Statistics and Probability Letters 80 (2010) 1551–1558 Contents lists available at ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro On mixed AR(1) time series model with approximated beta marginal Božidar V. Popović a,∗ , Tibor K. Pogány b , Saralees Nadarajah c a Statistical Office of the Republic of Serbia, Milana Rakića 5, 11000 Belgrade, Serbia b Faculty of Maritime Studies, University of Rijeka, Croatia c School of Mathematics, University of Manchester, UK article info Article history: Received 3 November 2009 Received in revised form 9 June 2010 Accepted 10 June 2010 Available online 18 June 2010 MSC: 62M10 33C15 62F10 Keywords: Power law distribution Two parameter beta distribution Kumaraswamy distribution Kummer function of the first kind Wright function First order autoregressive model abstract We consider the mixed AR(1) time series model X t =  αX t −1 w.p. α p β X t −1 + ξ t w.p. 1 − α p , α,β ∈ (0, 1), when X t has the two parameter beta distribution B 2 (p, q), p ∈ (0, 1], q > 1. Special at- tention is given to the case p = 1 when the marginal distribution is approximated by the power law distribution closely connected with the two parameter Kumaraswamy distribu- tion Kum 2 (p, q), p ∈ (0, 1], q > 1. Using the Laplace transform technique, we prove that for p = 1 the distribution of the innovation process is uniform discrete. For p ∈ (0, 1), the innovation process has a continuous distribution. We also consider estimation issues of the model. © 2010 Elsevier B.V. All rights reserved. 1. Introduction In standard time series analysis one assumes that its marginal distribution is Gaussian. However, a Gaussian distribution will not always be appropriate. In earlier works stationary non-Gaussian time series models were developed for variables with positive and highly skewed distributions. There still remains situations where Gaussian marginals are inappropriate, i.e. where the marginal time-series variable being modeled, although not skewed or inherently positive valued, has a large kurtosis and long-tailed distributions. There are plenty of real situations that cannot be modeled by the Gaussian distribution like in hydrology, meteorology, information theory, economics, etc. Simple models with exponential marginals or mixed exponential marginals are considered in (Gaver and Lewis, 1980; Jevremović, 1990; Lawrence, 1980; Lawrence and Lewis, 1980; Mališić et al., 1987), while other marginals like gamma – (Gaver and Lewis, 1980; Novković, 1997; Sim, 1986), Laplace –(Novković, 1997), uniform – (Chernick, 1981; Ristić and Popović, 2002) and Weibull (Novković, 1997; Sim, 1986) have been discussed. Finally, we point out autoregressive processes PBAR and NBAR constructed in (McKenzie, 1985) for positively and negatively correlated pairs of beta random variables employing certain properties of the B 2 (p, q) distribution. In this paper, we introduce a mixed autoregressive first order time series model with the two parameter beta distribution B 2 (p, q), p ∈ (0, 1], q > 1, whose Laplace transform is approximated when the transformation argument is large. The resulting approximation determines a new distribution and results in a discrete uniform distribution for the innovation ∗ Corresponding author. Tel.: +381 64 33 98 663. E-mail addresses: bozidar.popovic@stat.gov.rs, bpopovic@stat.gov.rs (B.V. Popović). 0167-7152/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2010.06.009