# Joint distribution of a two part model

Let $$Y$$ be a random variable defined on $$(0, +\infty)$$. In a univariate two part model, the distribution of $$Y$$ is defined as follows $$\begin{equation*} g ( y_i ) = \left\{ \begin{array} { l l } { \Pr[Y_i = 0] } & { \text{ if } y_i = 0 } \\ { \left( 1 - \Pr[Y_i = 0]\right) \times f(y_i | y_i > 0) , } & { \text{ if } y_i > 0 } \end{array} \right. \end{equation*}$$ where $$f(y_i | y_i > 0)$$ is the distribution for the positive values of $$y$$.

The book Predictive Modeling Applications in Actuarial Science by Frees, Derrig and Meyers, gives the definition of a bivariate two part model. Let $$Y_1$$ and $$Y_2$$ be two random variables on $$(0, +\infty) \times (0, +\infty)$$ and consider four scenarios: $$(Y_{1i} = 0, Y_{2i} = 0)$$, $$(Y_{1i} > 0, Y_{2i} = 0)$$, $$(Y_{1i} = 0, Y_{2i} > 0)$$, and $$(Y_{1i} > 0, Y_{2i} > 0)$$. The joint distribution is $$\begin{equation*}\label{key} g \left( y _ { i 1 } , y _ { i 2 } \right) = \left\{ \begin{array} { l l } { \operatorname { Pr } \left( Y _ { i 1 } = 0 , Y _ { i 2 } = 0 \right) } & { \text { if } y _ { i 1 } = 0 , y _ { i 2 } = 0 } \\ { \operatorname { Pr } \left( Y _ { i 1 } > 0 , Y _ { i 2 } = 0 \right) \times f _ { 1 } \left( y _ { i 1 } | y _ { i 1 } > 0 , y _ { i 2 } = 0 \right) } & { \text { if } y _ { i 1 } > 0 , y _ { i 2 } = 0 } \\ { \operatorname { Pr } \left( Y _ { i 1 } = 0 , Y _ { i 2 } > 0 \right) \times f _ { 2 } \left( y _ { i 2 } | y _ { i 1 } = 0 , y _ { i 2 } > 0 \right) } & { \text { if } y _ { i 1 } = 0 , y _ { i 2 } > 0 } \\ { \operatorname { Pr } \left( Y _ { i 1 } > 0 , Y _ { i 2 } > 0 \right) \times f \left( y _ { i 1 } , y _ { i 2 } | y _ { i 1 } > 0 , y _ { i 2 } > 0 \right) } & { \text { if } y _ { i 1 } > 0 , y _ { i 2 } > 0 } \end{array} \right. \end{equation*}$$ where $$f_1$$ and $$f_2$$ are the conditional densities and $$f$$ is the joint density of the positives.

My question is related to the construction of this bivariate distribution.

I understand there are four possible outcomes but I can't see how the joint distribution is constructed for each of the cases. Can anyone help?

• Because you have explicitly written down the joint distribution, it appears your aim is to write it in some different form. Could you please tell us what form you are looking for or what you mean by "derived"? – whuber Jan 15 at 13:34
• The expression for the distribution is taken from the book I mentioned. What I don't understand is how the expression for each of the cases is obtained. For example, when y_1 > 0 and y_2 = 0. – JJC Jan 15 at 13:43
• Does the book give a description of what they mean by a "bivariate two part model"? That would not only give respondents a useful point of departure, but also could explain why what is explicitly a four part model might be called "two part." – whuber Jan 15 at 13:45