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.     

Non-ergodic martingale estimating functions and related asymptotics

Well-known estimation methods such as conditional least squares, quasilikelihood and maximum likelihood (ML) can be unified via a single framework of martingale estimating functions (MEFs). Asymptotic distributions of estimates for ergodic processes use constant norm (e.g. square root of the sample size) for asymptotic normality. For certain non-ergodic-type applications, however, such as explosive autoregression and super-critical branching processes, one needs a random norm in order to get normal limit distributions. In this paper, we are concerned with non-ergodic processes and investigate limit distributions for a broad class of MEFs. Asymptotic optimality (within a certain class of non-ergodic MEFs) of the ML estimate is deduced via establishing a convolution theorem using a random norm. Applications to non-ergodic autoregressive processes, generalized autoregressive conditional heteroscedastic-type processes, and super-critical branching processes are discussed. Asymptotic optimality in terms of the maximum random limiting power regarding large sample tests is briefly discussed.     

Asymptotic bootstrap validity for finite markov chains

Two methods of bootstrap, viz., standard, and conditional, are presented for estimating the transition probabilities of a finite state Markov chain. Asymptotic validity of the bootstrap estimates are established for both methods. An applica- tion to a bootstrapped statistic for testing independence is briefly discussed together with some simulation results.     

First-order random coefficient integer-valued autoregressive processes

A first-order random coefficient integer-valued autoregressive (RCINAR(1)) model is introduced. Ergodicity of the process is established. Moments and autocovariance functions are obtained. Conditional least squares and quasi-likelihood estimators of the model parameters are derived and their asymptotic properties are established. The performance of these estimators is compared with the maximum likelihood estimator via simulation.     

PARSIMONIOUS PERIODIC TIME SERIES MODELING

This paper studies techniques for fitting parsimonious periodic time series models to periodic data. Large sample standard errors for the parameter estimates in a periodic autoregressive moving‐average time series model under parametric constraints are derived. Likelihood ratio statistics for hypothesis testing are examined. The techniques are applied in modeling daily temperatures at Griffin, Georgia, USA.     

MAXIMUM LIKELIHOOD ESTIMATION OF PARAMETERS IN RENEWAL AND MARKOV-RENEWAL PROCESSES

Sampling procedures using randomized observation-points are suggested for estimating parameters in renewal and Markov renewal models. The usual asymptotic properties of the maximum likelihood method are shown to hold. The method we suggest provides a solution to the ML estimation problem in either or both of the following situations: (i) observations on between-event intervals are unavailable, (ii) the interval densities are unknown or difficult to evaluate while their Laplace-Stieltjes transforms are known.     

RECENT TRENDS IN ASYMPTOTIC OPTIMAL INFERENCE FOR DEPENDENT OBSERVATIONS1

An overview of some recent developments in the area of asymptotic inference for non-ergodic type stochastic processes is presented. Both local and global formulations of the asymptotic model are given, and non-local optimality results are reviewed. Recent results on conditional inference are briefly discussed. Some open problems and possibilities for new developments are also mentioned.