International Journal of Statistika and Mathematika, ISSN: 2277- 2790 E-ISSN: 2249-8605, Volume 3, Issue 2, 2012 pp 77-81 On Half-Cauchy Distribution and Process Elsamma Jacob1 , K Jayakumar2 1Malabar Christian College, Calicut, Kerala, INDIA. 2Department of Statistics, University of Calicut, Kerala, INDIA. Corresponding addresses:
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[email protected] Research Article Abstract: A new form of half- Cauchy distribution using sin ξ cos ξ Marshall-Olkin transformation is introduced. The properties of () = − ξ, () = ξ, ≥ 0 the new distribution such as density, cumulative distribution ξ ξ function, quantiles, measure of skewness and distribution of the extremes are obtained. Time series models with half-Cauchy Remark 1.1. For the HC distribution the moments do distribution as stationary marginal distributions are not not exist. developed so far. We develop first order autoregressive process Remark 1.2. The HC distribution is infinitely divisible with the new distribution as stationary marginal distribution and (Bondesson (1987)) and self decomposable (Diedhiou the properties of the process are studied. Application of the (1998)). distribution in various fields is also discussed. Keywords: Autoregressive Process, Geometric Extreme Stable, Relationship with other distributions: Half-Cauchy Distribution, Skewness, Stationarity, Quantiles. 1. Let Y be a folded t variable with pdf given by 1. Introduction: ( ) > 0, (1.5) The half Cauchy (HC) distribution is derived from 2Γ( ) () = 1 + , ν ∈ , the standard Cauchy distribution by folding the ν ()√νπ curve on the origin so that only positive values can be observed. A continuous random variable X is said When ν = 1 , (1.5) reduces to 2 1 > 0 to have the half Cauchy distribution if its survival () = , function is given by 1 + (x)=1 − tan , > 0 (1.1) Thus, HC distribution coincides with the folded t distribution with = 1 degree of freedom.