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On the performance of a new bivariate pseudo Pareto distribution with application to drought data

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Abstract

A new bivariate pseudo Pareto distribution is proposed, and its distributional characteristics are investigated. The parameters of this distribution are estimated by the moment-, the maximum likelihood- and the Bayesian method. Point estimators of the parameters are presented for different sample sizes. Asymptotic confidence intervals are constructed and the parameter modeling the dependency between two variables is checked. The performance of the different estimation methods is investigated by using the bootstrap method. A Markov Chain Monte Carlo simulation is conducted to estimate the Bayesian posterior distribution for different sample sizes. For illustrative purposes, a real set of drought data is investigated.

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Acknowledgments

The authors are thankful to the Associate editor and the two referees for their valuable comments and suggestions which greatly helped to improve the paper. The first author is also thankful to the Higher Education Commission of Pakistan for their financial support for this project.

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Authors and Affiliations

  1. Department of Statistics, Alpen-Adria University, Klagenfurt, 9020, Austria

    Muhammad Mohsin, Gunter Spöck & Jürgen Pilz

Authors
  1. Muhammad Mohsin
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  2. Gunter Spöck
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  3. Jürgen Pilz
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Corresponding author

Correspondence to Muhammad Mohsin.

Appendix

Appendix

The Fisher information matrix \( Q\left( {\hat{g}} \right) \)is given by:

$$ Q\left( {\hat{g}} \right) = - \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} L\left( g \right)}}{{\partial \alpha^{2} }}} & {\frac{{\partial^{2} L\left( g \right)}}{\partial \alpha \partial \beta }} & 0 & 0 & 0 \\ {\frac{{\partial^{2} L\left( g \right)}}{\partial \beta \partial \alpha }} & {\frac{{\partial^{2} L\left( g \right)}}{{\partial \beta^{2} }}} & 0 & 0 & 0 \\ 0 & 0 & {\frac{{\partial^{2} L\left( g \right)}}{{\partial \gamma^{2} }}} & {\frac{{\partial^{2} L\left( g \right)}}{\partial \gamma \partial \delta }} & {\frac{{\partial^{2} L\left( g \right)}}{\partial \gamma \partial \eta }} \\ 0 & 0 & {\frac{{\partial^{2} L\left( g \right)}}{\partial \delta \partial \gamma }} & {\frac{{\partial^{2} L\left( g \right)}}{{\partial \delta^{2} }}} & {\frac{{\partial^{2} L\left( g \right)}}{\partial \delta \partial \eta }} \\ 0 & 0 & {\frac{{\partial^{2} L\left( g \right)}}{\partial \eta \partial \gamma }} & {\frac{{\partial^{2} L\left( g \right)}}{\partial \eta \partial \delta }} & {\frac{{\partial^{2} L\left( g \right)}}{{\partial \eta^{2} }}} \\ \end{array} } \right] $$

The corresponding entries are

$$ \frac{{\partial^{2} L\left( g \right)}}{{\partial \alpha^{2} }} = - \frac{n\beta }{{\alpha^{2} }} + \left( {\beta + 1} \right)\sum\limits_{i = 1}^{n} {\frac{1}{{\left( {\alpha + x_{i} } \right)^{2} }}} $$
$$ \frac{{\partial^{2} L\left( g \right)}}{\partial \alpha \partial \beta } = \frac{n}{\alpha } - \sum\limits_{i = 1}^{n} {\frac{1}{{\left( {\alpha + x_{i} } \right)}}} $$
$$ \frac{{\partial^{2} L\left( g \right)}}{{\partial \beta^{2} }} = - \frac{n}{{\beta^{2} }} $$
$$ \frac{{\partial^{2} L\left( g \right)}}{{\partial \gamma^{2} }} = - \frac{n}{{\gamma^{2} }} $$
$$ \frac{{\partial^{2} L\left( g \right)}}{\partial \gamma \partial \delta } = \sum\limits_{i = 1}^{n} {{ \ln }\left( {x_{i} } \right)} - \sum\limits_{i = 1}^{n} {\frac{{\eta x_{i}^{\delta } { \ln }\left( {x_{i} } \right)}}{{\left( {\eta x_{i}^{\delta } + y_{i} } \right)}}} $$
$$ \frac{{\partial^{2} L\left( g \right)}}{\partial \gamma\partial \eta } = \frac{n}{\eta} - \sum\limits_{i = 1}^{n} {\frac{{x_{i}^{\delta }}}{{\left( {\eta x_{i}^{\delta } + y_{i} } \right)}}} $$
$$ \frac{{\partial^{2} L\left( g \right)}}{\partial {\delta^{2}} } = - \left( {\gamma + 1} \right)\sum\limits_{i =1}^{n} {\frac {\eta {x_{i}^{\delta }{y_{i} {\left( { \ln }\left({x_{i} } \right)\right)^2}}}}{{\left( {\eta x_{i}^{\delta } +y_{i} } \right)^{2} }}} $$
$$ \frac{{\partial^{2} L\left( g \right)}}{\partial \delta \partial \eta } = - \left( {\gamma + 1} \right)\sum\limits_{i = 1}^{n} {\frac{{y_{i} x_{i}^{\delta } { \ln }\left( {x_{i} } \right)}}{{\left( {\eta x_{i}^{\delta } + y_{i} } \right)^{2} }}} $$
$$ \frac{{\partial^{2} L\left( g \right)}}{{\partial \eta^{2} }} = - \frac{n\gamma }{{\eta^{2} }} + \left( {\gamma + 1} \right)\sum\limits_{i = 1}^{n} {\frac{{x_{i}^{2\delta } }}{{\left( {\eta x_{i}^{\delta } + y_{i} } \right)^{2} }}} $$

The estimated standard deviations of \( \hat{\alpha },\,\hat{\beta },\,\hat{\gamma },\,\hat{\delta } \) and \( \hat{\eta } \) are:

$$ \sigma \left( {\hat{\alpha }} \right) = \hat{\alpha }\sqrt {\frac{n}{{\left[ {n\left\{ {n\hat{\beta } - \hat{\alpha }^{2} \left( {\hat{\beta } + 1} \right)\sum\nolimits_{i = 1}^{n} {\frac{1}{{\left( {\hat{\alpha } + x_{i} } \right)^{2} }}} } \right\} - \left\{ {\hat{\alpha }^{2} \hat{\beta }^{2} \left( {\sum\nolimits_{i = 1}^{n} {\frac{1}{{\left( {\hat{\alpha } + x_{i} } \right)}} - \frac{n}{{\hat{\alpha }}}} } \right)^{2} } \right\}} \right]}}} $$
$$ \sigma \left( {\hat{\beta }} \right) = \hat{\beta }\sqrt {\frac{{n\hat{\beta } - \hat{\alpha }^{2} \left( {\hat{\beta } + 1} \right)\sum\nolimits_{i = 1}^{n} {\frac{1}{{\left( {\hat{\alpha } + x_{i} } \right)^{2} }}} }}{{\left[ {n\left\{ {n\hat{\beta } - \hat{\alpha }^{2} \left( {\hat{\beta } + 1} \right)\sum\nolimits_{i = 1}^{n} {\frac{1}{{\left( {\hat{\alpha } + x_{i} } \right)^{2} }}} } \right\} - \left\{ {\hat{\alpha }^{2} \hat{\beta }^{2} \left( {\sum\nolimits_{i = 1}^{n} {\frac{1}{{\left( {\hat{\alpha } + x_{i} } \right)}} - \frac{n}{{\hat{\alpha }}}} } \right)^{2} } \right\}} \right]}}} $$
$$ \sigma \left( {\hat{\gamma }} \right) = \sqrt {\frac{{\left( {BE - C^{2} } \right)}}{{\left[ {A\left( {CD - AB} \right) - C\left( {\frac{n}{{\hat{\gamma }^{2} }}C - AD} \right) + E\left( {\frac{n}{{\hat{\gamma }^{2} }}B - D^{2} } \right)} \right]}}} $$
$$ \sigma \left( {\hat{\delta }} \right) = \sqrt {\frac{{\left( {\frac{n}{{\hat{\gamma }^{2} }}E - A^{2} } \right)}}{{\left[ {A\left( {CD - AB} \right) - C\left( {\frac{n}{{\hat{\gamma }^{2} }}C - AD} \right) + E\left( {\frac{n}{{\hat{\gamma }^{2} }}B - D^{2} } \right)} \right]}}} $$
$$ \sigma \left( {\hat{\eta }} \right) = \sqrt {\frac{{\left( {\frac{n}{{\hat{\gamma }^{2} }}B - D^{2} } \right)}}{{\left[ {A\left( {CD - AB} \right) - C\left( {\frac{n}{{\hat{\gamma }^{2} }}C - AD} \right) + E\left( {\frac{n}{{\hat{\gamma }^{2} }}B - D^{2} } \right)} \right]}}} $$

where

$$ A = \sum\limits_{i = 1}^{n} {\frac{{x_{i}^{{\hat{\delta }}} }}{{\left( {\hat{\eta }x_{i}^{{\hat{\delta }}} + y_{i} } \right)}}} - \frac{n}{{\hat{\eta }}} $$
$$ B = \left( {\hat{\gamma } + 1} \right)\sum\limits_{i = 1}^{n} {\frac{{\hat{\eta }x_{i}^{{\hat{\delta }}} y_{i} \left( {{ \ln }\left( {x_{i} } \right)} \right)^{2} }}{{\left( {\hat{\eta }x_{i}^{{\hat{\delta }}} + y_{i} } \right)^{2} }}} $$
$$ C = \left( {\hat{\gamma } + 1} \right)\sum\limits_{i = 1}^{n} {\frac{{x_{i}^{{\hat{\delta }}} y_{i} { \ln }\left( {x_{i} } \right)}}{{\left( {\hat{\eta }x_{i}^{{\hat{\delta }}} + y_{i} } \right)^{2} }}} $$
$$ D = \sum\limits_{i = 1}^{n} {\frac{{\hat{\eta }x_{i}^{{\hat{\delta }}} { \ln }\left( {x_{i} } \right)}}{{\left( {\hat{\eta }x_{i}^{{\hat{\delta }}} + y_{i} } \right)}} - \sum\limits_{i = 1}^{n} {{ \ln }\left( {x_{i} } \right)} } $$
$$ E = \frac{{n\hat{\gamma }}}{{\hat{\eta }^{2} }} - \left( {\hat{\gamma } + 1} \right)\sum\limits_{i = 1}^{n} {\frac{{x_{i}^{{2\hat{\delta }}} }}{{\left( {\hat{\eta }x_{i}^{{\hat{\delta }}} + y_{i} } \right)^{2} }}} $$

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Mohsin, M., Spöck, G. & Pilz, J. On the performance of a new bivariate pseudo Pareto distribution with application to drought data. Stoch Environ Res Risk Assess 26, 925–945 (2012). https://doi.org/10.1007/s00477-011-0529-x

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