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Optimal testing in a class of nonregular models

Yuya Shimizu, Taisuke Otsu

TL;DR

Addresses optimal hypothesis testing in nonregular econometric models with parameter-dependent support. Develops AUMP tests for one- and two-sided hypotheses via a limit experiment and monotone likelihood ratio, first in a benchmark case and then in a general setting with covariates and nuisance parameters. Uses sample-splitting and auxiliary data to estimate nuisance quantities and to enable construction of confidence sets for the nonregular parameter. Simulation results show favorable finite-sample performance, with controlled size and competitive power relative to Wald-type tests.

Abstract

This paper studies optimal hypothesis testing for nonregular econometric models with parameter-dependent support. We consider both one-sided and two-sided hypothesis testing and develop asymptotically uniformly most powerful tests based on a limit experiment. Our two-sided test becomes asymptotically uniformly most powerful without imposing further restrictions such as unbiasedness, and can be inverted to construct a confidence set for the nonregular parameter. Simulation results illustrate desirable finite sample properties of the proposed tests.

Optimal testing in a class of nonregular models

TL;DR

Addresses optimal hypothesis testing in nonregular econometric models with parameter-dependent support. Develops AUMP tests for one- and two-sided hypotheses via a limit experiment and monotone likelihood ratio, first in a benchmark case and then in a general setting with covariates and nuisance parameters. Uses sample-splitting and auxiliary data to estimate nuisance quantities and to enable construction of confidence sets for the nonregular parameter. Simulation results show favorable finite-sample performance, with controlled size and competitive power relative to Wald-type tests.

Abstract

This paper studies optimal hypothesis testing for nonregular econometric models with parameter-dependent support. We consider both one-sided and two-sided hypothesis testing and develop asymptotically uniformly most powerful tests based on a limit experiment. Our two-sided test becomes asymptotically uniformly most powerful without imposing further restrictions such as unbiasedness, and can be inverted to construct a confidence set for the nonregular parameter. Simulation results illustrate desirable finite sample properties of the proposed tests.
Paper Structure (16 sections, 10 theorems, 65 equations, 2 figures)

This paper contains 16 sections, 10 theorems, 65 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Benchmark case
  3. One-sided tests
  4. Two-sided test
  5. General case
  6. One-sided tests
  7. Two-sided test
  8. Simulation
  9. Mathematical appendix
  10. Proof of Theorem \ref{['thm:opt-']}
  11. Proof of Theorem \ref{['thm:opt+']}
  12. Proof of Theorem \ref{['thm:opt_ts']}
  13. Proof of Lemma \ref{['lem:weak_conv_nui']}
  14. Proof of Theorem \ref{['thm:opt-nui']}
  15. Proof of Theorem \ref{['thm:opt+nui']}
  16. ...and 1 more sections

Key Result

Lemma 1

(shao2003mathematical, Theorem 6.2.1) Suppose that a random variable $U_{h}$ has a distribution in $\mathcal{P}=\{P_{h}:h\in\mathcal{H}\subset\mathbb{R}\}$ that has monotone likelihood ratio in $S(U_{h})$. Consider the problem of testing $H_{0}:h\leq h_{0}$ against $H_{1}:h>h_{0}$, where $h_{0}$ is where $c$ and $\kappa$ are determined by $E_{h_{0}}[\phi(U_{h})]=\alpha$.

Figures (2)

  • Figure 1: Comparison of the proposed test with the CH test (Wald test by chernozhukov2004likelihood) with two sample sizes $n$ for $H_{0}^{-}:h\leq0$ against $H_{1}^{-}:h>0$.
  • Figure 2: Comparison of the proposed test with the CH test (Wald test by chernozhukov2004likelihood) with two sample sizes $n$ for $H_{0}^{+}:h\geq0$ against $H_{1}^{+}:h<0$.

Theorems & Definitions (12)

  • Definition 1: Monotone likelihood ratio
  • Lemma 1
  • Lemma 2
  • Definition 2: Asymptotically uniformly most powerful test
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 3
  • Theorem 4
  • Theorem 5
  • ...and 2 more