Asymptotic properties of the multivariate Sz\'{a}sz-Mirakyan estimator for cumulative distribution functions on the nonnegative orthant

TL;DR

This paper develops a comprehensive asymptotic theory for multivariate Szász-Mirakyan estimators of cdfs on the nonnegative orthant, highlighting differences between interior and boundary behaviors and demonstrating efficiency gains from Poisson smoothing.

Contribution

It provides explicit bias and variance expansions, establishes asymptotic efficiency, and analyzes boundary effects, advancing understanding of multivariate cdf estimation with Szász-Mirakyan estimators.

Findings

Poisson smoothing reduces variance in the interior, improving efficiency.

Boundary behavior differs; smoothing benefits diminish near the boundary.

Central limit theorems and uniform consistency are proved.

Abstract

The asymptotic properties of multivariate Sz\'{a}sz-Mirakyan estimators for cumulative distribution functions (cdf) 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 Poisson smoothing yields a non-negligible variance reduction relative to the empirical cdf, 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 $[0,\infty)^d$, 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 Sz\'{a}sz-Mirakyan cdf estimation and clarify the distinct roles of smoothing in the interior and boundary regions.