Optimal dependence on the count threshold

Determine whether the estimation bound for the count surprise probability functional can achieve linear, rather than quadratic, dependence on the count threshold parameter ζ.

Background

The count surprise probability functional estimates the probability that the unobserved next observation appears at most ζ times in the observed sample. The paper derives a leave-a-window-out estimator whose mean-squared error has quadratic dependence on ζ through the stability and variance terms.

The authors establish that the resulting rate is optimal in its dependence on the sample size and mixing-time parameter, but explicitly identify the dependence on ζ as unresolved. A sharper analysis or a different estimator would be needed to reduce the quadratic dependence to linear dependence.