Cramér Type Moderate Deviation Theorems for Self-Normalized Processes
نویسندگان
چکیده
Cramér type moderate deviation theorems quantify the accuracy of the relative error of the normal approximation and provide theoretical justifications for many commonly used methods in statistics. In this paper, we develop a new randomized concentration inequality and establish a Cramér type moderate deviation theorem for general self-normalized processes which include many well-known Studentized nonlinear statistics. In particular, a sharp moderate deviation theorem under optimal moment conditions is established for Studentized U -statistics.
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Let X1, X2, . . . be independent random variables with zero means and finite variances, and let Sn = ∑n i=1Xi and V 2 n = ∑n i=1X 2 i . A Cramér type moderate deviation for the maximum of the self-normalized sums max1≤k≤n Sk/Vn is obtained. In particular, for identically distributed X1, X2, · · · , it is proved that P(max1≤k≤n Sk ≥ xVn)/(1 − Φ(x)) → 2 uniformly for 0 < x ≤ o(n1/6) under the opt...
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