Statistical estimation of the multivariate parameter of spectral density. I
Let \(X_ t\), \(t=...,-1,0,1,...\), be a real random sequence with mean E \(X_ t=0\) and spectral density (s.d.) \(f(\lambda)\), which is stationary in the broad sense. That is \[ X_ t=\int^{\pi}_{- \pi}e^{it\lambda} d\zeta (\lambda),\quad t=...,-1,0,1,..., \] where \(\zeta(\lambda)\), \(-\pi\leq \lambda \leq \pi\), is a stochastic process with uncorrelated increments such that E \(d\zeta(\lambda)=0\) and \(E| d\zeta (\lambda)|^ 2=f(\lambda)d\lambda.\) Let us assume that we are interested in the values of the s.d. \(f(\lambda)\) on a fixed interval \(\lambda_ 1\leq \lambda \leq \lambda_ 2\), where \(0\leq \lambda_ 1<\lambda_ 2\leq \pi\) and that these values are uniquely determined by the multivariate parameter \(\theta =(\theta_ 1,\theta_ 2,...,\theta_ n)\in R^ n\) according to a rule which is known to us. Our problem will be to estimate this parameter from the ``observation'' \(X_ 1,X_ 2,...,X_ N\). Since the s.d. is an even function, \(f(-\lambda)=f(\lambda)\), it is sufficient to study it for \(\lambda\geq 0\) and so we restrict ourselves to the case \([\lambda_ 1,\lambda_ 2]\subseteq [0,\pi]\).
asymptotic properties of Whittle statistic
stochastic process with uncorrelated increments
scientific article; zbMATH DE number 4111863