ARMA Model Identification pp 29-42 | Cite as

# The Autocorrelation Methods

Chapter

## Abstract

It is known (see, e.g., T. W. Anderson [1971, pp. 463–495]) that, for large .

*T*, the bias of the sample ACRF is$$E\left( {{{\hat p}_k}} \right) - {p_k} = - \frac{1}{T} \sum\limits_{j = - \infty }^\infty {{p_j} + o\left( {\frac{1}{T}} \right)}.$$

## Keywords

Spectral Density Probability Density Function Maximum Entropy ARMA Model Maximum Entropy Method
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## Additional References

- (Section 2.1)For details about the bias, the variance, and the covariance of the sample ACRF, refer to Marriott and Pope (1954) and Kendall (1954).Google Scholar
- (Section 2.1)Some characteristics of the PACF have been discussed by Ramsey (1974) and Hamilton and Watts (1978). Theorem 2.1 was proven by Dixon (1944) and Quenouille (1949a, 1949b). Another simple proof can be found in Choi (1990b). The multivariate version of Theorem 2.1 was presented by Bartlett and Rajalakshman (1953), Morf, Vieira, and Kailath (1978), and Sakai (1981). Yajima (1985) investigated the asymptotic properties of the sample ACVF and of the PACF estimate of a multiplicative ARIMA model.Google Scholar
- (Section 2.2)For FFT algorithms, readers may refer to Runge (1903), Brigham and Morrow (1967), Cooley, Lewis, and Welch (1967, 1970a, 1970b), Bergland (1969), Brigham (1974), Silverman (1977), Brillinger (1981, pp. 64–66), Robinson (1982), Elliot and Rao (1982), and the references therein.Google Scholar
- (Section 2.2)Many have tried to prove Burg’s theorem. Smylie, Clarke, and Ulrych (1973, pp. 402–419) have established it by variational methods. Edward and Fitelson (1973) have proved it using the Lagrange multiplier method. Burg (1975), Ulrych and Bishop (1975), Haykin and Kesler (1979, pp. 16–21), and Robinson (1982) have followed Smylie ’s method. Ulrych and Ooe (1979) and McDonough (1979) have used Edward et al.’s proof. Van den Bos (1971) has tried to maximize the entropy h subject to the autocovariance constraints by differential calculus. But further argument is required to complete the proof. Also, refer to Feder and Weinstein (1984). In 1977, Akaike proved it using Kolmogorov’s identity (see Priestley [1981, pp. 604–606]). Grandell, Hamrud, and Toll (1980) have used the same idea to prove it. Choi and Cover (1984, 1987) have presented an information theoretic proof, which needs neither the stationarity assumption nor the Gaussian assumption. Choi (1986b) has proved it using Hadamard’s inequality and Choi (1991d) has proved it using the LU decomposition method. The last two proofs extend the maximum entropy spectrum problem to a probability density function problem subject to the autocovariance constraints.Google Scholar
- (Section 2.2)Properties of the maximum entropy spectral density have been studied by many time series analysts including Kromer (1969), Berk (1974), and Ensor and Newton (1988). Recently, Parzen (1983a) has reviewed the AR spectral density in detail. A multivariate maximum entropy spectral density has been given by Choi (1991e). Franke (1985b) has shown that an ARMA(p, q) process has the maximum entropy subject to the
*p*+1 autocovariances and the q impulse responses. Also, refer to Ihara (1984).Google Scholar - (Section 2.2)For recent developments in spectral analysis, readers may refer to the books edited by Childers (1978), Haykin (1979), Haykin and Cadzow (1982), Brillinger and Krishnaiah (1983), and Kesler (1986). Even though it is a little out of date, the survey paper by Kay and Marple (1981) is good enough to know what is going on with spectral analysis. Also, refer to Gutowski, Robinson, and Treitel (1978), Nitzberg (1979), Thomson (1981), Fuhrmann and Liu (1986), Kay and Shaw (1988), and the references therein.Google Scholar
- (Section 2.2)There are other IACF estimation methods presented by Battaglia (1983, 1986, 1988 ), Kanto (1987), and Subba Rao and Gabr (1989).Google Scholar

## Copyright information

© Applied Probability Trust 1992