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On some properties of Kies distribution

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Abstract

In this paper we study some important aspects of the Kies distribution by deriving expressions for its percentile measures, raw moments, reliability measures etc. The maximum likelihood estimation of the parameters of the distribution have been discussed and the distribution has been fitted to certain real life data sets. The asymptotic behaviour of maximum likelihood estimators are also studied by using simulated data sets.

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Acknowledgments

The authors would like to express their gratitude to the Editor-in-Chief and both the anonymous referees for their valuable comments on an earlier version of the paper which greatly improved the quality and presentation of the paper.

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Authors and Affiliations

  1. Department of Statistics, University of Kerala, Trivandrum, 695 581, India

    C. Satheesh Kumar & S. H. S. Dharmaja

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  1. C. Satheesh Kumar
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  2. S. H. S. Dharmaja
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Correspondence to C. Satheesh Kumar.

Appendix

Appendix

The second order partial derivatives of the loglikelihood function \(\ell \) given in (58) with respect to the parameters a, b, \(\lambda \) and \(\beta \) are respectively

$$\begin{aligned}&\displaystyle \frac{{\mathop \partial \nolimits ^2 \ell }}{{\mathop {\partial a}\nolimits ^2 }}\, = \,\frac{{ - n}}{{\mathop {\left( {b - a} \right) }\nolimits ^2 }}\, - \left( {\beta - 1} \right) \,\sum \limits _{i = 1}^n {\frac{1}{{\mathop {\left( {\mathop x\nolimits _i - a} \right) }\nolimits ^2 }}} \, - \lambda \,\beta \,\left( {\beta - 1} \right) \sum \limits _{i = 1}^n {\frac{{\mathop {\left( {\mathop x\nolimits _i - a} \right) }\nolimits ^{\beta - 2} }}{{\mathop {\left( {\mathop {b - x}\nolimits _i } \right) }\nolimits ^\beta }}}, \end{aligned}$$
(62)
$$\begin{aligned}&\displaystyle \frac{{\mathop \partial \nolimits ^2 \ell }}{{\partial a\partial b}} = \frac{{\mathop \partial \nolimits ^2 \ell }}{{\partial b\partial a}}\, = \,\frac{n}{{\mathop {\left( {b - a} \right) }\nolimits ^2 }}\,\, - \lambda \,\mathop \beta \nolimits ^2 \sum \limits _{i = 1}^n {\frac{{\mathop {\left( {\mathop x\nolimits _i - a} \right) }\nolimits ^{\beta - 1} }}{{\mathop {\left( {\mathop {b - x}\nolimits _i } \right) }\nolimits ^{\beta + 1} }}}, \end{aligned}$$
(63)
$$\begin{aligned}&\displaystyle \frac{{\mathop \partial \nolimits ^2 \ell }}{{\partial a\partial \lambda }} = \frac{{\mathop \partial \nolimits ^2 \ell }}{{\partial \lambda \partial a}}\, = \beta \sum \limits _{i = 1}^n {\frac{{\mathop {\left( {\mathop x\nolimits _i - a} \right) }\nolimits ^{\beta - 1} }}{{\mathop {\left( {\mathop {b - x}\nolimits _i } \right) }\nolimits ^\beta }}}, \end{aligned}$$
(64)
$$\begin{aligned}&\displaystyle \frac{{\mathop \partial \nolimits ^2 \ell }}{{\partial a\partial \beta }} = \frac{{\mathop \partial \nolimits ^2 \ell }}{{\partial \beta \partial a}} \nonumber \\&\displaystyle = \!-\! \sum \limits _{i = 1}^n {\frac{1}{{\left( {\mathop x\nolimits _i - a} \right) }}} \,\, + \lambda \,\sum \limits _{i = 1}^n {\frac{{\mathop {\left( {\mathop x\nolimits _i - a} \right) }\nolimits ^{\beta - 1} }}{{\mathop {\left( {\mathop {b - x}\nolimits _i } \right) }\nolimits ^\beta }}} + \lambda \,\beta \sum \limits _{i = 1}^n {\frac{{\mathop {\left( {\mathop x\nolimits _i - a} \right) }\nolimits ^{\beta - 1} }}{{\mathop {\left( {\mathop {b - x}\nolimits _i } \right) }\nolimits ^{\beta } }}} \,\log \left( {\frac{{\mathop x\nolimits _i - a}}{{\mathop {b - x}\nolimits _i }}} \right) , \end{aligned}$$
(65)
$$\begin{aligned}&\displaystyle \frac{{\mathop \partial \nolimits ^2 \ell }}{{\mathop {\partial b}\nolimits ^2 }}\, = \,\frac{{ - n}}{{\mathop {\left( {b - a} \right) }\nolimits ^2 }}\, + \left( {\beta + 1} \right) \,\sum \limits _{i = 1}^n {\frac{1}{{\mathop {\left( {\mathop {b - x}\nolimits _i } \right) }\nolimits ^2 }}} \, - \lambda \,\beta \,\left( {\beta + 1} \right) \sum \limits _{i = 1}^n {\frac{{\mathop {\left( {\mathop x\nolimits _i - a} \right) }\nolimits ^{\beta - 1} }}{{\mathop {\left( {\mathop {b - x}\nolimits _i } \right) }\nolimits ^{\beta + 2} }}}, \end{aligned}$$
(66)
$$\begin{aligned}&\displaystyle \frac{{\mathop \partial \nolimits ^2 \ell }}{{\partial b\partial \lambda }} = \frac{{\mathop \partial \nolimits ^2 \ell }}{{\partial \lambda \partial b}}\, = \beta \sum \limits _{i = 1}^n {\frac{{\mathop {\left( {\mathop x\nolimits _i - a} \right) }\nolimits ^{\beta - 1} }}{{\mathop {\left( {\mathop {b - x}\nolimits _i } \right) }\nolimits ^{\beta +1} }}}, \end{aligned}$$
(67)
$$\begin{aligned}&\displaystyle \frac{{\mathop \partial \nolimits ^2 \ell }}{{\partial b\partial \beta }} = \frac{{\mathop \partial \nolimits ^2 \ell }}{{\partial \beta \partial b}} \nonumber \\ \,&\displaystyle = - \sum \limits _{i = 1}^n {\frac{1}{{\left( {\mathop {b - x}\nolimits _i } \right) }}} \,\, + \lambda \,\sum \limits _{i = 1}^n {\frac{{\mathop {\left( {\mathop x\nolimits _i - a} \right) }\nolimits ^{\beta - 1} }}{{\mathop {\left( {\mathop {b - x}\nolimits _i } \right) }\nolimits ^{\beta + 1} }}} + \lambda \,\beta \sum \limits _{i = 1}^n {\frac{{\mathop {\left( {\mathop x\nolimits _i - a} \right) }\nolimits ^{\beta - 1} }}{{\mathop {\left( {\mathop {b - x}\nolimits _i } \right) }\nolimits ^{\beta + 1} }}} \,\log \left( {\frac{{\mathop x\nolimits _i - a}}{{\mathop {b - x}\nolimits _i }}} \right) , \nonumber \\ \end{aligned}$$
(68)
$$\begin{aligned}&\displaystyle \frac{{\mathop \partial \nolimits ^2 \ell }}{{\mathop {\partial \lambda }\nolimits ^2 }} = - \frac{n}{{\mathop \lambda \nolimits ^2 }}, \end{aligned}$$
(69)
$$\begin{aligned}&\displaystyle \frac{{\mathop \partial \nolimits ^2 \ell }}{{\partial \lambda \partial \beta }} = \frac{{\mathop \partial \nolimits ^2 \ell }}{{\partial \beta \partial \lambda }}\, = - \sum \limits _{i = 1}^n {\frac{{\mathop {\left( {\mathop x\nolimits _i - a} \right) }\nolimits ^\beta }}{{\mathop {\left( {\mathop {b - x}\nolimits _i } \right) }\nolimits ^\beta }}} \,\log \left( {\frac{{\mathop x\nolimits _i - a}}{{\mathop {b - x}\nolimits _i }}} \right) \end{aligned}$$
(70)

and

$$\begin{aligned} \frac{{\mathop \partial \nolimits ^2 \ell }}{{\mathop {\partial \beta }\nolimits ^2 }} = - \frac{n}{{\mathop \beta \nolimits ^2 }} - \lambda \sum \limits _{i = 1}^n {\frac{{\mathop {\left( {\mathop x\nolimits _i - a} \right) }\nolimits ^\beta }}{{\mathop {\left( {\mathop {b - x}\nolimits _i } \right) }\nolimits ^\beta }}} \,\mathop {\left( {\log \left( {\frac{{\mathop x\nolimits _i - a}}{{\mathop {b - x}\nolimits _i }}} \right) } \right) }\nolimits ^2 . \end{aligned}$$
(71)

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Satheesh Kumar, C., Dharmaja, S.H.S. On some properties of Kies distribution. METRON 72, 97–122 (2014). https://doi.org/10.1007/s40300-013-0018-8

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