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Multiple time series analysis of irregularly spaced data. (English) Zbl 0556.62063
Time series analysis of irregularly observed data, Proc. Symp., Texas A & M Univ., College Station/Tex. 1983, Lect. Notes Stat. 25, 276-289 (1984).
[For the entire collection see Zbl 0542.00014.]
A stochastic vector process $$Y(t)\in {\mathbb{R}}^ d$$ (t continuous or discrete) is observed at (possibly irregularly spaced or random) time points $$t_ 1<t_ 2<...<t_ N$$ and eventually single coordinate values may be missing. The observed vector $$Y=(Y_ 1,...,Y_ N)$$ is used for estimating or testing an unknown parameter vector $$\vartheta$$ which characterizes the distribution of $$\{$$ Y(t)$$\}$$. The Gaussian case $$(1)\quad Y(t)=\mu (t,\beta)+e(t)+\sum^{\infty}_{j=1}A_ j(\psi)e(t- j)$$ (with parameter $$\beta$$, scaling factors $$\sigma$$ for the errors e(t), and correlation parameters $$\psi)$$ and the stochastic differential equation case (2) $$\dot Y(t)=BY(t)+e(t)$$ are emphasized.
For (2), equally-spaced sampling cannot guarantee identifiability, but random sampling does (e.g., $$t_ n-t_{n-1}^ a$$renewal process with exponential distribution). Asymptotic formulas for the (conditional) ML- estimates of $$\beta$$, $$\sigma$$, $$\psi$$ are given in case (1); as for consistency and weak convergence, other papers are reviewed. For hypothesis testing, the Lagrange-multiplier test statistic seems to be superior, in some cases, to the Wald or LR statistic.
Reviewer: H.H.Bock

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62F12 Asymptotic properties of parametric estimators 62M07 Non-Markovian processes: hypothesis testing