Efron's asymptotic monotonicity property in the Gaussian stable domain of attraction

In Efron (1965), Efron studied the stochastic increasingness of the vector of independent random variables entering a sum, given the value of the sum. Precisely, he proved that log-concavity for the distributions of the random variables ensures that the vector becomes larger (in the sense of the usual multivariate stochastic order) when the sum is known to increase. This result is known as Efron's monotonicity property. Under the condition that the random variables entering in the sum have density functions with bounded second derivatives, we investigate whether Efron's monotonicity property generalizes when sums involve a large number of terms to which a central-limit theorem applies. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

The study shows that when the distribution of random variables exhibits log-concavity, the vector becomes larger as the sum increases, known as Efron's monotonicity property. Additionally, under the condition of random variables with density functions and bounded second derivatives, the research explores whether Efron's monotonicity property generalizes to sums involving a large number of terms.