Statistics and Probability Letters 82 (2012) 77–83 Contents lists available at SciVerse ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro Wrapped weighted exponential distributions Shongkour Roy ∗ , Mian Arif Shams Adnan Department of Statistics, Jahangirnagar University, Savar, Dhaka 1342, Bangladesh article info Article history: Received 4 February 2011 Received in revised form 27 August 2011 Accepted 29 August 2011 Available online 7 September 2011 Keywords: Directional data Weighted exponential distribution Wrapped distribution Sea star movements Circular data abstract We have developed a new class of circular distributions named wrapped weighted exponential distributions. The estimation of unknown parameters along with some characteristics of these distributions is also investigated. Some theorems that relate the distribution to some other circular distributions are established and we clarify their modeling potential using a classical data set on movements of sea stars. © 2011 Elsevier B.V. All rights reserved. 1. Introduction We introduce a class of non-symmetric circular distributions by wrapping an asymmetric weighted exponential distribution around the circumference of a unit circle. The wrapped distribution was introduced by Lévy (1939). Since then, a massive amount of work has been done to introduce a different trigonometric moment with a related parameter to a wrapped distribution. For example, Carlos and Coelho (2007) explored the wrapped gamma distribution and showed how this distribution can be mixed with other distributions for modeling directional data in biology and meteorology, the wrapped three-parameter gamma distribution and wrapped Weibull distribution have been defined with several properties (see Roy and Adnan (2010a,b), Rao et al. (2007)), and Rao et al. (2007) provided a nice modification of a wrapped model with various characteristics. Circular distributions play an important role in modeling directional data which arise in various fields. See, e.g., the references cited above, among others. While the wrapped Cauchy, normal, and stable distributions, and wrapped exponential and wrapped double-exponential (or Laplace) distributions have been studied extensively (see, e.g., Lévy (1939), Gatto and Jammalamadaka (in press) and Jammalamadaka and Kozubowski (2003, 2004)), relatively little has been done for the case of wrapped weighted exponential distributions, which produce equally interesting circular models. For the real line, Gupta and Kundu (2009) were the first to introduce the weighted exponential distribution and its different properties; however, the wrapped weighted exponential distributions have not been considered very much in previous work, although these distributions may be even better for modeling directional data. It is observed that the shape of the pdf of a wrapped weighted exponential is very similar to those of other generalizations of the wrapped exponential distribution, for example wrapped gamma, wrapped Weibull, etc. In this paper, we first obtain the probability density function along with the basic properties of the wrapped weighted exponential distribution in a much manageable form in Section 2. Section 3 makes explicit the characteristic function and the trigonometric moment, with related parameters. In Section 4, we provide parameter estimation and in Section 5, we briefly discuss an application in biology illustrating the modeling potential of wrapped weighted exponential laws. Finally, the conclusion appears in Section 6. ∗ Corresponding author. Tel.: +880 1674 749923. E-mail address: sankar1604@gmail.com (S. Roy). 0167-7152/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2011.08.023