Estimation and information in stationary time series

Abstract

Section (1) is devoted to a discussion of the model-fitting problem, which finds its explicit solution in equation (1.13). In section (2) the maximum likelihood, (ML), estimates of the model parameters are investigated, and for the class of series considered shown to possess the same optimum properties as in the case of independent series. Next, the covariance matrix of the parameter estimates is expressed in terms of the spectral function of the generating process (eq. 3.7). The last section is concerned with certain working approximations to the ML statistics.(1) The ultimate objects of any time series analysis are rarely more than two in number: firstly, to estimate, for its own interest, the stochastic relation generating the terms of the series, and secondly, to obtain a forecast by the use of this relation. If the spectrum of the process is known, then both of these problems may be solved by existing methods, under the assumption that the stochastic relation is a linear one (refs. 3, 10). Thus, if we limit ourselves to the case of linearity, the analysis is reduced to the estimation of the spectrum. The word "estimation" is here used in a fairly wide sense, since we require to estimate a function, not merely ~ finite set of parameters. In general we must specify a definite kind of function, and it is this necessity which compels the analyst to use some sort of test or decision procedure. However, we shall consider this aspect of the problem only in passing, since it has already been treated in ref. 9.Suppse, then, that we have a time series of N equidistant observations; xl, x2 .... XN, forming an N • 1 vector X. We shall assume that these observations constitute a part realisation of a real, discrete, stationary process, and our aim is then to obtain the best possible estimate of this process. As above, the generating process will for our purposes be considered as determined when we known its spectrum, F(y), defined by (see ref. 2) (1.1) If the spectrum is differentiable, we may define the spectral ]unction A (z) by A (e ~v) OF (y) _ ~ vs e isv.

Calibration of hydrological models in the spectral domain: An opportunity for scarcely gauged basins?

“…[8] The likelihood measure proposed by Whittle [1953] for the parameters of a generic model will be denoted in the following as L(q), where q is the model parameter vector. Note that the likelihood of a parameter set q is proportional to the probability of obtaining a correct model simulation when the model parameter set is q.…”

“…[6] The purpose of this paper is to propose the use of the maximum likelihood estimator introduced in the context of time series analysis by Whittle [1953] for calibrating hydrological model parameters. The estimator has good statistical properties, as it is asymptotically consistent.…”

Adjusted empirical likelihood for long-memory time-series models

“…We only consider the values of d between 0 and 0.5 since it is the most interesting long-memory scenario (Beran, 1994) and only under this condition the dependence structure of periodogram ordinates has been established (Yau, 2012). The parameter β = (φ 1 , ..., φ p , θ 1 , ..., θ q , d, σ 2 ) ∈ B is estimated by Whittle's method (Whittle, 1953) based on the periodogram where B is a compact subset of the k-dimensional Euclidean space (k = p + q + 2) .…”

“…Mykland (1995) established the connection between the dual likelihood and the empirical likelihood through the martingale estimating equations and applied it to time series model. Monti (1997) developed the idea of extending the EL method to short-memory stationary time series by using the Whittle's (1953) method to obtain an M-estimator of the periodogram ordinates of time series models which are asymptotically independent. However, his method can not be applied directly to long-time memory time series model.…”

Bayesian Methods for Function Estimation

“…Because the actual likelihood of f is difficult to handle, Whittle (1957Whittle ( , 1962 proposed a "quasi-likelihood" where ω l = 2l/n, ν is the greatest integer less than or equal to (n − 1)/2, and I n (ω) = | n t=1 X t e −itπω | 2 /(2πn) is the periodogram. A pseudo-posterior distribution may be obtained by updating the prior using this likelihood.…”