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Sampling, Embedding and Inference for CARMA Processes
Author(s) -
Brockwell Peter J.,
Lindner Alexander
Publication year - 2019
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/jtsa.12433
Subject(s) - mathematics , estimator , combinatorics , sequence (biology) , embedding , stochastic differential equation , discrete mathematics , statistics , artificial intelligence , computer science , genetics , biology
A CARMA( p , q ) process Y is a strictly stationary solution Y of the pth ‐order formal stochastic differential equation a ( D ) Y t = b ( D ) DL t , where L is a two‐sided Lévy process, a ( z ) and b ( z ) are polynomials of degrees p and q respectively, with p > q , and D denotes differentiation with respect to t . Since estimation of the coefficients of a ( z ) and b ( z ) is frequently based on observations of the Δ‐sampled sequence Y Δ : = ( Y n Δ ) n ∈ Z , for some Δ > 0, it is crucial to understand the relation between Y and Y Δ . If E L 1 2 < ∞ then Y Δ is an ARMA sequence with coefficients depending on those of Y and the crucial problems for estimation are the determination of the coefficients of Y Δ from those of Y ( the sampling problem ) and the determination of the coefficients of Y from those of Y Δ ( the embedding problem ). In this article we consider both questions and use the results to determine the asymptotic distribution, as n → ∞ , with Δ fixed, ofn Δ ( β ^ − β ) , where β ^ is the quasi‐maximum‐likelihood estimator of the vector of coefficients of a ( z ) and b ( z ), based on n consecutive observations of Y Δ .