Hyperbolastic functions

Let ${\displaystyle Y}$ be a binary outcome variable which can assume one of two mutually exclusive values, success or failure. If we code success as ${\displaystyle Y=1}$ and failure as ${\displaystyle Y=0}$, the hyperbolastic success probability of type I as a function of ${\displaystyle K}$ explanatory variables ${\displaystyle X_{1},\ X_{2},\ldots ,\ X_{k}}$ is given by:

${\displaystyle P(Y=1)={\frac {1}{1+e^{-(\beta _{0}+\beta _{1}x_{1}+\beta _{2}x_{2}+\ldots +\beta _{k}x_{k})-\operatorname {arsinh} (\beta _{0}+\beta _{1}x_{1}+\beta _{2}x_{2}+\ldots +\beta _{k}x_{k})}}}}$,

where ${\displaystyle \beta _{0},\beta _{1},\ldots ,\beta _{k}}$ are model parameters. The odds of success is the ratio of probability of success to the probability of failure. For hyperbolastic regression of type I, the odds of success is denoted by ${\displaystyle Odds_{H1}}$ and expressed by the equation:

${\displaystyle Odds_{H1}=e^{-(\beta _{0}+\beta _{1}x_{1}+\beta _{2}x_{2}+\ldots +\beta _{k}x_{k})-\operatorname {arsinh} (\beta _{0}+\beta _{1}x_{1}+\beta _{2}x_{2}+\ldots +\beta _{k}x_{k})}}$.

The logarithm of ${\displaystyle Odds_{H1}}$ is called the logit of Hyperbolastic of type I. The logit transformation is denoted by ${\displaystyle L_{H1}}$ and can be written as:

${\displaystyle L_{H1}=\beta _{0}+\beta _{1}x_{1}+\beta _{2}x_{2}+\ldots +\beta _{k}x_{k}+\operatorname {arsinh} (\beta _{0}+\beta _{1}x_{1}+\beta _{2}x_{2}+\ldots +\beta _{k}x_{k})}$.

For the binary outcome variable ${\displaystyle Y}$, the hyperbolastic success probability of type II as a function of ${\displaystyle K}$ explanatory variables ${\displaystyle X_{1},X_{2},\ldots ,X_{k}}$ is:

${\displaystyle P(Y=1)={\frac {1}{1+\operatorname {arsinh} (e^{-(\beta _{0}+\beta _{1}x_{1}+\beta _{2}x_{2}+\ldots +\beta _{k}x_{k})})}}}$ ,

For the hyperbolastic regression of type II, the odds of success is denoted by ${\displaystyle Odds_{H2}}$ and is given by:

${\displaystyle Odds_{H2}={\frac {1}{\operatorname {arsinh} (e^{-(\beta _{0}+\beta _{1}x_{1}+\beta _{2}x_{2}+\ldots +\beta _{k}x_{k})})}}.}$

The logit transformation is denoted by ${\displaystyle L_{H2}}$ and is given by:

${\displaystyle L_{H2}=-\log {(\operatorname {arsinh} e^{\beta _{0}+\beta _{1}x_{1}+\beta _{2}x_{2}+\ldots +\beta _{k}x_{k}})}}$