On the Extinction Time of a Continuous time Markov Branching Process

Several authors have studied and derived bounds for the mean extinction time of a discrete branching process. In this note, we obtain the mean and variance of the extinction time, given extinction of a continuous time Markov branching process. We also obtain bounds for the mean extinction time in terms of moments of the offspring distribution.     

Estimation in second order branching processes with application to swine flu data

Abstract

This paper discusses inferential issues related to estimation of offspring mean and variance in a second order branching process, when both the offspring distributions are assumed to have identical mean and variance. Estimating equation approach is used to find the estimator of the offspring mean and the fact that a second order branching process model can be modeled as an autoregressive process is utilized to obtain the estimator of the offspring variance. Both the estimators are shown to be consistent and asymptotically normal. The second order branching process model is applied to H1N1 data for Pune, India, and Mexico and is found to be a suitable model. The estimates obtained from this model are used to compute the proportion of vaccination required for elimination of the disease.     

Estimation of extinction probability of a multitype markov branching process

Maximum likelihood estimation of the probability of ultimate extinction of a multitype Markov branching process is studied when the process is observed completely over a fixed time interval [0,t]. The asymptotic properties of the estimator are discussed.     

On asymptotic posterior normality for controlled branching processes

For a controlled branching process (CBP) with offspring distribution belonging to the power series family, the asymptotic normality of the posterior distribution of the basic parameter and the offspring mean is proved. As practical applications, we calculate asymptotic high probability density credibility sets for the offspring mean and we provide a rule to make inference about the value of this parameter. Moreover, the asymptotic posterior normality of the respective parameters of two classical branching models, namely the standard Galton–Watson process and the Galton–Watson process with immigration, is derived as particular cases of the CBP.     

REMARKS ON OPTIMAL INFERENCE FOR MARKOV BRANCHING PROCESSES: A SEQUENTIAL APPROACH

Use of a suitable stopping rule yields exact uniformly most powerful tests and minimum variance unbiased estimators of various parameters of a Markov branching model with or without immigration. The population model discussed includes the pure birth, simple epidemic, immigration-death, M/M/ 1 queue, linear birth-death and a branching diffusion process, among others, as special cases.     

On the genealogy and coalescence times of Bienaymé–Galton–Watson branching processes

Coalescence processes have received a lot of attention in the context of conditional branching processes with fixed population size and non-overlapping generations. Here, we focus on similar problems in the context of the standard unconditional Bienaymé–Galton–Watson branching processes, either (sub)-critical or supercritical. Using an analytical tool, we derive the structure of some counting aspects of the ancestral genealogy of such processes, including: the transition matrix of the ancestral count process and an integral representation of various coalescence times distributions, such as the time to most recent common ancestor of a random sample of arbitrary size, including full size. We illustrate our results on two important examples of branching mechanisms displaying either finite or infinite reproduction mean, their main interest being to offer a closed form expression for their probability generating functions at all times. Large time behaviors are investigated.     

Functionals of Markovian Branching D-BMAPS

Abstract

We consider a stochastic system in which Markovian customer attribute processes are initiated at customer arrivals in a discrete batch Markovian arrival process (D-BMAP). We call the aggregate a Markovian branching D-BMAP. Each customer attribute process is an absorbing discrete time Markov chain whose parameters depend both on the phase transition, of the driving D-BMAP, and the number of arrivals taking place at the customer's arrival instant. We investigate functionals of Markovian branching D-BMAPs that may be interpreted as cumulative rewards collected over time for the various customers that arrive to the system, in the transient and asymptotic cases. This is achieved through the derivation of recurrence relations for expected values and Laplace transforms in the former case, and Little's law in the latter case.     

The multitype branching random walk,I

The limiting behaviour of the multitype branching random walk is studied. A limit theorem is proven for the supercritical process. Steady-state distributions are shown to exist for the subcritical process with immigration, and for the critical transient process beginning with Poisson random fields. An analogue of the exponential limit law is proven for the critical process whose migration process is Brownian motion in two dimensions.     

Non-parametric Estimation of the Death Rate in Branching Diffusions

We consider finite systems of diffusing particles in with branching and immigration. Branching of particles occurs at position dependent rate. Under ergodicity assumptions, we estimate the position-dependent branching rate based on the observation of the particle process over a time interval [0, t ]. Asymptotics are taken as t  → ∞. We introduce a kernel-type procedure and discuss its asymptotic properties with the help of the local time for the particle configuration. We compute the minimax rate of convergence in squared-error loss over a range of Hölder classes and show that our estimator is asymptotically optimal.     

Large Sample Inference for Markovian Exponential Families with Application to Branching Processes with Immigration

This paper presents limit distributions for the score and likelihood-ratio (L.R.) statistic for testing a composite hypothesis involving the mean of the offspring distribution of the Bienaymé-Galton-Watson branching process with immigration (BPWI) when the process is subcritical, critical or supercritical. The BPWI is shown to be a member of a certain Markovian exponential family.     

A test for randomness of the environments in a branching process

This paper discusses an approximate score test for testing randomness of environments in a branching process without observing the environments. Using an appropriate martingale central limit theorem the asymptotic null distribution of test statistic is shown to be normal. When the offspring distribution is Poisson, the detail derivation of asymptotic distribution of the test statistic is presented.     

On semi-markov compartmental models with branching particles under the influence of disasters

A semi-Markov multi-compart mental system in which particles reproduce similar particles as a Markov branching process and being subjected to disasters is studied. Expressions for the mean number of particles alive at time t in each compartment are obtained. The results concerning irreversible, mammillarian and catenary compartmental systems have been discussed.     

Probabilistic Approach to Solution of the Neumann Problem for Some Nonlinear Equation

In this article we will consider the Neumann boundary-value problem for the nonlinear Helmholtz equation ? Δ?u + a?u = gexp?(u) + f₀. We will assume that there exists the solution to our problem and this permits us to construct an unbiased estimator on the trajectories of certain branching processes. To do so, we apply Green’s formula and an elliptic mean value theorem. This allows us to derive a special integral equation that gives the value of the function u(x) at the point x, with its integral over the domain D and on boundary of the domain ?D = G. The solution of the problem in the form of a mathematical expectation of some random variable is also obtained. In accordance with the probabilistic representation, a branching process is constructed and an unbiased estimator of the solution of the problem is built on its trajectories. The derived unbiased estimator has finite variance. The proposed branching process has a finite average number of branches, and easily simulated. We provide numerical results based on numerical experiments carried out with these algorithms.     

On Bounds for Probability Generating Functions

Brook (1966) gave an upper bound for the moment generating function (m.g.f.) of a positive random variable (r.v.) in terms of its moments, and used this to obtain an upper bound for the probability generating function (p.g.f.) and hence the extinction probability of a simple branching process. Agresti (1974) rederived this bound of the p.g.f. and used it to obtain a lower bound of the expectation of extinction time of a branching process. In both of these applications the random variable is integer valued, and for this class we improve on Brook's bound by deriving the best upper bound of the p.g.f. Our method, which is a variant of Brook's (1966) is used later to obtain the lower bound of the p.g.f. when the third moment is also known.     

Modelling Variability in the Branching Structure of Strawberry Inflorescences

The branching structure of inflorescences of the cultivated strawberry ( Fragaria × ananassa Duch.) is very variable. This paper demonstrates that some aspects of this variability are well described by a simple stochastic model of branching that has two adjustable parameters. The model is shown to provide a good fit to data from a set of almost 700 inflorescences of the cultivar Elsanta, collected over two successive years. For one parameter the maximum likelihood estimator is a moment estimator which is fully efficient even if the detailed branching structure of the inflorescences is not recorded. This parameter provides a convenient summary of branching vigour. The maximum likelihood estimator of the second parameter must be determined iteratively and can be quite inefficient unless the full branching structure is recorded. The model demonstrates that branching structure is affected by the order in which inflorescences emerge on the plant.     

ASYMPTOTIC TESTS FOR THE CONTINUOUS TIME MARKOV BRANCHING PROCESS WITH IMMIGRATION

This paper presents limit distributions for the modified score and the likelihood-ratio (LR) statistic for testing a composite hypothesis involving the split intensity and mean of the offspring distribution of the supercritical continuous time Markov branching process allowing immigration (CBPI). The immigration intensity and mean are treated as nuisance parameters.     

A two-dimensional branching process with migration

A two-dimensional branching process with both immigration and emigration 1s studied. The probability generating function of the population size at different generation points is obtained. The result is used to derive the expressions for the expected population vector under different conditions of immigration and emigration which leads to the ultimate structure of the population.     

A nonparametric estimation procedure for the Hawkes process: comparison with maximum likelihood estimation

In earlier work, Kirchner [An estimation procedure for the Hawkes process. Quant Financ. 2017;17(4):571–595], we introduced a nonparametric estimation method for the Hawkes point process. In this paper, we present a simulation study that compares this specific nonparametric method to maximum-likelihood estimation. We find that the standard deviations of both estimation methods decrease as power-laws in the sample size. Moreover, the standard deviations are proportional. For example, for a specific Hawkes model, the standard deviation of the branching coefficient estimate is roughly 20% larger than for MLE – over all sample sizes considered. This factor becomes smaller when the true underlying branching coefficient becomes larger. In terms of runtime, our method clearly outperforms MLE. The present bias of our method can be well explained and controlled. As an incidental finding, we see that also MLE estimates seem to be significantly biased when the underlying Hawkes model is near criticality. This asks for a more rigorous analysis of the Hawkes likelihood and its optimization.     

Estimation of the offspring mean in a branching process with non stationary immigration

ABSTRACT

In the paper, we consider a natural estimator of the offspring mean of a branching process with non stationary immigration based on observation of population sizes and number of immigrating individuals to each generation. We demonstrate that using a central limit theorem for multiple sums of dependent random variables it is possible to derive asymptotic distributions for the estimator without prior knowledge about the behavior (criticality) of the reproduction process. Before the three cases of criticality have been considered separately. Assuming that the immigration mean and variance vary regularly, conditions guaranteeing the strong consistency of the proposed estimator is also derived.