A generalization of the slashed distribution via alpha skew normal distribution

Abstract

In this paper, we introduce a new class of the slash distribution, an alpha skew normal slash distribution. The proposed model is more flexible in terms of its kurtosis than the slashed normal distribution and can efficiently capture the bimodality. Properties involving moments and moment generating function are studied. The distribution is illustrated with a real application.

Appendix A: Appendix

1.1 Appendix A.1: Score vector and Hessian matrix

Suppose $y_1, y_2,\ldots , y_n$ is a random sample drawn from the alpha skew normal slash distribution $\textit{ASNS}(\alpha ,q,\mu ,\sigma )$, then the log-likelihood function is given by (21). The elements of the score vector are obtained by differentiation

$$\begin{aligned}&\displaystyle l_{\alpha }=\sum _{i\!=\!1}^n \frac{\displaystyle \int _0^1\frac{2\alpha ^2\left( \frac{y_i\!-\!\mu }{\sigma }\right) t\!+\!4\alpha \left( \left( \frac{y_i\!-\!\mu }{\sigma }\right) ^2t^2\!-\!1\right) \!-\!4\left( \frac{y_i\!-\!\mu }{\sigma }\right) t}{(2\!+\!\alpha ^2)^2}\phi \left[ \left( \frac{y_i\!-\!\mu }{\sigma }\right) t\right] t^qdt}{\displaystyle \int _0^1\frac{\left[ 1\!-\!\alpha \left( \frac{y_i\!-\!\mu }{\sigma }\right) t\right] ^2\!+\!1}{2\!+\!\alpha ^2}\phi \left[ \left( \frac{y_i\!-\!\mu }{\sigma }\right) t\right] t^qdt} \\&\displaystyle l_{q}\!=\!\frac{n}{q}\!+\!\sum _{i\!=\!1}^n \frac{\displaystyle \int _0^1\frac{\left[ 1\!-\!\alpha \left( \frac{y_i-\mu }{\sigma }\right) t\right] ^2+1}{2+\alpha ^2}\phi \left[ \left( \frac{y_i-\mu }{\sigma }\right) t\right] t^q\log tdt }{ \displaystyle \int _0^1\frac{\left[ 1-\alpha \left( \frac{y_i-\mu }{\sigma }\right) t\right] ^2+1}{2+\alpha ^2}\phi \left[ \left( \frac{y_i-\mu }{\sigma }\right) t\right] t^qdt} \\&\displaystyle l_{\mu }=\sum _{i=1}^n \frac{\displaystyle \int _0^1\frac{t^{q+1}}{(2+\alpha ^2)\sigma }\left\{ \left[ 1+\alpha -\alpha \left( \frac{y_i-\mu }{\sigma }\right) t\right] ^2+1-\alpha ^2\right\} \phi \left[ \left( \frac{y_i-\mu }{\sigma }\right) t\right] dt }{\displaystyle \int _0^1\frac{\left[ 1-\alpha \left( \frac{y_i-\mu }{\sigma }\right) t\right] ^2+1}{2+\alpha ^2}\phi \left[ \left( \frac{y_i-\mu }{\sigma }\right) t\right] t^qdt}\\&\displaystyle l_{\sigma }=-\frac{n}{\sigma }+ \sum _{i\!=\!1}^n \frac{\displaystyle \int _0^1\frac{(y_i\!-\!\mu )t^{q\!+\!1}}{(2\!+\!\alpha ^2)\sigma ^2}\left\{ 2\alpha \left[ 1\!-\!\alpha \left( \frac{y_i\!-\!\mu }{\sigma }\right) t\right] \!+\!(\left[ 1-\alpha \left( \frac{y_i\!-\!\mu }{\sigma }\right) t\right] ^2\!+\!1)(\frac{y_i\!-\!\mu }{\sigma })\right\} \phi \left[ \left( \frac{y_i\!-\!\mu }{\sigma }\right) t\right] dt }{\displaystyle \int _0^1\frac{\left[ 1-\alpha \left( \frac{y_i\!-\!\mu }{\sigma }\right) t\right] ^2\!+\!1}{2\!+\!\alpha ^2}\phi \left[ \left( \frac{y_i\!-\!\mu }{\sigma }\right) t\right] t^qdt} \end{aligned}$$

The Hessian matrix, second partial derivatives of the log-likelihood, is given by

$$\begin{aligned} H= \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} l_{\alpha \alpha } &{}l_{\alpha q} &{} l_{\alpha \mu } &{} l_{\alpha \sigma } \\ l_{q \alpha } &{}l_{q q} &{} l_{q \mu } &{} l_{q\sigma } \\ l_{\mu \alpha } &{}l_{\mu q} &{} l_{\mu \mu } &{} l_{\mu \sigma } \\ l_{\sigma \alpha } &{}l_{\sigma q} &{} l_{\sigma \mu } &{} l_{\sigma \sigma } \\ \end{array} \right) \end{aligned}$$

where

$$\begin{aligned}&l_{\alpha \alpha } \!=\! \sum _{i\!=\!1}^n \frac{\displaystyle \int _0^1 \left( \begin{array}{c} \frac{2 t^{q\!+\!2} \left( y_i\!-\!\mu \right) {}^2 \phi [(\frac{y_i\!-\!\mu }{\sigma })t]}{ \left( \alpha ^2\!+\!2\right) \sigma ^2}\!+\!\frac{8 \alpha t^{q\!+\!1} \left( y_i\!-\!\mu \right) \phi [(\frac{y_i\!-\!\mu }{\sigma })t] \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{ \left( \alpha ^2\!+\!2\right) ^2 \sigma }\\ \!+\!\frac{8 \alpha ^2 t^q \phi [(\frac{y_i\!-\!\mu }{\sigma })t] \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) }{\left( \alpha ^2\!+\!2\right) ^3 }\!-\!\frac{2 t^q \phi [(\frac{y_i\!-\!\mu }{\sigma })t] \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) }{\left( \alpha ^2\!+\!2\right) ^2 } \end{array}\right) \, dt}{\displaystyle \int _0^1 \frac{t^q \phi [(\frac{y_i\!-\!\mu }{\sigma })t] \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) }{\left( \alpha ^2\!+\!2\right) } \, dt}\\&\qquad \!-\!\sum _{i=1}^n \frac{\left( \displaystyle \int _0^1 \left( -\frac{2 t^{q\!+\!1} \left( y_i\!-\!\mu \right) \phi [(\frac{y_i\!-\!\mu }{\sigma })t] \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{ \left( \alpha ^2\!+\!2\right) \sigma }\!-\!\frac{2 \alpha t^q \phi [(\frac{y_i\!-\!\mu }{\sigma })t] \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) }{\left( \alpha ^2\!+\!2\right) ^2 }\right) \, dt\right) {}^2}{\left( \displaystyle \int _0^1 \frac{t^q \phi [(\frac{y_i\!-\!\mu }{\sigma })t] \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) }{\left( \alpha ^2\!+\!2\right) } \, dt\right) {}^2}\\&l_{\alpha q}\!=\!l_{q \alpha }\\&\quad \!=\! \sum _{i\!=\!1}^n \frac{\displaystyle \int _0^1 \left( -\frac{2 t^{q\!+\!1} \log (t) \left( y_i\!-\!\mu \right) \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma }\!-\!\frac{2 \alpha t^q \log (t) \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) ^2}\right) \, dt}{\displaystyle \int _0^1 \frac{t^q \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\alpha ^2\!+\!2} \, dt}\\&\qquad \!-\!\sum _{i\!=\!1}^n \frac{\displaystyle \int _0^1 \left( \begin{array}{c} \!-\!\frac{2 t^{q\!+\!1} \left( y_i\!-\!\mu \right) \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma }\\ \!-\!\frac{2 \alpha t^q \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) ^2} \end{array}\right) \, dt \left( \displaystyle \int _0^1 \frac{t^q \log (t) \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\alpha ^2\!+\!2} \, dt\right) }{\left( \displaystyle \int _0^1 \frac{t^q \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\alpha ^2\!+\!2} \, dt\right) {}^2}\\&l_{\alpha \mu }\!=\!l_{\mu \alpha } \\&\quad \!=\!\sum _{i\!=\!1}^n \frac{\displaystyle \int _0^1 \left( \begin{array}{c} \frac{2 t^{q\!+\!2} \left( y_i\!-\!\mu \right) \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi '\left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^2}\!-\!\frac{2 \alpha t^{q\!+\!2} \left( y_i\!-\!\mu \right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^2}\\ \!+\!\frac{2 \alpha t^{q\!+\!1} \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi '\left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) ^2 \sigma }\!+\!\frac{2 t^{q\!+\!1} \left( 1\!-v\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma }\\ \!-\! \frac{4 \alpha ^2 t^{q\!+\!1} \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi \left( \frac{t \left( y_i-\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) ^2 \sigma } \end{array}\right) \, dt}{\displaystyle \int _0^1 \frac{t^q \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\alpha ^2\!+\!2} \, dt}\\&\qquad \!-\!\sum _{i\!=\!1}^n \frac{\left( \displaystyle \int _0^1 \left( \begin{array}{c} \!-\!\frac{2 t^{q\!+\!1} \left( y_i\!-\!\mu \right) \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma }\\ \!-\!\frac{2 \alpha t^q \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) ^2} \end{array}\right) \, dt\right) \displaystyle \int _0^1 \left( \begin{array}{c} \frac{2 \alpha t^{q\!+\!1} \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma }\\ \!-\!\frac{t^{q\!+\!1} \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi '\left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma } \end{array}\right) \, dt}{\left( \displaystyle \int _0^1 \frac{t^q \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\alpha ^2\!+\!2} \, dt\right) {}^2}\\&l_{\alpha \sigma }\!=\!l_{\sigma \alpha }\\&\quad \!=\!\sum _{i\!=\!1}^n \frac{\displaystyle \int _0^1 \left( \begin{array}{c} \frac{2 t^{q\!+\!2} \left( y_i\!-\!\mu \right) {}^2 \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi '\left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^3}\!-\!\frac{2 \alpha t^{q\!+\!2} \left( y_i\!-\!\mu \right) {}^2 \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^3}\\ \!+\!\frac{2 \alpha t^{q\!+\!1} \left( y_i\!-\!\mu \right) \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi '\left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) ^2 \sigma ^2}\!+\!\frac{2 t^{q\!+\!1} \left( y_i\!-\!\mu \right) \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^2}\\ \!-\!\frac{4 \alpha ^2 t^{q\!+\!1} \left( y_i\!-\!\mu \right) \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) ^2 \sigma ^2} \end{array}\right) \, dt}{\displaystyle \int _0^1 \frac{t^q \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\alpha ^2\!+\!2} \, dt}\\&\qquad \!-\!\sum _{i\!=\!1}^n \frac{ \displaystyle \int _0^1 \left( \begin{array}{c} \!-\!\frac{2 t^{q\!+\!1} \left( y_i\!-\!\mu \right) \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma }\\ \!-\!\frac{2 \alpha t^q \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) ^2} \end{array}\right) \, dt \displaystyle \int _0^1 \left( \begin{array}{c} \frac{2 \alpha t^{q\!+\!1} \left( y_i\!-\!\mu \right) \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^2}\\ \!-\! \frac{t^{q\!+\!1} \left( y_i\!-\!\mu \right) \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi '\left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^2} \end{array}\right) \, dt}{\left( \displaystyle \int _0^1 \frac{t^q \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\alpha ^2\!+\!2} \, dt\right) {}^2}\\&l_{q q}\!=\!-\sum _{i\!=\!1}^n \frac{\left( \displaystyle \int _0^1 \frac{t^q \log (t) \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\alpha ^2\!+\!2} \, dt\right) {}^2}{\left( \displaystyle \int _0^1 \frac{t^q \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2v+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\alpha ^2\!+\!2} \, dt\right) {}^2}-\frac{n}{q^2}\\&\qquad \sum _{i\!=\!1}^n \frac{\displaystyle \int _0^1 \frac{t^q \log ^2(t) \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\alpha ^2\!+\!2} \, dt}{\displaystyle \int _0^1 \frac{t^q \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\alpha ^2\!+\!2} \, dt}\\&l_{q \mu }\!=\!l_{\mu q}\\&\quad \!=\!\underset{i\!=\!1}{\overset{n}{\!-\!\sum }}\frac{\displaystyle \int _0^1 \left( \begin{array}{c} \frac{2 \alpha t^{q\!+\!1} \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma }\\ \!-\!\frac{t^{q\!+\!1} \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi '\left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma } \end{array}\right) \, dt \left( \displaystyle \int _0^1 \frac{t^q \log (t) \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\alpha ^2\!+\!2} \, dt\right) }{\left( \displaystyle \int _0^1 \frac{t^q \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\alpha ^2\!+\!2} \, dt\right) {}^2}\\&\qquad \sum _{i\!=\!1}^n \frac{\displaystyle \int _0^1 \left( \begin{array}{c} \frac{2 \alpha t^{q\!+\!1} \log (t) \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma }\\ \!-\! \frac{t^{q\!+\!1} \log (t) \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi '\left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma } \end{array}\right) \, dt}{\displaystyle \int _0^1 \frac{t^q \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\alpha ^2\!+\!2} \, dt}\\&l_{q \sigma }=l_{\sigma q}\\&\quad \!=\!\!-\!\sum _{i\!=\!1}^n \frac{\displaystyle \int _0^1 \left( \begin{array}{c} \frac{2 \alpha t^{q\!+\!1} \left( y_i\!-\!\mu \right) \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^2}\\ \!-\! \frac{t^{q\!+\!1} \left( y_i\!-\!\mu \right) \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi '\left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^2} \end{array}\right) \, dt \left( \displaystyle \int _0^1 \frac{t^q \log (t) \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\alpha ^2\!+\!2} \, dt\right) }{\left( \displaystyle \int _0^1 \frac{t^q \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\alpha ^2\!+\!2} \, dt\right) {}^2}\\&\qquad \sum _{i\!=\!1}^n \frac{\displaystyle \int _0^1 \left( \begin{array}{c} \frac{2 \alpha t^{q\!+\!1} \log (t) \left( y_i\!-\mu \right) \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^2}\\ \!-\!\frac{t^{q\!+\!1} \log (t) \left( y_i\!-\!\mu \right) \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi '\left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^2} \end{array}\right) \, dt}{\displaystyle \int _0^1 \frac{t^q \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\alpha ^2\!+\!2} \, dt}\\&l_{\mu \mu }\!=\!\!-\!\sum _{i\!=\!1}^n \frac{\left( \displaystyle \int _0^1 \left( \begin{array}{c} \frac{2 \alpha t^{q\!+\!1} \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma }\\ - \frac{t^{q\!+\!1} \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi '\left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma } \end{array}\right) \, dt\right) {}^2}{\left( \displaystyle \int _0^1 \frac{t^q \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\alpha ^2\!+\!2} \, dt\right) {}^2}\\&\qquad \!+\! \sum _{i\!=\!1}^n \frac{\displaystyle \int _0^1 \left( \begin{array}{c} \frac{t^{q\!+\!2} \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi ''\left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^2}\\ - \frac{4 \alpha t^{q\!+\!2} \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi '\left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^2}\!+\!\frac{2 \alpha ^2 t^{q\!+\!2} \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^2} \end{array}\right) \, dt}{\displaystyle \int _0^1 \frac{t^q \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\alpha ^2\!+\!2} \, dt} \end{aligned}$$

$$\begin{aligned}&l_{\mu \sigma }\!=\!l_{\sigma \mu }\\&\quad \!=\!\underset{i\!=\!1}{\overset{n}{\!-\!\sum }}\frac{\left( \displaystyle \int _0^1 \left( \begin{array}{c} \frac{2 \alpha t^{q\!+\!1} \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma }\\ \!-\! \frac{t^{q\!+\!1} \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi '\left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma } \end{array}\right) \, dt\right) \displaystyle \int _0^1 \left( \begin{array}{c} \frac{2 \alpha t^{q\!+\!1} \left( y_i\!-\!\mu \right) \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^2}\\ - \frac{t^{q\!+\!1} \left( y_i\!-\!\mu \right) \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi '\left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^2} \end{array}\right) \, dt}{\left( \displaystyle \int _0^1 \frac{t^q \left( \left( 1\!-\!\frac{\alpha t \left( y_i-\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\alpha ^2\!+\!2} \, dt\right) {}^2}\\&\qquad \sum _{i\!=\!1}^n \frac{\displaystyle \int _0^1 \left( \begin{array}{c} \frac{t^{q\!+\!2} \left( y_i\!-\!\mu \right) \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi ''\left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^3}-\frac{4 \alpha t^{q\!+\!2} \left( y_i\!-\!\mu \right) \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi '\left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^3}\\ \!+\! \frac{2 \alpha ^2 t^{q\!+\!2} \left( y_i\!-\!\mu \right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^3}\!+\!\frac{t^{q\!+\!1} \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi '\left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^2}\\ \!-\! \frac{2 \alpha t^{q\!+\!1} \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^2} \end{array}\right) \, dt}{\displaystyle \int _0^1 \frac{t^q \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\alpha ^2\!+\!2} \, dt}\\&l_{\sigma \sigma }\!=\!\frac{n}{\sigma ^2}\!-\!\sum _{i\!=\!1}^n \frac{\left( \displaystyle \int _0^1 \left( \begin{array}{c} \frac{2 \alpha t^{q\!+\!1} \left( y_i\!-\!\mu \right) \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^2}\\ - \frac{t^{q+1} \left( y_i\!-\!\mu \right) \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi '\left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^2} \end{array}\right) \, dt\right) {}^2}{\left( \displaystyle \int _0^1 \frac{t^q \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i-\mu \right) }{\sigma }\right) }{\alpha ^2\!+\!2} \, dt\right) {}^2}\\&\qquad \!+\! \sum _{i\!=\!1}^n \frac{\displaystyle \int _0^1 \left( \begin{array}{c} \frac{t^{q+2} \left( y_i\!-v\mu \right) {}^2 \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi ''\left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^4}\!-\!\frac{4 \alpha t^{q\!+\!2} \left( y_i\!-\!\mu \right) {}^2 \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi '\left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^4}\\ \!+\! \frac{2 \alpha ^2 t^{q\!+\!2} \left( y_i\!-\!\mu \right) {}^2 \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^4}\!+\!\frac{2 t^{q\!+\!1} \left( y_i\!-\!\mu \right) \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi '\left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^3}\\ \!-\! \frac{4 \alpha t^{q\!+\!1} \left( y_i\!-\!\mu \right) \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\left( \alpha ^2\!+\!2\right) \sigma ^3} \end{array}\right) \, dt}{\displaystyle \int _0^1 \frac{t^q \left( \left( 1\!-\!\frac{\alpha t \left( y_i\!-\!\mu \right) }{\sigma }\right) {}^2\!+\!1\right) \phi \left( \frac{t \left( y_i\!-\!\mu \right) }{\sigma }\right) }{\alpha ^2\!+\!2} \, dt} \end{aligned}$$