Sampling errors and confidence intervals for order statistics: Implementing the family support act and welfare reform

Federal regulations recommend that states reimburse the child care costs of recipients of cash assistance and other low-income families, up to the 75th percentile of the market price of care in the relevant local market. The regulations recommend that states carry out surveys to estimate these 75th pcrcentiles. This estimation problem raises two major statistical issues: (1) picking a sample design that will allow one to estimate the 75th percentiles cheaply, efficiently and equitably; and (2) assessing the sampling variability of the estimates obtained. For the state of Massachusetts, we developed a sampling design that equalized the standard errors of the estimated percentiles across 65 distinct local markets. This design was selected because state administrators felt the public, child care providers and child advocates would find it equitable, thus limiting costly appeals. Estimation of standard errors for the sample 75th pcrcentiles requires estimation of the density of the population at the 75th percentile. We implement and compare a number of parametric and nonparametric methods of density estimation. A kernel estimator appeared to provide the most reasonable estimates. On the basis of the mean integrated squared error criterion, we selected the Epanechnikov kernel and the Shearther-Jones automatic bandwidth selection procedure. Because some of our sample sizes were too small to rely on asymptotics, we also constructed nonparametric confidence intervals using the hypergeometric distribution. For most of our samples, these confidence intervals were similar to those based on the asymptotic standard errors. Substantively, we find wide variation in the price of child care, depending on age of the child, type of care and geographic location. For full-time care, the estimated 75th percentiles ranged from $242 per week for infants in child care centers in Boston to $85 per week for family child care in western Massachusetts.