Asymptotic properties of multivariate Szász-Mirakyan distribution estimators on the nonnegative orthant

The asymptotic properties of multivariate Szász–Mirakyan estimators for distribution functions supported on the nonnegative orthant are investigated. Explicit bias and variance expansions are derived on compact subsets of the interior, yielding sharp mean squared error characterizations and optimal smoothing rates. The analysis shows that the proposed Pois-son smoothing yields a non-negligible variance reduction relative to the empirical distribution function, leading to asymptotic efficiency gains that can be quantified through local and global deficiency measures. The behavior of the estimator near the boundary of its support is examined separately. Under a boundary-layer scaling that preserves nondegenerate Poisson smoothing as the evaluation point approaches the boundary of [0, ∞) d , bias and variance expansions are obtained that differ fundamentally from those in the interior region. In particular, the variance reduction mechanism disappears at leading order, implying that no asymptotically optimal smoothing parameter exists in the boundary regime. Central limit theorems and almost sure uniform consistency are also established. Together, these results provide a unified asymptotic theory for multivariate Szász–Mirakyan distribution estimation and clarify the distinct roles of smoothing in the interior and boundary regions.