# J-Shaped Distributions and Their Applications

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Mohammad Ahsanullah, PhDÂ – Professor Emeritus, Rider University, Lawrenceville, New Jersey, USA
Mohammad Shakil – Department of Mathematics/Statistics, Miami Dade College, Miami, Florida, USA

Series: Mathematics Research Developments; Life Sciences Research and Development
BISAC: SCI086000
DOI: https://doi.org/10.52305/KILR7482

#### Target Audience

We hope the book will be useful for statistics students and applied statisticians for model building, data analysis and other applications.

#### Description

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 Toppâ€“Leoneâ€™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.

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Preface

Acknowledgments

Chapter 1. Introduction

1. Introduction
1.1. Some Real-Life Examples
1.2. Some More Real-Life Examples
1.3. Literature Review
1.4. Remark
1.5. Conclusion

Chapter 2. Mathematical Formulations
2. Introduction
2.1. Topp-Leoneâ€™s J-Shaped Distribution
2.2. General Form of J-Shaped Distribution
2.3. A Solution of the Generalized Pearson Differential Equation
2.4. Some Variations of Toppâ€“Leoneâ€™s J-Shaped Distribution
2.5. Failure Rate Function
2.6. Conclusion

Chapter 3. Distributional Properties
3. Introduction
3.1. Moments
3.2. Integer Order Moments in Terms of Special Functions
3.3. Explicit Expressions for the Integer Order Moments in Terms of Gamma Functions
3.4. Shannon Entropy
3.5. Computations of Shannon Entropy
3.6. Shapes of Shannon Entropy
3.7. Conclusion

Chapter 4. Order Statistics
4. Introduction
4.1. Order Statistics
4.2. Distributional Properties
4.3. First and Second Moments
4.4. Product Moments
4.5. Computations of Mean, Variance and Covariance
4.6. Conclusion

Chapter 5. Record Values
5. Introduction
5.1. Basic Ideas, Definitions and Notations of Record Values
5.2. Distributions of Record Values
5.3. Probability Density and Cumulative Distribution FunctionsÂ  of Upper Record Values
5.4. Probability Density and Cumulative Distribution FunctionsÂ  of Lower Record Values
5.5. Probability Density Functions of Joint and ConditionalÂ  Record Values
5.6. Moments of Record Values
5.7. Percentage Points
5.8. Conclusion

Chapter 6. Sum, Product and Ratio
6. Introduction
6.1. Literature Review
6.2. Distributions of and |X + Y| , |XY| and ||When X and Y Belong to Different Families
6.3. Some Preliminaries on Distributions of the Sum , Product , and Ratio
6.4. The Distribution of the Sum, Product and Ratio of IndependentÂ  J-Shaped Random Variables
6.5. Conclusion

Chapter 7. Characterizations
7. Introduction
7.1. A Solution of the Generalized Pearson Differential Equation
7.2. Characterizations
7.3. Conclusion

Chapter 8. Percentile Points
8. Introduction
8.1. Percentile of Order p
8.2. Percentile Points
8.3. Conclusion

Chapter 9. Conclusion
9. Introduction
9.1. Conclusion

Appendix
A. Useful Formulas and Results
A.1. Gamma Function
A.2. Digamma Function
A.3. Error Function
A.4. Some Useful Properties of Gamma Function