Alexis Derumigny, Lucas Girard, Yannick Guyonvarch
arXiv 10 Apr 2019 · Mathematics — Statistics Theory
arXiv:1904.07111 · PDF · DOI · OpenAlex · Extracted main text
In econometrics, many parameters of interest can be written as ratios of expectations. The main approach to construct confidence intervals for such parameters is the delta method. However, this asymptotic procedure yields intervals that may not be relevant for small sample sizes or, more generally, in a sequence-of-model framework that allows the expectation in the denominator to decrease to $0$ with the sample size. In this setting, we prove a generalization of the delta method for ratios of expectations and the consistency of the nonparametric percentile bootstrap. We also investigate finite-sample inference and show a partial impossibility result: nonasymptotic uniform confidence intervals can be built for ratios of expectations but not at every level. Based on this, we propose an easy-to-compute index to appraise the reliability of the intervals based on the delta method. Simulations and an application illustrate our results and the practical usefulness of our rule of thumb.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | J.-M. Dufour (1997) Some impossibility theorems in econometrics with applications to structural and dynamic models | 0.950 | 7 | 5 | 86% |
| 2 | R. R. Bahadur and L. J. Savage (1956) The nonexistence of certain statistical procedures in nonparametric problems | 0.737 | 3 | 2 | 100% |
| 3 | O. Catoni (2012) Challenging the empirical mean and empirical variance: a deviation study | 0.511 | 4 | 2 | 25% |
| 4 | A. Gut (1992) Complete convergence for arrays | 0.511 | 3 | 2 | 33% |
| 5 | I. Pinelis and R. Molzon (2016) Optimal-order bounds on the rate of convergence to normality in the multivariate delta method | 0.511 | 2 | 1 | 100% |
| 6 | J. P. Romano and M. Wolf (2000) Finite sample nonparametric inference and large sample efficiency | 0.511 | 2 | 1 | 100% |
| 7 | T. W. Anderson and H. Rubin (1949) Estimation of the parameters of a single equation in a complete system of stochastic equations | 0.405 | 1 | 1 | 100% |
| 8 | I. Andrews, J. Stock, and L. Sun (2019) Weak instruments in IV regression: Theory and practice | 0.405 | 1 | 1 | 100% |
| 9 | S. Boucheron, G. Lugosi, and P. Massart (2013) Concentration inequalities: A nonasymptotic theory of independence | 0.405 | 1 | 1 | 100% |
| 10 | D. Staiger and J. Stock (1997) Instrumental variables regression with weak instruments | 0.405 | 1 | 1 | 100% |
Showing the top 10 of 13 scored citations.