 Sequential response selection model

Suppose outcome collection stops at the first success or after $m$ unsuccessful attempts, and let $A$ count unsuccessful attempts. Conditional on the outcome $y$, put $q_j(y)=\mathbb{P}(A\ge j\mid A\ge j-1,Y=y)$. For $k<m$, the joint density of $A=k$ and an observed $Y=y$ isFor $A=m$, the unobserved outcome is integrated out, giving $\int f_{\beta }(y){\prod }_{j=1}^m{q}_j(y)dy$. These probabilities sum to one by telescoping and integration. Dependence of $q_j$ on $y$ models missing not at random. If all $q_j$ are independent of $y$ conditional on the regression predictors, the outcome density among responders remains $f_{\beta }$, as required for complete-case analysis of that conditional model.