Exact bounds on the truncated-tilted mean, with applications

Exact upper bounds for ${\mathbf{E} Xe^{h(X\wedge w)}}/{\mathbf{E} e^{h(X\wedge w)}}$, which is the expectation of the Cramér transform of the so-called Winsorized-tilted mean of a random variable, are given in terms of its first two moments. Such results are needed in work with nonuniform Berry–Esseen-type bounds for general nonlinear statistics. As another application, optimal upper bounds on the Bayes posterior mean are provided. Certain monotonicity properties of the tilted mean are also presented.