A new quantile regression model with application to human development index

Abstract

A new odd log-logistic unit omega distribution is defined and studied, and some of its structural properties are obtained. A quantile regression model based on the new re-parameterized distribution is constructed, and the estimation is conducted by the maximum likelihood method. Monte Carlo simulations are used to assess the accuracy of the estimators. The flexibility, practical relevance and applicability of the proposed regression are proved by means of Human Development Index data from the cities of the state of São Paulo (Brazil).

Appendices

Appendix: Simulation plots

See Figs. 11 and 12.

Fig. 11

MSEs versus n for some $\tau$ values

Fig. 12

Biases versus n for some $\tau$ values

Appendix: Score equations

$$\begin{aligned} \frac{\partial l({\varvec{\theta }})}{\partial \beta }= & {} \frac{n}{\beta } + \sum _{i=1}^{n} \log (x_i) -\alpha (\nu -1) \sum _{i=1}^{n} \Bigg \{\Bigg [1- \Bigg (\frac{ 1+x_i^{\beta }}{1-x_i^{\beta }}\Bigg )^{-\alpha } \Bigg ]^{-1} \Bigg (\frac{ 1+x_i^{\beta }}{1-x_i^{\beta }}\Bigg )^{-\alpha -1} \Bigg [\frac{2 x_{i}^{\beta }\log (x_i) }{(1-x_{i}^{\beta })^{2}} \Bigg ] \Bigg \} \\{} & {} -\alpha (\nu -1) \sum _{i=1}^{n} \Bigg \{ \Bigg (\frac{ 1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg )^{-1} \Bigg [\frac{2 x_{i}^{\beta }\log (x_i) }{(1-x_{i}^{\beta })^{2}} \Bigg ] \Bigg \}+ \sum _{i=1}^{n} \frac{2x_{i}^{2\beta } \log (x_i)}{1-x_{i}^{2\beta }}\\{} & {} -2 \sum _{i=1}^{n} \Bigg [ \Bigg \{\Bigg [ 1- \Bigg ( \frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg )^{-\alpha } \Bigg ]^{\nu } + \Bigg (\frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg )^{-\alpha \,\nu }\Bigg \}^{ -1} \\{} & {} \times \Bigg \{ \nu \alpha \Bigg [1-\Bigg ( \frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg )^{-\alpha } \Bigg ]^{\nu -1} \Bigg ( \frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg )^{-\alpha -1} \Bigg [\frac{2 x_{i}^{\beta }\log (x_i) }{(1-x_{i}^{\beta })^{2}} \Bigg ] \\{} & {} -\alpha \nu \Bigg ( \frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg )^{-\alpha \nu -1} \Bigg [\frac{2 x_{i}^{\beta }\log (x_i) }{(1-x_{i}^{\beta })^{2}} \Bigg ] \Bigg \} \Bigg ].\\ \frac{\partial l({\varvec{\theta }})}{\partial \alpha }= & {} \frac{n}{\alpha } +(\nu -1)\sum _{i=1}^{n} \Bigg \{ \Bigg [ 1- \Bigg ( \frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg )^{-\alpha } \Bigg ]^{-1} \Bigg ( \frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg )^{-\alpha } \log \Bigg ( \frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg ) \Bigg \} \\{} & {} - (\nu -1) \sum _{i=1}^{n} \log \Bigg ( \frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg )-2 \sum _{i=1}^{n} \Bigg [ \Bigg \{\Bigg [ 1- \Bigg ( \frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg )^{-\alpha } \Bigg ]^{\nu } + \Bigg (\frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg )^{-\alpha \,\nu } \Bigg \}^{ -1} \\{} & {} \times \Bigg \{ \nu \Bigg [ 1- \Bigg ( \frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg )^{-\alpha } \Bigg ]^{\nu -1} \Bigg ( \frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg )^{-\alpha } \log \Bigg ( \frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg ) - \Bigg ( \frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg )^{-\alpha \nu } \log \Bigg [ \Bigg ( \frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg )^{\nu } \Bigg ] \Bigg \} \Bigg ].\\ \frac{\partial l({\varvec{\theta }})}{\partial \nu }= & {} \frac{n}{\nu } +\sum _{i=1}^{n} \log \Bigg [ 1- \Bigg ( \frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg )^{-\alpha } \Bigg ] -\alpha \sum _{i=1}^{n} \log \Bigg ( \frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg ) \\{} & {} -2 \sum _{i=1}^{n} \Bigg [ \Bigg \{\Bigg [ 1- \Bigg ( \frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg )^{-\alpha } \Bigg ]^{\nu } + \Bigg (\frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg )^{-\alpha \,\nu } \Bigg \}^{ -1} \Bigg \{ \Bigg [ 1- \Bigg ( \frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg )^{-\alpha } \Bigg ]^{\nu }\\{} & {} \times \log \Bigg [ 1- \Bigg ( \frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg )^{-\alpha } \Bigg ] + \Bigg ( \frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg )^{-\alpha \nu } \log \Bigg [\Bigg ( \frac{1+x_i^{\beta }}{1-x_i^{\beta }} \Bigg )^{-\alpha } \Bigg ] \Bigg \} \Bigg ]. \end{aligned}$$