Exploration of some distributional properties, parameter estimates and applications of the wrapped quasi Lindley distribution

Abstract

A circular distribution called Wrapped Quasi Lindley distribution with two parameters has been recently proposed, but apart from the expressions for pdf, cdf, their circular representations, characteristic function and the maximum likelihood equations for the proposed distribution, no other properties of the distribution as well as the characteristics of the parameter estimates were explored by the authors. A slight error has also been observed in the expression for pdf of the distribution. Also, the application of the distribution in modeling real life data was not exhibited. Further, the form of the characteristic function in the paper is not compact and there is no closed form of the expression of the trigonometric moments. This paper thus aims to rectify the expression for pdf and explore a few descriptive measures and distributional properties of the Wrapped Quasi Lindley distribution and derive closed form expressions for the characteristic function and hence the trigonometric moments using an identity. it is found that the operations of wrapping and convoluting linear distributions around unit circle are commutative. The maximum likelihood estimates of the parameters of the distribution are shown to be consistent through a simulation study. The utility of the Wrapped Quasi Lindley model to a real-life data set on orientations is shown and the goodness-of-fit of the distribution is assessed and compared to that of the Wrapped Exponential and Wrapped Lindley distribution with the help of the log-likelihood, AIC and BIC measures. Further, the probabilities of the orientations to lie in a certain interval are estimated on the basis of the fitted Wrapped Quasi Lindley distribution. The distribution is found to be more appropriate in modeling the situations where the directions having lower magnitude have higher likelihood of occurrence.