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Exponentiated Generalized Kumaraswamy Distribution with Applications

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Abstract

In this article, we introduced and studied exponentiated generalized Kumaraswamy distribution. We derived mathematical properties including quantile function, moment generating function, ordinary moments, probability weighted moments, incomplete moments, and Rényi entropy. The expressions of order statistics are also derived. Here we discuss the parameter estimation by using the method of maximum likelihood. We showed resilience of the introduced distribution over existing some well-known distributions by using real dataset applications.

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Author information

Authors and Affiliations

  1. Vice Presidency for Graduate Studies and Scientific Research, Jeddah University, Jeddah, Saudi Arabia

    M. Elgarhy

  2. Institute of Statistical Studies and Research (ISSR), Cairo University, Giza, Egypt

    M. Elgarhy

  3. College of Statistical and Actuarial Sciences, University of the Punjab, Lahore, Pakistan

    Muhammad Ahsan ul Haq

  4. Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan

    Qurat ul Ain

Authors
  1. M. Elgarhy
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  2. Muhammad Ahsan ul Haq
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  3. Qurat ul Ain
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Corresponding author

Correspondence to Muhammad Ahsan ul Haq.

Appendix

Appendix

The elements of the observed Fisher information matrix \(I^{-1}(\Theta )\) are given by

$$\begin{aligned} U_{aa}= & {} -\,\frac{n}{a^{2}}+-\left( {b\alpha -1} \right) \sum _{i=1}^n {\left\{ {\frac{x_i ^{a}\left( {\ln x_i } \right) ^{2}}{\left[ {1-x_i ^{a}} \right] }+\left[ {\frac{x_i ^{a}\left( {\ln x_i } \right) }{\left[ {1-x_i ^{a}} \right] }} \right] ^{2}} \right\} } \\&\quad +\,b\alpha \left( {\beta -1} \right) \sum _{i=1}^n {\left\{ {\frac{-\left( {b\alpha -1} \right) \left( {\ln x_i } \right) ^{2}\left[ {1-x_i ^{a}} \right] ^{b\alpha -2}x_i ^{a}}{\left[ {1-\left[ {1-x_i ^{a}} \right] ^{b\alpha }} \right] }} \right. } \\&\quad \left. +\,\frac{\left( {\ln x_i } \right) ^{2}\left[ {1-x_i ^{a}} \right] ^{b\alpha -1}x_i ^{a}}{\left[ {1-\left[ {1-x_i ^{a}} \right] ^{b\alpha }} \right] } {-\,b\alpha \left[ {\frac{\ln x_i \left[ {1-x_i ^{a}} \right] ^{b\alpha -1}x_i ^{a}}{\left[ {1-\left[ {1-x_i ^{a}} \right] ^{b\alpha }} \right] }} \right] ^{2}} \right\} \end{aligned}$$
$$\begin{aligned} U_{ab}= & {} -\,\alpha \sum _{i=1}^n {\frac{x_i ^{a}\left( {\ln x_i } \right) }{1-x_i ^{a}}} +\alpha \left( {\beta -1} \right) \sum _{i=1}^n {\frac{\left[ {1-x_i ^{a}} \right] ^{b\alpha -1}x_i ^{a}\ln x_i }{1-\left[ {1-x_i ^{a}} \right] ^{b\alpha }}} \\&+\,b\alpha ^{2}\left( {\beta -1} \right) \sum _{i=1}^n {\left\{ {\frac{\left[ {1-x_i ^{a}} \right] ^{b\alpha -1}\ln \left( {1-x_i ^{a}} \right) x_i ^{a}\ln x_i }{1-\left[ {1-x_i ^{a}} \right] ^{b\alpha }}} \right. } \\&\left. {+\,\frac{\left[ {1-x_i ^{a}} \right] ^{2b\alpha -1}\ln \left( {1-x_i ^{a}} \right) x_i ^{a}\ln x_i }{\left[ {1-\left[ {1-x_i ^{a}} \right] ^{b\alpha }} \right] ^{2}}} \right\} \\ U_{a\alpha }= & {} -b\sum _{i=1}^n {\frac{x_i ^{a}\left( {\ln x_i } \right) }{1-x_i ^{a}}} +b\left( {\beta -1} \right) \sum _{i=1}^n {\frac{\left[ {1-x_i ^{a}} \right] ^{b\alpha -1}x_i ^{a}\ln x_i }{1-\left[ {1-x_i ^{a}} \right] ^{b\alpha }}} \\&+\,b^{2}\alpha \left( {\beta -1} \right) \sum _{i=1}^n {\left\{ {\frac{\left[ {1-x_i ^{a}} \right] ^{b\alpha -1}\ln \left( {1-x_i ^{a}} \right) x_i ^{a}\ln x_i }{1-\left[ {1-x_i ^{a}} \right] ^{b\alpha }}} \right. }\\&\left. {+\,\frac{\left[ {1-x_i ^{a}} \right] ^{2b\alpha -1}\ln \left( {1-x_i ^{a}} \right) x_i ^{a}\ln x_i }{\left[ {1-\left[ {1-x_i ^{a}} \right] ^{b\alpha }} \right] ^{2}}} \right\} \\ U_{a\beta }= & {} b\alpha \sum _{i=1}^n {\frac{\left( {\ln x_i } \right) \left[ {1-x_i ^{a}} \right] ^{b\alpha -1}x_i ^{a}}{\left[ {1-\left[ {1-x_i ^{a}} \right] ^{b\alpha }} \right] }}\\ \end{aligned}$$
$$\begin{aligned} U_{bb}= & {} -\,\frac{n}{b^{2}}-\alpha ^{2}\left( {\beta -1} \right) \sum _{i=1}^n \left\{ \frac{\left( {\ln \left( {1-x_i ^{a}} \right) } \right) ^{2}\left[ {1-x_i ^{a}} \right] ^{b\alpha }}{\left[ {1-\left[ {1-x_i ^{a}} \right] ^{b\alpha }} \right] }\right. \\&\left. +\left[ {\frac{\ln \left( {1-x_i ^{a}} \right) \left[ {1-x_i ^{a}} \right] ^{b\alpha }}{\left[ {1-\left[ {1-x_i ^{a}} \right] ^{b\alpha }} \right] }} \right] ^{2} \right\} \\ U_{b\alpha }= & {} \sum _{i=1}^n {\ln \left( {1-x_i ^{a}} \right) } -\left( {\beta -1} \right) \sum _{i=1}^n {\left\{ {\frac{\ln \left( {1-x_i ^{a}} \right) \left[ {1-x_i ^{a}} \right] ^{b\alpha }}{\left[ {1-\left[ {1-x_i ^{a}} \right] ^{b\alpha }} \right] }} \right\} } \\&-\alpha b\left( {\beta -1} \right) \sum _{i=1}^n {\left\{ {\frac{\left( {\ln \left( {1-x_i ^{a}} \right) } \right) ^{2}\left[ {1-x_i ^{a}} \right] ^{b\alpha }}{\left[ {1-\left[ {1-x_i ^{a}} \right] ^{b\alpha }} \right] }} \right. } \left. {+\left[ {\frac{\ln \left( {1-x_i ^{a}} \right) \left[ {1-x_i ^{a}} \right] ^{b\alpha }}{\left[ {1-\left[ {1-x_i ^{a}} \right] ^{b\alpha }} \right] }} \right] ^{2}} \right\} \\ U_{b\beta }= & {} -\alpha \sum _{i=1}^n {\frac{\left[ {1-x_i ^{a}} \right] ^{b\alpha }\ln \left( {1-x_i ^{a}} \right) }{\left[ {1-\left[ {1-x_i ^{a}} \right] ^{b\alpha }} \right] }}\\ \end{aligned}$$
$$\begin{aligned} U_{\alpha \alpha }= & {} -\,\frac{n}{\alpha ^{2}}-b^{2}\left( {\beta -1} \right) \sum _{i=1}^n {\left\{ {\frac{\left( {\ln \left( {1-x_i ^{a}} \right) } \right) ^{2}\left[ {1-x_i ^{a}} \right] ^{b\alpha }}{\left[ {1-\left[ {1-x_i ^{a}} \right] ^{b\alpha }} \right] }} \right. } \\&\quad \left. {+\,\left[ {\frac{\ln \left( {1-x_i ^{a}} \right) \left[ {1-x_i ^{a}} \right] ^{b\alpha }}{\left[ {1-\left[ {1-x_i ^{a}} \right] ^{b\alpha }} \right] }} \right] ^{2}} \right\} \\ U_{\alpha \beta }= & {} -b\sum _{i=1}^n {\frac{\left[ {1-x_i ^{a}} \right] ^{b\alpha }\ln \left( {1-x_i ^{a}} \right) }{\left[ {1-\left[ {1-x_i ^{a}} \right] ^{b\alpha }} \right] }} \\ U_{\beta \beta }= & {} \frac{-n}{\beta ^{2}} \end{aligned}$$

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Elgarhy, M., Haq, M.A.u. & ul Ain, Q. Exponentiated Generalized Kumaraswamy Distribution with Applications. Ann. Data. Sci. 5, 273–292 (2018). https://doi.org/10.1007/s40745-017-0128-x

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