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Estimation of Parameters of the Truncated Normal Distribution with Unknown Bounds

Dylan Borchert, Semhar Michael, Christopher Saunders

TL;DR

This paper tackles estimating all four parameters $(\mu,\sigma,\tau_l,\tau_u)$ of a truncated normal distribution when the truncation bounds are unknown. It introduces an expectation-solution (ES) algorithm that solves a system of estimating equations derived from regression on order statistics and unbiased truncation-bound estimation, incorporating latent counts for the truncation limits. The authors prove convergence and consistency, derive Weibull limits for the bounds and asymptotic normality for $(\mu,\sigma)$ (with a key conjecture for joint behavior), and validate the approach via extensive simulations and a real iris-data discriminant analysis demonstrating robustness to unseen classes. The work advances parameter estimation under unknown truncation and offers a principled framework for open-set classification problems in forensic contexts and beyond.

Abstract

Estimators of parameters of truncated distributions, namely the truncated normal distribution, have been widely studied for a known truncation region. There is also literature for estimating the unknown bounds for known parent distributions. In this work, we develop a novel algorithm under the expectation-solution (ES) framework, which is an iterative method of solving nonlinear estimating equations, to estimate both the bounds and the location and scale parameters of the parent normal distribution utilizing the theory of best linear unbiased estimates from location-scale families of distribution and unbiased minimum variance estimation of truncation regions. The conditions for the algorithm to converge to the solution of the estimating equations for a fixed sample size are discussed, and the asymptotic properties of the estimators are characterized using results on M- and Z-estimation from empirical process theory. The proposed method is then compared to methods utilizing the known truncation bounds via Monte Carlo simulation.

Estimation of Parameters of the Truncated Normal Distribution with Unknown Bounds

TL;DR

This paper tackles estimating all four parameters of a truncated normal distribution when the truncation bounds are unknown. It introduces an expectation-solution (ES) algorithm that solves a system of estimating equations derived from regression on order statistics and unbiased truncation-bound estimation, incorporating latent counts for the truncation limits. The authors prove convergence and consistency, derive Weibull limits for the bounds and asymptotic normality for (with a key conjecture for joint behavior), and validate the approach via extensive simulations and a real iris-data discriminant analysis demonstrating robustness to unseen classes. The work advances parameter estimation under unknown truncation and offers a principled framework for open-set classification problems in forensic contexts and beyond.

Abstract

Estimators of parameters of truncated distributions, namely the truncated normal distribution, have been widely studied for a known truncation region. There is also literature for estimating the unknown bounds for known parent distributions. In this work, we develop a novel algorithm under the expectation-solution (ES) framework, which is an iterative method of solving nonlinear estimating equations, to estimate both the bounds and the location and scale parameters of the parent normal distribution utilizing the theory of best linear unbiased estimates from location-scale families of distribution and unbiased minimum variance estimation of truncation regions. The conditions for the algorithm to converge to the solution of the estimating equations for a fixed sample size are discussed, and the asymptotic properties of the estimators are characterized using results on M- and Z-estimation from empirical process theory. The proposed method is then compared to methods utilizing the known truncation bounds via Monte Carlo simulation.
Paper Structure (19 sections, 10 theorems, 104 equations, 9 figures, 2 tables, 1 algorithm)

This paper contains 19 sections, 10 theorems, 104 equations, 9 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methods
  3. Estimating Equations for $\mu$ and $\sigma$
  4. The Distributions of $N_l$ and $N_u$
  5. Estimating Equations for $\tau_l$ and $\tau_u$
  6. ES Algorithm
  7. Convergence to Root
  8. Consistency of Parameter Estimates
  9. Asymptotic Distributions of Parameter Estimates
  10. Simulation
  11. Setup
  12. Results
  13. Real Data Application
  14. Conclusion
  15. Standard Truncated Normal Distributions
  16. ...and 4 more sections

Key Result

Lemma 1

Let $H$ be a $k\times k$ matrix with spectral radius $\rho(H) \equiv \lambda < 1$ and let $r:\mathcal{M} \subset \mathbb{R}^k \rightarrow \mathbb{R}^k$ be a mapping from a neighborhood of the origin, $\mathcal{M}$, into $\mathbb{R}^l$ such that Then given any constant $\epsilon$ satisfying $\lambda < \lambda+\epsilon<1$, there is an open neighborhood of the origin $\mathcal{M}'$ and a positive co

Figures (9)

  • Figure 1: Visualization of the different cases used for simulation, comparing the distribution function for the truncated normal distribution (black solid line) to the distribution function of the uniform distribution with the same bounds (red dashed line).
  • Figure 2: Results of the simulations illustrating the distribution of $\hat{\mu}_n$ as $n$ increases.
  • Figure 3: Results of the simulations illustrating the distribution of $\hat{\sigma}_n$ as $n$ increases.
  • Figure 4: Results of the simulations illustrating the distribution of $\hat{\tau}_{nl}$ as $n$ increases.
  • Figure 5: Results of the simulations illustrating the distribution of $\hat{\tau}_{nu}$ as $n$ increases.
  • ...and 4 more figures

Theorems & Definitions (22)

  • Lemma 1: Ortega-difference1966
  • Theorem 1
  • proof
  • Lemma 2: Shorack and Wellner Shorackandwellner
  • Lemma 3
  • proof
  • Lemma 4: Van der Vaart vdVaart_1998)
  • Theorem 2
  • proof
  • Lemma 5: Gnedenko as Presented in David-orderstats2003
  • ...and 12 more