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.     

Characterization by invariance under length-biasing and random scaling

A positive random variable X with law L(X) and finite moment of order r > 0 has an induced length-biased law of order r, denoted by L(X_r). Let V ⩾ 0 be independent of X_r. A characterization problem seeks solution pairs (L(X), L(V)) for the “in-law” equation X ≅ VX_r, where ≅ denotes equality in law. A renewal process interpretation asks when is the random rescaling of the stationary total lifetime VX₁ equal in law to a typical lifetime X? Solutions are known in special cases.A comprehensive existence/uniqueness theory is presented, and many consequences are explored. Unique solutions occur when − log X and − log V have spectrally positive infinitely divisible laws. Particular cases are explored.Connections with the stationary lifetime law of renewal theory also are investigated.     

Critical Bellman–Harris branching processes with infinite variance allowing state-dependent immigration

The branching processes with state-dependent immigration are considered as alternating regenerative processes. The main purpose is to demonstrate some new “regenerative” methods. Critical Bellman–Harris branching processes with state-dependent immigration are investigated and new limit theorems are obtained in the case of an infinite offspring variance and possibly infinite mean of the immigrants.     

The estimation of adopted mortality and morbidity rates using model and the phase type law: the Turkish case

Abstract

This paper aims to estimate mortality rate, morbidity-mortality rates of a chronic disease utilizing phase type law in the frame of two and three state processes. The application on commonly used mortality tables in Turkey are adopted to process to estimate the future mortalities with respect to phase type distribution for the purpose of justifying. Using one absorbing state, two and three state Models calculate the time until absorbing of the death and death by phase type distribution for each gender. Consequently, the 3-state probabilities in estimating the mortality-morbidity rates of IHD for Turkish population yield a significant information on the health management and pricing health insurance products.     

Plug-in estimators for the mean value and variance functions in delayed renewal processes

Abstract

In this paper, we deal with the problem of estimating the delayed renewal and variance functions in delayed renewal processes. Two parametric plug-in estimators for these functions are proposed and their unbiasedness, asymptotic unbiasedness and consistency properties are investigated. The asymptotic normality of these estimators are established. Further, a method for the computation of the estimators is given. Finally, the performances of the estimators are evaluated for small sample sizes by a simulation study.     

A Note on Estimation of Parameters for Branching Processes with Immigration

It is proved that under certain conditions the conditional least-squares estimator of the offspring mean for a sequence of nearly critical Ji?ina processes with immigration is consistent but not asymptotically normal and the conditional least-squares estimator of the immigration mean is not consistent. These results differ from the existing results in the case where the initial values are zero (see Ispány et al., 2005 Ispány , M. , Pap , G. , Zuijlen , M. V. ( 2005 ). Fluctuation limit of branching processes with immigration and estimation of the means . Adv. Appl. Probab. 37 : 523 – 538 . [Google Scholar]).     

The covariance of the backward and forward recurrence times in a renewal process: the stationary case and asymptotics for the ordinary case

In a Poisson process, it is well-known that the forward and backward recurrence times at a given time point t are independent random variables. In a renewal process, although the joint distribution of these quantities is known (asymptotically), it seems that very few results regarding their covariance function exist. In the present paper, we study this covariance and, in particular, we state both necessary and sufficient conditions for it to be positive, zero or negative in terms of reliability classifications and the coefficient of variation of the underlying inter-renewal and the associated equilibrium distribution. Our results apply either for an ordinary renewal process in the steady state or for a stationary process.     

Time series models associated with Mittag-Leffler type distributions and its properties

ABSTRACT

We have provided a fractional generalization of the Poisson renewal processes by replacing the first time derivative in the relaxation equation of the survival probability by a fractional derivative of order α(0 < α ? 1). A generalized Laplacian model associated with the Mittag-Leffler distribution is examined. We also discuss some properties of this new model and its relevance to time series. Distribution of gliding sums, regression behaviors, and sample path properties are studied. Finally we introduce the q-Mittag-Leffler process associated with the q-Mittag-Leffler distribution.     

Conditional Least Squares Estimators for the Offspring Mean in a Subcritical Branching Process with Immigration

Consider a Bienayme–Galton–Watson process with generation-dependent immigration, whose mean and variance vary regularly with non negative exponents α and β, respectively. We study the estimation problem of the offspring mean based on an observation of population sizes. We show that if β <2α, the conditional least squares estimator (CLSE) is strongly consistent. Conditions which are sufficient for the CLSE to be asymptotically normal will also be derived. The rate of convergence is faster than n ^?1/2, which is not the case in the process with stationary immigration.     

Moderate deviations for the total population arising from a nearly unstable sub-critical Galton-Watson process with immigration

Abstract

In this paper, we consider the moderate deviation of the nearly unstable sub-critical Galton-Waston process with immigration for the centered total population arising. We discuss the main influence factors for the form of moderate deviation in different stochastic processes. Moreover, we compute the exact rate function in every different situation of MDP.     

On Erlang(2) Risk Process Perturbed by Diffusion

ABSTRACT

In this article, we consider an Erlang(2) risk process perturbed by diffusion. From the extreme value distribution of Brownian motion with drift and the renewal theory, we show that the survival probability satisfies an integral equation. We then give the bounds for the ultimate ruin probability and the ruin probability caused by claim. By introducing a random walk associated with the proposed risk process, we define an adjustment-coefficient. The relation between the adjustment-coefficient and the bound is given and the Lundberg-type inequality for the bound is obtained. Also, a formula of Pollaczek–Khinchin type for the bound is derived. Using these results, the bound can be calculated when claim sizes are exponentially distributed.     

Efficient sequential estimation in a markov branching process with immigration

Sequential estimation of parameters In a continuous time Markov branching process with Immigration with split rate λ₁ Immigration rate λ₂, offspring distribution {p₁j≥O) and Immigration distribution {p₂j≥l} is considered. A sequential version of the Cramér-Rao type information inequality is derived which gives a lower bound on the variances of unbiased estimators for any function of these parameters. Attaining the lower bounds depends on whether the sampling plan or stopping rule S, the estimator f, and the parametric function g = E(f) are efficient. All efficient triples (S,f,g) are characterized; It Is shown that for i = 1,2, only linear combinations of λ_ip_{ij j}'s or their ratios are efficiently estimable. Applications to a Yule process, a linear birth and death process with immigration and an M/M/∞ queue are also considered     

Zuverlässigkeitsmodelle auf der Grundlage stochastischer Modelle von Verschleiß prozessen

Reliability of products of mechanical engineering is often decided by wear processes.

Suitable stcohastic precesses (cumulative stochastic precesses, Wiener-process with drift and related multiplicative processes) are applied to modelling of such wear processes. Then the lifetime is the random time to first crossing a given limiting wear level or wear reserve. Some lifetime distributions which are founded in such a wag for constant and random wear reserves are discussed (Birnbaum-Saunders-distribution, Inverse-Gaussian-distribution, special mixtures of distributions).

For some of these models a favourable statistical approach to lifetime distribution arises from samples of the wear process. Process. Possibilities to calculate characteristics or ordinary renewal processes are dealt with. In essential cases such characteristics can be expressed explicity.

By an example the application of the results is demonstrated beginning with samples from the wear process followed by choosing a wear model, estimating the parameters, testing goodness of fit, calculating characteristics of reliability up to optimal designing of block replacement in preventive maintenance and calculating the technical-founded demand for replacement parts.     

Long-Range Dependence of Markov Renewal Processes

This paper examines long‐range dependence (LRD) and asymptotic properties of Markov renewal processes generalizing results of Daley for renewal processes. The Hurst index and discrepancy function, which is the difference between the expected number of arrivals in (0, t] given a point at 0 and the number of arrivals in (0, t] in the time stationary version, are examined in terms of the moment index. The moment index is the supremum of the set of r > 0 such that the rth moment of the first return time to a state is finite, employing the solidarity results of Sgibnev. The results are derived for irreducible, regular Markov renewal processes on countable state spaces. The paper also derives conditions to determine the moment index of the first return times in terms of the Markov renewal kernel distribution functions of the process.     

Approximation to two independent Gaussian processes from a unique Lévy process and applications

Abstract

In this article, we construct two families of processes, from a unique Lévy process, the finite dimensional distributions of which converge in law towards the finite dimensional distributions of the two independent Gaussian processes. As applications of this result, we obtain families of processes that converge in law towards fractional Brownian motion, sub-fractional Brownian motion and bifractional Brownian motion, respectively.     

A Branching Process with Immigration in Varying Environments

Bienaymé–Galton–Watson branching processes with varying offspring variance and an immigration component are studied in the critical case. The asymptotic formulas for the probability for non extinction are derived, in dependence of immigration component. A limit theorem is proved too.     

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.     

The discretely observed immigration-death process: Likelihood inference and spatiotemporal applications

ABSTRACT

We consider a stochastic process, the homogeneous spatial immigration-death (HSID) process, which is a spatial birth-death process with as building blocks (i) an immigration-death (ID) process (a continuous-time Markov chain) and (ii) a probability distribution assigning iid spatial locations to all events. For the ID process, we derive the likelihood function, reduce the likelihood estimation problem to one dimension, and prove consistency and asymptotic normality for the maximum likelihood estimators (MLEs) under a discrete sampling scheme. We additionally prove consistency for the MLEs of HSID processes. In connection to the growth-interaction process, which has a HSID process as basis, we also fit HSID processes to Scots pine data.     

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.