SOME PROPERTIES OF CONDITIONAL QUASI-LIKELIHOOD FUNCTIONS FOR TIME SERIES MODEL FITTING

Abstract. We consider fitting a parametric model to a time series and obtain the maximum likelihood estimates of unknown parameters included in the model by regarding the time series as a Gaussian process satisfying the model. We evaluate the asymptotic value of the conditional quasi-likelihood function when the number of observations tends to infinity. We show what properties of the time series we can find by examining the behaviour of the conditional quasi-likelihood function, even when the time series does not necessarily satisfy the model and is not necessarily Gaussian.     

ESTIMATION OF COEFFICIENTS OF TIME SERIES REGRESSION WITH A NONSTATIONARY ERROR PROCESS

Abstract. We treat a problem of estimating unknown coefficients of a time series regression when the variance of the error changes with time, i.e. when a process which the error term obeys is nonstationary. First, we show the weak consistency of the ordinary least squares estimator for the coefficients of a polynomial regression under some assumptions on the covariance structure of the error process. Next, we propose a nonparametric method for estimating the variance of the error process and a weighted least squares estimator of the regression coefficients, which is constructed by using the estimator of the variance. We investigate statistical properties of our proposed estimator in the following way. We consider the prediction of a future value of a linear trend by using our proposed estimator and evaluate its prediction error. By simulation studies, we compare the prediction error of the predictor constructed by using our proposed estimator with the prediction errors obtained for other estimators including the ordinary least squares estimator when the variance of the error process increases with time and the sample sizes are small. As a result, our proposed estimator seems to be reasonable.     

ESTIMATION OF COEFFICIENTS OF AN AUTOREGRESSIVE PROCESS BY USING A HIGHER ORDER MOMENT

Abstract. When we use the estimators, obtained by solving Yule-Walker equations, of the coefficients of an autoregressive process, we cannot discriminate X _t and Y _t where all the solutions of the associated polynomial equation of X _t are less than 1 in the absolute value and, at least, one of the solutions of that of Y _t is greater than 1 in the absolute value. To discriminate between X _t and Y _t Rosenblatt proposed a method. We propose another method by using a higher order moment.     

Embedding a Gaussian discrete‐time autoregressive moving average process in a Gaussian continuous‐time autoregressive moving average process

Abstract. Embedding a discrete‐time autoregressive moving average (DARMA) process in a continuous‐time ARMA (CARMA) process has been discussed by many authors. These authors have considered the relationship between the autocovariance structures of continuous‐time and related discrete‐time processes. In this article, we treat the problem from a slightly different point of view. We define embedding in a more rigid way by taking account of the probability structure. We consider Gaussian processes. First we summarize the necessary and sufficient condition for a DARMA process to be able to be embedded in a CARMA process. Secondly, we show a concrete condition such that a DARMA process can be embeddable in a CARMA process. This condition is new and general. Thirdly, we show some special cases including new examples. We show how we can examine embeddability for these special cases.     

On the Timing Information Disappearance of Digital Transmission Systems

Probability distribution of a timing-wave envelope level in a regenerative repeater, which transmits a random binary sequence, is analyzed. The result obtained is that the probability of timing information disappearance in a repeated line can be disregarded compared with that of failure of a repeater.