# On the construction of confidence intervals for ratios of expectations

Alexis Derumigny, Lucas Girard, Yannick Guyonvarch

Research output: Working paper

### Abstract

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.
Original language English Published - 10 Apr 2019 Yes

### Fingerprint

Delta Method
Confidence interval
Interval
Percentile
Small Sample Size
Denominator
Econometrics
Bootstrap
Sample Size
Partial
Decrease
Delta method
Simulation
Sample size
Model

### Keywords

• Delta-method
• Confidence regions
• Uniformly valid inference
• Sequence of models
• Nonparametric percentile bootstrap

### Cite this

@techreport{4964b6b4930e47d7867797f84262609c,
title = "On the construction of confidence intervals for ratios of expectations",
abstract = "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.",
keywords = "Delta-method, Confidence regions, Uniformly valid inference, Sequence of models, Nonparametric percentile bootstrap",
author = "Alexis Derumigny and Lucas Girard and Yannick Guyonvarch",
note = "59 pages",
year = "2019",
month = "4",
day = "10",
language = "English",
type = "WorkingPaper",

}

On the construction of confidence intervals for ratios of expectations. / Derumigny, Alexis; Girard, Lucas; Guyonvarch, Yannick.

2019.

Research output: Working paper

TY - UNPB

T1 - On the construction of confidence intervals for ratios of expectations

AU - Derumigny, Alexis

AU - Girard, Lucas

AU - Guyonvarch, Yannick

N1 - 59 pages

PY - 2019/4/10

Y1 - 2019/4/10

N2 - 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.

AB - 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.

KW - Delta-method

KW - Confidence regions

KW - Uniformly valid inference

KW - Sequence of models

KW - Nonparametric percentile bootstrap

M3 - Working paper

BT - On the construction of confidence intervals for ratios of expectations

ER -