Multiple time series analysis of irregularly spaced data. (English) Zbl 0556.62063

Robinson, P. M.

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

Keywords:

missing data; vector process; Gaussian case; stochastic differential equation; equally-spaced sampling; identifiability; random sampling; consistency; weak convergence; Lagrange-multiplier test