Second Order Chebyshev–Edgeworth-Type Approximations for Statistics Based on Random Size Samples

This article completes our studies on the formal construction of asymptotic approximations for statistics based a random number observations. Second order Chebyshev–Edgeworth expansions asymptotically normally or chi-squared distributed from samples with negative binomial Pareto-like sample sizes are obtained. The results can have applications wide spectrum chi-square statistics. Random, non-random, and mixed scaling factors each studied produce three different limit distributions. In addition to expected normal distributions, Student’s t-, Laplace, Fisher, gamma, weighted sums generalized gamma distributions also occur.