The irregular and slope disturbances, εt and ζt, respectively, are mutually independent and the notation NID(0, σ2) denotes normally and independently distributed with mean zero and variance σ2. The signal-noise ratio, q = σ2 ζ/σ 2 ε , plays the key role in determining how observations should be weighted for prediction and signal extraction. The higher is q, the more past observations are disco...

Methods of estimating unknown parameters of a trend function for trend-renewal processes are investigated in the case when the renewal distribution function is unknown. If the renewal distribution is unknown, then the likelihood function of the trend-renewal process is unknown and consequently the maximum likelihood method cannot be used. In such a situation we propose three other methods of es...

where it is assumed that the expected value of X(t) is zero. There is no commonly accepted precise definition for trend, but it is usually spoken of as a nonrandom (deterministic) smooth function representing long-term movement or systematic variations in a series (for example, Priestley (1981) refers to a trend as “. . . a tendency to increase (or decrease) steadily over time . . . [or to] flu...

Suppose that the data is modeled as replicated realizations of a pdimensional random vector whose mean μ is a trend of interest and whose covariance matrix Σ is unknown, positive definite. REACT estimators for the trend involve transformation of the data to a new basis, estimating the risks of a class of candidate linear shrinkage estimators, and selecting the candidate estimator with smallest ...