Modeling and large sample estimation for multi-casting autoregression

Hwang, Sun Young; Choi, M.S.

doi:10.1016/j.spl.2009.06.005

Cited by 8 publications

(10 citation statements)

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“…A tri-casting (m = 3) data is illustrated in Figure 2.1. See for instance, Hwang and Choi (2009) and Hwang and Kang (2012). We consider a multicast autoregressive (MAR, for short) model to accommodate this class of multicast data.…”

Section: From the Bifurcating Tree To Multicast And Branching Treementioning

confidence: 99%

“…Consider the non-stationary case of |θ| > 1. Based on the data X 1 , · · · , X n , the conditional least squares estimator θ n of θ is given by where the norming constant c n is given by c n = 2(2θ 2 − 1) −1 ( √ 2θ) n and U and V are random variables as described in Lemmas 4.1 and 4.2 of Hwang and Choi (2009). The unit root case of |θ| = 1 is now under investigation.…”

Section: Non-stationary Casesmentioning

confidence: 99%

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Contemporary review on the bifurcating autoregressive models : Overview and perspectives

Hwang¹

2014

Journal of the Korean Data and Information Science Society

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Since the bifurcating autoregressive (BAR) model was developed by Cowan and Staudte (1986) to analyze cell lineage data, a lot of research has been directed to BAR and its generalizations. Based mainly on the author's works, this paper is concerned with a contemporary review on the BAR in terms of an overview and perspectives. Specifically, bifurcating structure is extended to multi-cast tree and to branching tree structure. The AR(1) time series model of Cowan and Staudte (1986) is generalized to tree structured random processes. Branching correlations between individuals sharing the same parent are introduced and discussed. Various methods for estimating parameters and related asymptotics are also reviewed. Consequently, the paper aims to give a contemporary overview on the BAR model, providing some perspectives to the future works in this area.

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Section: From the Bifurcating Tree To Multicast And Branching Treementioning

confidence: 99%

Section: Non-stationary Casesmentioning

confidence: 99%

Contemporary review on the bifurcating autoregressive models : Overview and perspectives

Hwang¹

2014

Journal of the Korean Data and Information Science Society

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“…Most of the research on the multi-casting case of σ 2 η = 0 has been directed to identification of the conditional mean function of the models. For traditional issues on the multi-casting tree such as the estimation of mean parameters and stability of the models, we refer to, for instance, Hwang and Choi (2009), Baek et al (2011), Hwang and Basawa (2011.…”

Section: Motivation Of the Studymentioning

confidence: 99%

“…It is noted that there are exactly two offspring X 2t and X 2t+1 from the common mother X t . Extending (2.1) to the multicasting case of m offspring (m ≥ 3), Hwang and Choi (2009) proposed the following multi-casting AR(MCAR) model generated by the m equations.…”

Section: Branching Heteroscedasticity(bh) and Illustrative Examplesmentioning

confidence: 99%

Preliminary Identification of Branching-Heteroscedasticity for Tree-Indexed Autoregressive Processes

Hwang¹,

Choi²

2011

Communications for Statistical Applications and Methods

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A tree-indexed autoregressive(AR) process is a time series defined on a tree which is generated by a branching process and/or a deterministic splitting mechanism. This short article is concerned with conditional heteroscedastic structure of the tree-indexed AR models. It has been usual in the literature to analyze conditional mean structure (rather than conditional variance) of tree-indexed AR models. This article pursues to identify quadratic conditional heteroscedasticity inherent in various tree-indexed AR models in a unified way, and thus providing some perspectives to the future works in this area. The identical conditional variance of sisters sharing the same mother will be referred to as the branching heteroscedasticity(BH, for short). A quasilikelihood but preliminary estimation of the quadratic BH is discussed and relevant limit distributions are derived.Keywords: Branching heteroscedasticity, quasilikelihood, tree-indexed AR. Motivation of the StudyWe first construct a tree-index on which an AR time series (X) of interest is defined. Following the lines as in Hwang and Basawa (2011), consider the successive generation sizes {Z t } with the initial size Z 0 = 1. In particular, the super critical G-W(Galton-Watson) branching process {Z t } is defined bywhere {η t j , t = 1, 2, . . . , j = 1, 2, . . .} is an array of iid non-negative integer-valued random variables with common (offspring) mean m > 1 and variance σ 2 η ≥ 0. Let X t ( j) denote the observation on the j th individual in t th generation. In addition, let X t−1 (t( j)) denote the observation on immediate mother of the j th individual in t th generation. It is noticed that X t−1 (t( j)) is an observation made in the (t − 1) th generation. In Figure 1, note that x 2 (3(1)) = x 2 (1); x 2 (3(10)) = x 2 (6). As with e.g., Hwang (2011), one can consider two cases separately according as σ Figure 1 illustrates a tree consisting of three generations. It is noted in Figure 1 that there are random number of individuals in each generation. On the other hand, the case of σ 2 η = 0 (see Figure 2) is referred to as a multi-casting tree where each individual (mother) gives rise to exactly m-offspring (daughters) in the next generation. When m = 2, the multi-casting tree reduces to a bifurcating case, i.e., a binary-splitting tree studied by several authors including Cowan and Staudte (1986) and Basawa and Zhou (2004) among others. Most of the research on the multi-casting case of σ 2 η = 0 has been directed to identification of the conditional mean function of the models. For traditional issues on the 1 Corresponding author: Professor,

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“…For instance, in an internet traffic data, a package (signal) may be sent along m links and delay times X over each link may be of interest. As a natural extension of the BAR(1), Hwang and Choi (2009) have recently introduced an AR(1) model defined on a multi-casting tree where each individual gives rise to exactly m-offspring in the next generation. They referred to such an AR(1) as the multicast autoregressive model of order 1 (MCAR(1), for short).…”

Section: Introductionmentioning

confidence: 99%

Asymptotics for a class of generalized multicast autoregressive processes

Hwang

Kang

2012

Journal of the Korean Statistical Society

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a b s t r a c tWe propose a new class of generalized multicast autoregressive (GMCAR, for short, hereafter) models indexed by a multi-casting tree where each individual produces exactly the same number of offspring. This class includes standard bifurcating autoregressive processes (BAR, cf. Cowan and Staudte (1986)) and multicast autoregressive (MCAR, cf. Hwang and Choi (2009)) models as special cases. Accommodating non-Gaussian, non-negative and count data, the class includes various models such as nonlinear autoregression, conditionally heteroscedastic process and conditional exponential family. The pathwise stationarity of the GMCAR model is discussed. A law of large numbers and a central limit theorem are established which are in turn used to derive asymptotic distributions associated with martingale estimating functions.

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Modeling and large sample estimation for multi-casting autoregression

Cited by 8 publications

References 10 publications

Contemporary review on the bifurcating autoregressive models : Overview and perspectives

Contemporary review on the bifurcating autoregressive models : Overview and perspectives

Preliminary Identification of Branching-Heteroscedasticity for Tree-Indexed Autoregressive Processes

Asymptotics for a class of generalized multicast autoregressive processes

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