ABSTRACT

A three-dimensional random vector $(X_0,X_1,X_3)$ with independent components is transformed into a bivariate vector $(Y_1,Y_2)=(\phi_1(X_0,X_1),\phi_2(X_0,X_2))$. We want to identify the distributions of $X_i$'s having observed $Y_i$'s. For coding functions $\phi_i$'s falling into a semigroup scheme described in Kotlarski and Sasvari (1992) such an identification is possible even in quite abstract settings. A thorough review is given in Prakasa Rao's (1992) monograph. Here we consider a new coding method - independent random choices (with the same unknown probability): $X_0$ or $X_1$ for $Y_1$ and $X_0$ or $X_2$ for $Y_2$. It appears that in this case the full identification of the model is possible. Also this new approach will be combined with standard coding functions used earlier. A somewhat related question of identification of a finite bivariate mixtures has been treated recently in Hall and Zhou (2001) (earlier studied also by Luboi\'nska and Niemiro (1991)).