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Inference for concave distribution functions under measurement error

Mohammed Es-Salih Benjrada, Cecile Durot, Tommaso Lando

TL;DR

This work develops estimation and testing procedures for a concave cumulative distribution function under measurement error. It introduces a constrained estimator formed by the least concave majorant of a deconvolution estimator, proving uniform consistency and a $\sqrt{n}$-rate in $\ell_\infty(\mathbb{R})$, and derives its limiting distribution via a Hadamard directional derivative. A nonparametric test for concavity on $[0,\infty)$ is proposed, calibrated with a bootstrap procedure that relies on a concave surrogate and a Donsker-type limit theorem combined with the functional delta method; the test is shown to have correct asymptotic level under the null and power tending to one under any fixed non-concave alternative. Simulation studies confirm finite-sample gains of the constrained estimator and demonstrate the test’s effectiveness across varying noise-to-signal ratios and distribution shapes, including cases where measurement error can mask the true concavity. The methods advance nonparametric inference under shape constraints in deconvolution models and provide practical tools for validating concavity in the presence of measurement error.

Abstract

We propose an estimator of a concave cumulative distribution function under the measurement error model, where the non-negative variables of interest are perturbed by additive independent random noise. The estimator is defined as the least concave majorant on the positive half-line of the deconvolution estimator of the distribution function. We show its uniform consistency and its square root convergence in law in $\ell_\infty(\mathbb R)$. To assess the validity of the concavity assumption, we construct a test for the nonparametric null hypothesis that the distribution function is concave on the positive half-line, against the alternative that it is not. We calibrate the test using bootstrap methods. The theoretical justification for calibration led us to establish a bootstrap version of Theorem 1 in Söhl and Trabs (2012), a Donsker-type result from which we obtain, as a special case, the limiting behavior of the deconvolution estimator of the distribution function in a bootstrap setting with measurement error. Combining this Donsker-type theorem with the functional delta method, we show that the test statistic and its bootstrap version have the same limiting distribution under the null hypothesis, whereas under the alternative, the bootstrap statistic is stochastically smaller. Consequently, the power of the test tends to one, for any fixed alternative, as the sample size tends to infinity. In addition to the theoretical results for the estimator and the test, we investigate their finite-sample performance in simulation studies.

Inference for concave distribution functions under measurement error

TL;DR

This work develops estimation and testing procedures for a concave cumulative distribution function under measurement error. It introduces a constrained estimator formed by the least concave majorant of a deconvolution estimator, proving uniform consistency and a -rate in , and derives its limiting distribution via a Hadamard directional derivative. A nonparametric test for concavity on is proposed, calibrated with a bootstrap procedure that relies on a concave surrogate and a Donsker-type limit theorem combined with the functional delta method; the test is shown to have correct asymptotic level under the null and power tending to one under any fixed non-concave alternative. Simulation studies confirm finite-sample gains of the constrained estimator and demonstrate the test’s effectiveness across varying noise-to-signal ratios and distribution shapes, including cases where measurement error can mask the true concavity. The methods advance nonparametric inference under shape constraints in deconvolution models and provide practical tools for validating concavity in the presence of measurement error.

Abstract

We propose an estimator of a concave cumulative distribution function under the measurement error model, where the non-negative variables of interest are perturbed by additive independent random noise. The estimator is defined as the least concave majorant on the positive half-line of the deconvolution estimator of the distribution function. We show its uniform consistency and its square root convergence in law in . To assess the validity of the concavity assumption, we construct a test for the nonparametric null hypothesis that the distribution function is concave on the positive half-line, against the alternative that it is not. We calibrate the test using bootstrap methods. The theoretical justification for calibration led us to establish a bootstrap version of Theorem 1 in Söhl and Trabs (2012), a Donsker-type result from which we obtain, as a special case, the limiting behavior of the deconvolution estimator of the distribution function in a bootstrap setting with measurement error. Combining this Donsker-type theorem with the functional delta method, we show that the test statistic and its bootstrap version have the same limiting distribution under the null hypothesis, whereas under the alternative, the bootstrap statistic is stochastically smaller. Consequently, the power of the test tends to one, for any fixed alternative, as the sample size tends to infinity. In addition to the theoretical results for the estimator and the test, we investigate their finite-sample performance in simulation studies.
Paper Structure (40 sections, 15 theorems, 142 equations, 2 figures, 2 tables)

This paper contains 40 sections, 15 theorems, 142 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Context
  3. Notations
  4. Model, objective and basic estimators
  5. Assumptions
  6. Concave estimators
  7. Definition of the concave estimator and consistency
  8. Limit distribution
  9. The test statistic
  10. Calibration of the test
  11. Bootstrap calibration
  12. About least favorable hypothesis
  13. Simulations
  14. Performance of $\mathcal{M}\hat{F}_n^d$ in $\mathcal{H}_0$
  15. Rejection rates for $T(F_n^d)$
  16. ...and 25 more sections

Key Result

Lemma 1

Let $f_n^d$ and $F_n^d$ be defined by fn and Fn. Assume that $h_n^{ 2+2\beta }n\rightarrow \infty$, $\mathcal{F} K$ is supported on $[-1,1]$, there exist positive $\beta$ and $C$ such that $|1/\varphi_\varepsilon(t)|\leq C(1+|t|^2)^{\beta/2}$ for all $t\in\mathbb R$, $f$ is bounded and $\int|\mathca

Figures (2)

  • Figure 1: Simulation results. Mean squared error (MSE) for the Beta $B(0.75,1)$ and Weibull $W(0.75,1)$ models, across NSR levels of $10\%$, $20\%$, and $50\%$ (from left to right) and sample sizes $n = 20, 50,$ and $100$.
  • Figure 2: Rejection rates for $\mathrm{Beta}(a, 1)$ and $W(a, 1)$ distributions for different values of $a$, NSR, and $n$.

Theorems & Definitions (16)

  • Lemma 1
  • Definition 1
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3: Brendan2017, Proposition 2.1
  • Theorem 4
  • Theorem 5
  • Lemma 2
  • Theorem 6
  • ...and 6 more