Abstract
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.
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