The bifurcating autoregressive model has been used previously to model cell lineage data. A feature of this model is that each line of descendants from an initial cell follows an AR(1) model, and that the environmental effects on sisters are correlated. However, this model concentrates on modelling the correlations between mother and daughter cells and between sister cells, and does not explain the large correlations between more distant relatives observed by some authors. Here the model is extended, firstly by allowing lines of descent to follow an ARMA(p,q) model rather than an AR(1) model, and secondly by allowing correlations between the environmental effects of relatives more distant than sisters. The models are applied to several data sets consisting of independent cell lineage trees.

An important model in communications is the stochastic FM signal st = A cos , where the message process {mt} is a stochastic process. In this paper, we investigate the linear models and limit distributions of FM signals. Firstly, we show that this non-linear model in the frequency domain can be converted to an ARMA (2, q + 1) model in the time domain when {mt} is a Gaussian MA (q) sequence. The spectral density of {St} can then be solved easily for MA message processes. Also, an error bound is given for an ARMA approximation for more general message processes. Secondly, we show that {St} is asymptotically strictly stationary if {mt} is a Markov chain satisfying a certain condition on its transition kernel. Also, we find the limit distribution of st for some message processes {mt}. These results show that a joint method of probability theory, linear and non-linear time series analysis can yield fruitful results. They also have significance for FM modulation and demodulation in communications.

We consider records from the sequence Yj = Xj, + cj, j ≧ 1 where c > 0 if upper records are of interest and c < 0 if lower records are studied. If {Xj} is stationary the record rate is asymptotically constant and if {Xj} is i.i.d. the record rate is, in addition, asymptotically normal. These results are illustrated by analysis of the times in the mile run.

The problem initially considered is that of testing whether ρ = 0 in a model y(n) = x(n)+ η (n), x(n) = ρx(n – 1) + ξ (n) where only y(n) is observed and η (n), ξ (n) are white noise. This is equivalent to distinguishing between an ARMA (1, 1) model and white noise. The asymptotic distribution of the likelihood ratio criterion is derived. This is shown to be of an unusual form. This result is then used to discuss the asymptotic properties of Akaike's procedure for estimating (p, q) in an ARMA (p, q) model. If p0, q0 are the true values and p0 < P, q0< Q, when P, Q are the maximum values considered, then it is shown that, in a certain asymptotic sense, the procedure is sure to overestimate p0, q0. However, the asymptotic situation may be very far from that relevant in a practical case. The relevance of overestimation is briefly discussed.

Three related estimators are considered for the parametrized spectral density of a discrete-time process X(n), n = 1, 2, · · ·, when observations are not available for all the values n = 1(1)N. Each of the estimators is obtained by maximizing a frequency domain approximation to a Gaussian likelihood, although they do not appear to be the most efficient estimators available because they do not fully utilize the information in the process a(n) which determines whether X(n) is observed or missed. One estimator, called M3, assumes that the second-order properties of a(n) are known; another, M2, lets these be known only up to an unknown parameter vector; the third, M1, requires no model for a(n). Under representative sets of conditions, which allow for both deterministic and stochastic a(n), the strong consistency and asymptotic normality of M1, M2, and M3 are established. The conditions needed for consistency when X(n) is an autoregressive moving-average process are discussed in more detail. It is also shown that in general M1 and M3 are equally efficient asymptotically and M2 is never more efficient, and may be less efficient, than M1 and M3.