We consider the problem of estimation in a partially observed discrete-time Galton-Watson branching process, focusing on the first two moments of the offspring distribution. Our study is motivated by modelling the counts of new cases at the onset of a stochastic epidemic, allowing for the facts that only a part of the cases is detected, and that the detection mechanism may affect the evolution of the epidemic. In this setting, the offspring mean is closely related to the spreading potential of the disease, while the second moment is connected to the variability of the mean estimators. Inference for branching processes is known for its nonstandard characteristics, as compared with classical inference. When, in addition, the true process cannot be directly observed, the problem of inference suffers significant further perturbations. We propose nonparametric estimators related to those used when the underlying process is fully observed, but suitably modified to take into account the intricate dependence structure induced by the partial observation and the interaction scheme. We show consistency, derive the limiting laws of the estimators, and construct asymptotic confidence intervals, all valid conditionally on the explosion set.

We consider statistical inference for a parametric cooperative sequential adsorption model for spatial time series data, based on maximum likelihood. We establish asymptotic normality of the maximum likelihood estimator in the thermodynamic limit. We also perform and discuss some numerical simulations of the model, which illustrate the procedure for creating confidence intervals for large samples.

Results on asymptotic normality for the maximum likelihood estimate in hidden Markov models are extended in two directions. The stationarity assumption is relaxed, which allows for a covariate process influencing the hidden Markov process. Furthermore, a class of estimating equations is considered instead of the maximum likelihood estimate. The basic ingredients are mixing properties of the process and a general central limit theorem for weakly dependent variables.

The limit behaviour in probability of realised quadratic variation is discussed under a relatively simple ambit process setting. The relation of this to the underlying volatility/intermittency field is in focus, especially as concerns the question of no volatility/intermittency memory.

We consider the problem of estimating a latent point process, given the realization of another point process. We establish an expression for the conditional distribution of a latent Poisson point process given the observation process when the transformation from the latent process to the observed process includes displacement, thinning, and augmentation with extra points. Our original analysis is based on an elementary and self-contained random measure theoretic approach. This simplifies and complements previous derivations given in Mahler (2003), and Singh, Vo, Baddeley and Zuyev (2009).

We consider the following stochastic model for a mobile service scenario. Consider a stationary Poisson process in R d , with its points radially ordered with respect to the origin (the anchor); if d = 2, the points may correspond to locations of, e.g. restaurants. A user, with a location different from the origin, asks for the location of the first Poisson point and keeps asking for the location of the next Poisson point until the first time that he/she can be completely certain that he/she knows which Poisson point is his/her nearest neighbour. This waiting time is the communication cost, while the inferred privacy region is a random set obtained by an adversary who only knows the anchor and the points received from the server, where the adversary ‘does the best’ to infer the possible locations of the user. Probabilistic results related to the communication cost and the inferred privacy region are established for any dimension d ≥ 1. Furthermore, special results when d = 1 and particularly when d = 2 are derived.

Suppose that {X t } is a Markov chain such as the state space model for a threshold GARCH time series. The regularity assumptions for a drift condition approach to establishing the ergodicity of {X t } typically are ϕ-irreducibility, aperiodicity, and a minorization condition for compact sets. These can be very tedious to verify due to the discontinuous and singular nature of the Markov transition probabilities. We first demonstrate that, for Feller chains, the problem can at least be simplified to focusing on whether the process can reach some neighborhood that satisfies the minorization condition. The results are valid not just for the transition kernels of Markov chains but also for bounded positive kernels, opening the possibility for new ergodic results. More significantly, we show that threshold GARCH time series and related models of interest can often be embedded into Feller chains, allowing us to apply the conclusions above.

Customers arrive sequentially at times x 1 < x 2 < · · · < x n and stay for independent random times Z 1, …, Z n > 0. The Z-variables all have the same distribution Q. We are interested in situations where the data are incomplete in the sense that only the order statistics associated with the departure times x i + Z i are known, or that the only available information is the order in which the customers arrive and depart. In the former case we explore possibilities for the reconstruction of the correct matching of arrival and departure times. In the latter case we propose a test for exponentiality.

Contemporaneous aggregation of N independent copies of a random-coefficient AR(1) process with random coefficient a ∈ (−1, 1) and independent and identically distributed innovations belonging to the domain of attraction of an α-stable law (0 < α < 2) is discussed. We show that, under the normalization N 1/α, the limit aggregate exists, in the sense of weak convergence of finite-dimensional distributions, and is a mixed stable moving average as studied in Surgailis, Rosiński, Mandrekar and Cambanis (1993). We focus on the case where the slope coefficient a has probability density vanishing regularly at a = 1 with exponent b ∈ (0, α − 1) for α ∈ (1, 2). We show that in this case, the limit aggregate {X̅ t } exhibits long memory. In particular, for {X̅ t }, we investigate the decay of the codifference, the limit of partial sums, and the long-range dependence (sample Allen variance) property of Heyde and Yang (1997).

In this paper we describe methods for randomly thinning certain classes of spatial point processes. In the case of a Markov point process, the proposed method involves a dependent thinning of a spatial birth-and-death process, where clans of ancestors associated with the original points are identified, and where we simulate backwards and forwards in order to obtain the thinned process. In the case of a Cox process, a simple independent thinning technique is proposed. In both cases, the thinning results in a Poisson process if and only if the true Papangelou conditional intensity is used, and, thus, can be used as a graphical exploratory tool for inspecting the goodness-of-fit of a spatial point process model. Several examples, including clustered and inhibitive point processes, are considered.

For a time series, a plot of sample covariances is a popular way to assess its dependence properties. In this paper we give a systematic characterization of the asymptotic behavior of sample covariances of long-memory linear processes. Central and noncentral limit theorems are obtained for sample covariances with bounded as well as unbounded lags. It is shown that the limiting distribution depends in a very interesting way on the strength of dependence, the heavy-tailedness of the innovations, and the magnitude of the lags.

In a repulsive point process, points act as if they are repelling one another, leading to underdispersed configurations when compared to a standard Poisson point process. Such models are useful when competition for resources exists, as in the locations of towns and trees. Bertil Matérn introduced three models for repulsive point processes, referred to as types I, II, and III. Matérn used types I and II, and regarded type III as intractable. In this paper an algorithm is developed that allows for arbitrarily accurate approximation of the likelihood for data modeled by the Matérn type-III process. This method relies on a perfect simulation method that is shown to be fast in practice, generating samples in time that grows nearly linearly in the intensity parameter of the model, while the running times for more naive methods grow exponentially.

Let {Z t }t≥0 be a Lévy process with Lévy measure ν, and let τ(t)=∫0 t r(u) d u, where {r(t)}t≥0 is a positive ergodic diffusion independent from Z. Based upon discrete observations of the time-changed Lévy process X t ≔Z τt during a time interval [0,T], we study the asymptotic properties of certain estimators of the parameters β(φ)≔∫φ(x)ν(d x), which in turn are well known to be the building blocks of several nonparametric methods such as sieve-based estimation and kernel estimation. Under uniform boundedness of the second moments of r and conditions on φ necessary for the standard short-term ergodic property limt→ 0 E φ(Zt )/t = β(φ) to hold, consistency and asymptotic normality of the proposed estimators are ensured when the time horizon T increases in such a way that the sampling frequency is high enough relative to T.

We consider a model for a time series of spatial locations, in which points are placed sequentially at random into an initially empty region of ℝd , and given the current configuration of points, the likelihood at location x for the next particle is proportional to a specified function βk of the current number (k) of points within a specified distance of x. We show that the maximum likelihood estimator of the parameters βk (assumed to be zero for k exceeding some fixed threshold) is consistent in the thermodynamic limit where the number of points grows in proportion to the size of the region.

Consider the additive effects outliers (A.O.) model where one observes , with The sequence of r.v.s is independent of and , are i.i.d. with d.f. , where the d.f.s Ln, n ≦ 0, are not necessarily known and εj's are i.i.d.. This paper discusses the asymptotic behavior of functional least squares estimators under the above model. Uniform consistency and uniform strong consistency of these estimators are proven. The weak convergence of these estimators to a Gaussian process and their asymptotic biases are also discussed under the above A.O. model.

We establish the uniform almost sure convergence of the partitioning estimate, which is a histogram-like mean regression function estimate, under ergodic conditions for a stationary and unbounded process. The main application of our results concerns time series analysis and prediction in the Markov processes case.

The paper gives a spectral representation for a class of random fields which are bounded in mean square almost surely. A characterisation of the corresponding spectral measure in the representation is obtained based on Beurling's duality theory and generalised Fourier transforms. A representation for the covariance function of asymptotically stationary random fields is also derived.

The Bernstein-von Mises theorem, concerning the convergence of suitably normalized and centred posterior density to normal density, is proved for a certain class of linearly parametrized parabolic stochastic partial differential equations (SPDEs) as the number of Fourier coefficients in the expansion of the solution increases to infinity. As a consequence, the Bayes estimators of the drift parameter, for smooth loss functions and priors, are shown to be strongly consistent, asymptotically normal and locally asymptotically minimax (in the Hajek-Le Cam sense), and asymptotically equivalent to the maximum likelihood estimator as the number of Fourier coefficients become large. Unlike in the classical finite dimensional SDEs, here the total observation time and the intensity of noise remain fixed.

This paper considers a large class of non-stationary random fields which have fractal characteristics and may exhibit long-range dependence. Its motivation comes from a Lipschitz-Holder-type condition in the spectral domain.

The paper develops a spectral theory for the random fields, including a spectral decomposition, a covariance representation and a fractal index. From the covariance representation, the covariance function and spectral density of these fields are defined. These concepts are useful in multiscaling analysis of random fields with long-range dependence.