摘要:

Abstract.Consider a stationary autoregressive process given byXt=b1Xt‐1+…+bpXt‐p+Yt, where theYtare independent identically distributed positive variables andb1,…,bpare non‐negative parameters. Let the variablesX1,…,Xnbe given. Ifp= 1 then it is known thatb1*= min(Xt/Xt‐1) is a strongly consistent estimator forb1under very general conditions. In this paper the casep= 2 is analysed in detail. It is proved that min(Xt/Xt‐1)→b1almost surely (a.s.) and min(Xt/Xt‐2)→b2+b12a.s. asn→ 8. The convergence is very slow. Denote byb1* andb2* values ofb1andb2respectively which maximizeb2+b2under the conditionsXt‐b1Xt‐1‐b2Xt‐2≥ 0 fort= 3,…,n. We prove thatb1*b1andb2*b2a.s. Simulations show thatb1* andb2* are better than the least‐squares estimators of the autoregressive coefficients whe

ISSN:0143-9782    

DOI:10.1111/j.1467-9892.1989.tb00011.x

出版商:Blackwell Publishing Ltd    

年代:1989

数据来源: WILEY

摘要:

Abstract.The problem is that of determining the parameters in a trigonometric polynomial when it is observed with added stationary noise. The frequencies, in particular, must be determined and the situation especially considered is that where these are close together. A similar problem arises if an angular frequency is close to zero or π. The method of estimation is the maximization of the regression sum of squares as a function of the unknown frequencies. In the asymptotic theory, the closely adjacent frequencies are separated by an amount that is of the order T‐1, whereTis the length of the series. Simulations show that this asymptotic treatment gives a better approximation in cases where the separation is of this magnitude than that obtained by treating the frequencies as fix

ISSN:0143-9782    

DOI:10.1111/j.1467-9892.1989.tb00012.x

出版商:Blackwell Publishing Ltd    

年代:1989

数据来源: WILEY

摘要:

Abstract.A sufficient condition is derived for the existence of a strictly stationary solution of some bilinear time series which may have infinite variance innovations. This condition is equivalent to the condition that a polynomial of degreerhas no zeros within the unit circle. In the special case when the innovations have finite variance, the computational effort involved in checking this condition is significantly reduced compared with checking the stationarity conditions given by Bhaskara Raoet al.and Liu and Brockwell which requires a knowledge of the maximum eigenvalue in the absolute value of anr2xr2matrix.

ISSN:0143-9782    

DOI:10.1111/j.1467-9892.1989.tb00013.x

出版商:Blackwell Publishing Ltd    

年代:1989

数据来源: WILEY

摘要:

Abstract.The purpose of this paper is to discuss several fundamental issues in the theory of time‐dependent spectra for univariate and multivariate non‐stationary processes. The general framework is provided by Priestley's evolutionary spectral theory which is based on a family of stochastic integral representations. A particular spectral density function can be obtained from the Wold—Cramér decomposition, as illustrated by several examples. It is shown why the coherence is time invariant in the evolutionary theory and how the theory can be generalized so that the coherence becomes time dependent. Statistical estimation of the spectrum is also considered. An improved upper bound for the bias due to non‐stationarity is obtained which does not rely on the characteristic width of the process. The results obtained in the paper are illustrated using time series simulated from an evolving bivariate autoregressive moving‐average process of order (1, 1) with a highly time‐varyi

ISSN:0143-9782    

DOI:10.1111/j.1467-9892.1989.tb00014.x

出版商:Blackwell Publishing Ltd    

年代:1989

数据来源: WILEY

摘要:

Abstract.For the strictly stationary AR(k) processZt=Λ(Zt‐1) +αt, withΛ:Rk→R,Zt‐1= [Zt‐1,Zt‐2,…,Zt‐k] and {αt} an independent identically distributed white noise process, we partially characterize theΛfor which the stationary distribu

ISSN:0143-9782    

DOI:10.1111/j.1467-9892.1989.tb00015.x

出版商:Blackwell Publishing Ltd    

年代:1989

数据来源: WILEY

摘要:

Abstract.A procedure based on the automatic information criterion procedure of Akaike is presented for estimating the number of sinusoidal terms present in a time series. The procedure is shown to produce a strongly consistent estimator.

ISSN:0143-9782    

DOI:10.1111/j.1467-9892.1989.tb00016.x

出版商:Blackwell Publishing Ltd    

年代:1989

数据来源: WILEY

摘要:

Abstract.A data analysis estimation method called structural persistence is presented in this paper. Prediction applications of the method to time series with trend and seasonal components are discussed. The basic underlying assumption is that the structure of a given series does not change in the forecasting range. Therefore, when forecasts are made, the values which measure the structure (e.g. the trend shape) of the original and of the forecast‐extended time series should be the same. We propose relaxed ‘model‐free’ use of the principle which nevertheless provides an explicit prediction formula for the one‐stepahead prediction Pn+1. Missing data can also be estimated using this method. The procedure is applied to some previously published time series data and the prediction results are compared with those obtained using the Box‐Jenki

ISSN:0143-9782    

DOI:10.1111/j.1467-9892.1989.tb00017.x

出版商:Blackwell Publishing Ltd    

年代:1989

数据来源: WILEY