J-Shaped Distributions and Their Applications

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In this book, we have studied the J-shaped distributions and their applications. As a motivation, we have discussed several real-world examples that can be modeled through a J-shaped distribution.


Made In: United States (US)

Supplier: Nova Science Publishers SKU: ISBN: 979-8-89113-013-5 Categories: , Tag: More From:

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.

We have presented the mathematical formulation of the family of J-shaped probability distributions 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 J-shaped distributions’ sum, product, and ratio are provided. Characterizations of the J-shaped distribution are given using the truncated moment method, order statistics, and record values.

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