A new non-linear AR(1) time series model having approximate beta marginals |
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Authors: | Božidar V. Popović Saralees Nadarajah Miroslav M. Ristić |
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Affiliation: | 1. University of Donja Gorica, Podgorica, Montenegro 2. School of Mathematics, University of Manchester, Manchester, M13 9PL, UK 3. Department of Mathematics and Informatics, Faculty of Sciences and Mathematics, University of Ni?, Ni?, Serbia
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Abstract: | We consider the mixed AR(1) time series model $$X_t=left{begin{array}{ll}alpha X_{t-1}+ xi_t quad {rm w.p.} qquad frac{alpha^p}{alpha^p-beta ^p}, beta X_{t-1} + xi_{t} quad {rm w.p.} quad -frac{beta^p}{alpha^p-beta ^p} end{array}right.$$ for ?1 < β p ≤ 0 ≤ α p < 1 and α p ? β p > 0 when X t has the two-parameter beta distribution B2(p, q) with parameters q > 1 and ${p in mathcal P(u,v)}$ , where $$mathcal P(u,v) = left{u/v : u < v,,u,v,{rm odd,positive,integers} right}.$$ Special attention is given to the case p = 1. Using Laplace transform and suitable approximation procedures, we prove that the distribution of innovation sequence for p = 1 can be approximated by the uniform discrete distribution and that for ${p in mathcal P(u,v)}$ can be approximated by a continuous distribution. We also consider estimation issues of the model. |
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