Re-Examining Confidence Intervals for Ratios of Parameters

This paper considers the problem of constructing confidence intervals (CIs) for nonlinear functions of parameters, particularly ratios of parameters a common issue in econometrics and statistics. Classical CIs (such as the Delta method and the Fieller method) often fail in small samples due to biased parameter estimators and skewed distributions. We extended the Delta method using the Edgeworth expansion to correct for skewness due to estimated parameters having non-normal and asymmetric distributions. The resulting bias-corrected confidence intervals are easy to compute and have a good coverage probability that converges to the nominal level at a rate of O(n−1/2) where n is the sample size. We also propose bias-corrected estimators based on second-order Taylor expansions, aligning with the “almost unbiased ratio estimator” . We then correct the CIs according to the Delta method and the Edgeworth expansion. Thus, our new methods for constructing confidence intervals account for both the bias and the skewness of the distribution of the nonlinear functions of parameters. We conduct a simulation study to compare the confidence intervals of our new methods with the two classical methods. The methods evaluated include Fieller’s interval, Delta with and without the bias correction interval, and Edgeworth expansion with and without the bias correction interval. The results show that our new methods with bias correction generally have good performance in terms of controlling the coverage probabilities and average length intervals. They should be recommended for constructing confidence intervals for nonlinear functions of estimated parameters.