J-Shaped Distributions and Their Applications

 "In  many fields of research, such as, biology, computer science, control  theory, economics, engineering, genetics, hydrology, medicine, number  theory, statistics, physics, psychology, reliability, risk management,  etc., the shapes of probability distributions of non-normal data exhibit  J-shaped distributions. The shapes of such distributions may be skewed  to the left or the right depending on whether a large percentage of data  is at the lower or upper extreme. In this book, we have studied the  J-shaped distributions and their applications. As a motivation, we have  discussed several real-world examples which can be modeled through  J-shaped distribution. We have presented the mathematical formulation of  the family of J-shaped probability distributions which was first  proposed by Topp and Leone (1955). We also have discussed several  variations of Topp-Leone's family of J-shaped distribution. We have  considered the general form of J-shaped distribution and derived its  moments independently. We also have discussed other distributional  properties of the J-shaped distribution. Some distributional properties  of order statistics of the J-shaped distribution such as moment,  variance, product moments, and covariance are also provided. To describe  the shapes of the J-shaped distribution, the plots of the and for  various values of the parameter have been provided. Entropy provides an  excellent tool to quantify the amount of information (or uncertainty)  contained in a random observation regarding its parent distribution  (population). A large value of entropy implies greater uncertainty in  the data. As such, Shannon entropy of the J-shaped distribution is  provided. The distributional properties of order statistics of the  J-shaped distribution such as moment, variance, product moments, and  covariance, have also been presented. The numerical computations of  these for selected values of the parameters are provided. The  distributional properties of the record values of the J-shaped  distribution are also investigated. Some discussions on the sum, product  and ratio of the J-shaped distributions are provided. Characterizations  of the J-shaped distribution are given by using the method of truncated  moment, order statistics and record values"--