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 $ \boldsymbol {X} $ 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 $ \boldsymbol{X} $ X. Using their order statistics, the PMQE method matches the linear combination of random variables $ \boldsymbol{\beta} ^{T} \boldsymbol{X} $ βTX to $ {\rm log}(Y) $ log(Y). We further improve the performance of the proposed method by $ \ell _1 $ ℓ1 penalizing $ \boldsymbol{\beta} $ β, 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.