Mean-zero Process Research Articles

Linear invariant estimation of clock phase and trend from noisy phase data

We show how to solve two problems of optimal linear estimation from a set of noisy measurements of clock phase. The phase is modelled as a process with stationary dth increments, e.g. a sum of processes with power-law spectra. The additive measurement noise can be any mean-zero process with a known autocovariance function. The estimation targets are the phase at a given time and the coefficient of the overall trend, e.g. the frequency drift. A feasibility condition called ‘invariance’ is imposed according to the target and the degree of non-stationarity of the phase model. The solution of a set of linear equations gives the regression coefficients and the mean squared error of the best linear invariant estimate. Proofs of these results are available from the online version of this journal.

I.i.d. representations for the bivariate product limit estimators and the bootstrap versions

Let F( s, t) = P( X > s, Y > t) be the bivariate survival function which is subject to random censoring. Let F ̂ n(s, t) be the bivariate product limit estimator (PL-estimator) by Campbell and Földes (1982, Proceedings International Colloquium on Non-parametric Statistical Inference, Budapest 1980, North-Holland, Amsterdam). In this paper, it was shown that F ̂ n(s, t) − F(s, t) = n −1 Σ i = 1 n ζ i(s, t) + R n(s, t) , where { ζ i ( s, t)} is i.i.d. mean zero process and R n ( s, t) is of the order O((n −1 log n) 3 4 ) a.s. uniformly on compact sets. Weak convergence of the process { n −1 Σ i = 1 n ζ i ( s, t)} to a two-dimensional-time Gaussian process is shown. The covariance structure of the limiting Gaussian process is also given. Corresponding results are also derived for the bootstrap estimators. The result can be extended to the multivariate cases and are extensions of the univariate case of Lo and Singh (1986, Probab. Theory Relat. Fields, 71, 455–465). The estimator F ̂ n(s, t) is also modified so that the modified estimator is closer to the true survival function than F ̂ n(s, t) in supnorm.