Latent structure models in time series
Adelchi Azzalini · Spiral (Imperial College London) · 1982
In the first three chapters, the analysis of a number, say n, of independent time series is considered in a Normal theory context.Chapter 1 deals with the simplest case of n replicates of a first or second order autoregressive stationary process.The emphasis is on asymptotics as n tends to infinity and the length of the series is fixed.The likelihood conditional on the initial values of each of the n series produces estimates of the parameters which are appreciably less efficient than the ones derived from the unconditional likelihood.Chapter 2 and 3 consider more complex situations (allowing for non-stationarity and non-identical distributions) with special attention for Markov processes, using the unconditional likelihood.Chapter 2 deals with systematic effect models, while Chapter 3 deals with latent structure models.In Chapter 4, a stationary Markov process {9^}, not necessarily Normal, is observable only indirectly, via a process {Y^}.More precisely, the current value is sampled from a distribution determined by the current value of and it is conditionally independent of all past values of {9^} an d ^ }.A procedure is proposed for approximating the conditional densities of and given A method for approximating the likelihood of a .stationarytime series is studied in Chapter 5. General results on the asymptotic properties of the associated estimates are given, and numerical comparisons with other estimators are carried out in special cases.In Appendix A, it is proved that the likelihood of a stationary Normal Markov process with unknown mean, variance and autocorrelation is unimodal.Appendix B studies a Markov process with Beta marginal distribution; its transition law and the autocovariance function are determined.that, for each i, {y^ t for t=l,..., T^} is a section of a first order autoregressive processes and these processes are identically distributed Assume T^ •> 2 for all i, and define T=^T^/n.First, we consider the case of zero mean.We therefore assume the model y i,t = a y i,t-l + £ i,t ' l a ' <:L ' t=2 »---» T i; i=l,...,n; (1) where, for each i, the e^, t are independent and identically distributed (i.i.d.) N(0,a 2 ), i.e. {e.} is Gaussian white noise.If the initial l, t values y. -(i=l,...,n) are regarded as sampled from the stationary 1 j -L distribution N(0, o 2 /(l-a 2 )) the loglikelihood for the unknown parameters (a 2 ,a) is