Matching a discrete distribution by Poisson matching quantiles estimation

Analyzing the data collected from different sources requires unpaired data analysis to account for the absence of correspondence between the random variable Y and the covariates $X$ . Several attempts have been made to analyze continuous Y, but it may follow a discrete distribution, which previous methodologies have overlooked. To address these limitations, we propose Poisson matching quantiles estimation (PMQE), the first unpaired data analysis method designed to examine the discrete Y and the unpaired continuous covariates $X$ . Using their order statistics, the PMQE method matches the linear combination of random variables ${\beta }^{T}X$ to $\log \left(Y\right)$ . We further improve the performance of the proposed method by ${\ell }_1$ penalizing $\beta$ , leading to the PMQE LASSO. An effective algorithm and simulation results are presented, along with the convergence results. We illustrate the practical application of PMQE using real data.