Inference and Density Estimation with Interval Statistics

Individual data from a continuous distribution are often partitioned into a collection of intervals defined by either fixed interval limits or sample quantiles. In this study, we derive asymptotic distribution of interval statistics for both cases, allowing multiple sample statistics for each interval. Under fixed intervals, the covariance matrix is singular. We identify a computationally simple non-singular generalized inverse for corresponding χ2 test and density estimator. For quantile intervals, the interval limits are stochastic, which complicates the asymptotic distribution. We use an influence function approach to derive the joint distribution of interval limits and other interval statistics. We the propose minimum distance estimators of the underlying distribution based on interval statistics. Asymptotic properties of the proposed estimators are established. Monte Carlo simulations suggest that the proposed estimators provide good finite sample performance. An empirical example on income distribution estimation based on interval statistics is presented. JEL Classification: C13,C16,C25