Confidence intervals in Cavalieri sampling

The aim of this study is to derive a formula that allows the prediction, from a Cavalieri data sample, of an appropriate confidence interval for a parameter Q. Two different approaches are used to address the problem. The first approach is to investigate whether the asymptotic distribution of the Cavalieri estimator exists when the sampling period T tends to zero. In particular, the distribution of the standardized version of the Cavalieri estimator z(T) is analysed for a measurement function f whose smoothness constant is an integer number m. The analysis reveals that when the first noncontinuous derivative of f, f((m)), exhibits a unique discontinuity, the asymptotic distribution of z(T) exists and it has a bounded support. An analytical expression of the distribution is derived for the cases m = 0 and 1. However, when f((m)) has two or more discontinuities, the asymptotic distribution of z(T) does not exist and its support may be unbounded. In the second approach, a generalized version of the refined Euler Mac-Laurin summation formula, valid for measurement functions with a fractional, rather than just an integer, smoothness constant, is applied to the Cavalieri estimator. As a result, a formula that predicts a lower and upper bound for the true parameter is derived for small T. This bound prediction formula is applied to Cavalieri data samples of human cerebral cortex. In particular, for sample sizes n = 8, 12 and 16, the true volume of cerebral cortex is bounded by relative distances 8%, 4% and 2% of the Cavalieri estimate, respectively.