.L.S.Prakasa Rao

MSC:

62M15 Inference from stochastic processes and spectral analysis
62E20 Asymptotic distribution theory in statistics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bartlett M. S., Publ. Inst. Statist. Univ. Paris 3 pp 119– (1954)
[2] Bartlett M. S., An Introduction to Stochastic Processes with Special Reference to Methods and Applications, 2. ed. (1966) · Zbl 0196.18301
[3] Bary N., Treatise on Trigonometric Series (1964) · Zbl 0129.28002
[4] Bentkus R., Liet. Matem. Rink. 16 (4) pp 501– (1976)
[5] Billingsley P., Convergence of Probability Measures (1968) · Zbl 0172.21201
[6] Bloomfield P., Fourier Analysis of Time Series; An Introduction (1976) · Zbl 0353.62051
[7] Brillinger D. R., Time Series:Data Analysis and Theory (1975)
[8] Dahlhaus R., J. Time Ser. Anal. 4 (3) pp 163– (1983)
[9] Dahlhaus R., Stoch. Proc. Appl. 18 (1984)
[10] Dahlhaus R., J. Multivariate Anal. 16 (1985)
[11] Feller W., An Introduction to Probability Theory and Its Applications, 2. ed. (1966) · Zbl 0138.10207
[12] Ganssler P., Wahrscheinlichkeitstheorie (1977) · doi:10.1007/978-3-642-66749-7
[13] Grenander U., Statistical Analysis of Stationary Time Series (1984) · Zbl 0575.62080
[14] DOI: 10.1137/1126029 · Zbl 0481.60032 · doi:10.1137/1126029
[15] DOI: 10.1070/SM1981v038n04ABEH001455 · Zbl 0462.60079 · doi:10.1070/SM1981v038n04ABEH001455
[16] Priestley M. B., Spectral Analysis and Time Series 1 (1981) · Zbl 0537.62075
[17] Zhurbenko I. G., Ukrain. Matem. Z. 27 (4) pp 452– (1975)
[18] Zygmund A., Trigonometric Series (1968)
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.