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

Guanjie Lyu, Frédéric Ouimet, Cindy Feng

TL;DR

This work develops a multivariate Szász–Mirakyan smoothing framework for nonnegative-support distribution estimation on the orthant $[0, abla)^d$, and provides a complete asymptotic theory that separates interior and boundary behaviors. It derives explicit second-order bias and variance expansions, identifies optimal smoothing rates (notably $m_{ m opt}(m x)\sim n^{2/3}$ in the interior), and shows asymptotic efficiency gains over the empirical c.d.f via deficiency metrics. A boundary-layer analysis reveals that smoothing loses its leading variance-reduction effect near the boundary, yielding distinct, boundary-appropriate MSE rates and no local smoothing parameter with interior-like optimality. Central limit theorems and almost-sure uniform consistency complete the asymptotic picture, while simulations validate the theoretical predictions and demonstrate practical smoothing parameter selection via least-squares cross-validation.

Abstract

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 Poisson 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 $[0,\infty)^d$, 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 Szász-Mirakyan distribution estimation and clarify the distinct roles of smoothing in the interior and boundary regions.

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

TL;DR

This work develops a multivariate Szász–Mirakyan smoothing framework for nonnegative-support distribution estimation on the orthant , and provides a complete asymptotic theory that separates interior and boundary behaviors. It derives explicit second-order bias and variance expansions, identifies optimal smoothing rates (notably in the interior), and shows asymptotic efficiency gains over the empirical c.d.f via deficiency metrics. A boundary-layer analysis reveals that smoothing loses its leading variance-reduction effect near the boundary, yielding distinct, boundary-appropriate MSE rates and no local smoothing parameter with interior-like optimality. Central limit theorems and almost-sure uniform consistency complete the asymptotic picture, while simulations validate the theoretical predictions and demonstrate practical smoothing parameter selection via least-squares cross-validation.

Abstract

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 Poisson 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 , 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.
Paper Structure (20 sections, 10 theorems, 181 equations, 2 figures, 3 tables, 1 algorithm)

This paper contains 20 sections, 10 theorems, 181 equations, 2 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Interior properties
  3. Bias and variance expansions
  4. Mean squared error and optimal smoothing
  5. Asymptotic deficiency of the empirical c.d.f.
  6. Limit distributions and uniform consistency
  7. Near boundary properties
  8. Simulation study
  9. Models, evaluation region, and performance criteria
  10. Smoothing parameter selection via least-squares cross-validation
  11. Results and discussion
  12. Concluding remarks
  13. Proofs of results
  14. Proof of Proposition \ref{['prop:SM-bias-vector']}
  15. Proof of Theorem \ref{['thm:szasz-bias-var']}
  16. ...and 5 more sections

Key Result

Proposition 2.2

Suppose Assumption ass:C2-vector holds. Then, for every compact set $S\subset (0,\infty)^d$, as $m_{\min}=\min_{1\le j\le d} m_j\to\infty$, uniformly for $\bm x\in S$.

Figures (2)

  • Figure 1: Schematic Poisson smoothing surfaces for the bivariate Szász--Mirakyan estimator evaluated at a fixed interior point $\bm x$.
  • Figure 2: Mean $\mathrm{ISE}_{S_\delta}$ (Monte Carlo average) versus sample size $n$ on a log--log scale: empirical c.d.f. versus Szász--Mirakyan with LSCV-selected smoothing.

Theorems & Definitions (17)

  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Corollary 2.5
  • Theorem 2.6
  • proof
  • Theorem 2.7
  • proof
  • ...and 7 more