Exact lower bounds on the exponential moments of min(y,X) and XI{X<y} are provided given the first two moments of a random variable X. These bounds are useful in work on large deviations probabilities and nonuniform Berry-Esseen bounds, when the Cram\ér tilt transform may be employed. Asymptotic properties of these lower bounds are presented. Comparative advantages of the Winsorization min(y,X) over the truncation XI{X<y} are demonstrated.
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