Non-ergodic martingale estimating functions and related asymptotics

Martingale Estimating Functions for Stochastic Processes: A Review Toward a Unifying Tool

“…Instead, a random norm is required to get normal limit distributions for nonergodic processes. Asymptotics of various statistics normalized by random norms in a broad context have recently been discussed by Pena et al (2009) andHwang et al (2013a). Refer to Basawa and Scott (1983) for various nonergodic processes including normal mixture models, explosive autoregressive processes and branching processes.…”

“…Refer to Hwang et al (2013a) for further details. Cowan and Stuadte (1986) introduced (BAR) processes indexed by a binary-splitting tree for cell lineage study.…”

“…Via establishing a convolution theorem using a random norm, it will be shown that the ML estimator continues to be asymptotically optimum in a sense of "minimum variance" within a class of estimators obtained from nonergodic MEFs. Most of the contents in this section are adapted from those in Hwang et al (2013a) and Hwang and Basawa (2011a) and therefore we provide streamlined outlines only, omitting some details.…”

Contemporary review on the bifurcating autoregressive models : Overview and perspectives

“…Denote the (2 × 2) conditional variance-covariance matrix of a i (θ) given the first ancestor Hwang et al, 2014a). The conditional expectation of the derivative matrix of a i (θ) is given by…”

“…Unlike the classical three tests, it is not obvious that QL based tests are asymptotically efficient. Readers refer to Hwang et al (2014a) and Hwang and Basawa (2011) for a rigorous discussion on the certain asymptotic optimality of QL based tests Q 1n and Q 2n .…”