Nonparametric Density Estimation Methods with Application to the U.S. Crop Insurance Program
Theoretically, the bias reduction and the variance reduction nonparametric density estimators could have but not yet been combined. Practically, accurate estimation of premiums is a necessary condition for the financial solvency of the U.S. crop insurance program as well as to mitigate, to the extent possible, problems of moral hazard and adverse selection. Typically, premiums are derived from crop yield density which is parametrically or nonparametrically estimated by historical data. Unfortunately, valid historical yield data is limited. To meet these challenges, I develop two novel nonparametric density estimators denoted as Comb1 and Comb2 which selectively borrow information from extraneous sources. They have the advantage to reduce not only the estimation bias but also variance. By combining bias reduction and variance reduction estimators in different ways, Comb1 unifies the standard kernel estimator, Jones’ bias reduction estimator and Ker’s possibly similar estimator while Comb2 unifies the standard kernel estimator, Ker’s possibly similar estimator, and different from Comb1, the conditional estimator. Numerical simulations suggest the two proposed estimators outperform a number of existing methods: if the true densities are known, Comb1 and Comb2 are far ahead with no obvious peers; if the true densities are assumed to be unknown and bandwidths are selected by maximum likelihood cross-validation, Comb1 and Comb2 still have promising performance, especially Comb2. Finally, the two estimators are applied to rate crop insurance contracts over the alternatives methods in an out-of-sample simulation game. Statistically and economically significant improvements are found. Given the size of the crop insurance program, updating the government’s density estimation method to Comb1 or Comb2 may potentially save enormous amount of taxpayers’ money. Sensitivity analysis where only data of the most recent 25 or 15 years is used suggests the findings are robust to missing historical yield data, implying that by adopting Comb1 or Comb2 the Supplemental Coverage Option, a new crop insurance option that provides additional coverage to farmer, could potentially be expanded to crops and areas with significantly less historical data.