The strong interest in recent years in analyzing chaotic dynamical systems
according to their asymptotic behavior has led to various definitions of fractal
dimension and corresponding methods of statistical estimation. In this paper we
first provide a rigorous mathematical framework for the study of dimension,
focusing on pointwise dimension a(x) and the generalized Renyi dimensions
D(q), and give a rigorous proof of inequalities first derived by Grassberger and
Procaccia and Hentschel and Procaccia. We then specialize to the problem of
statistical estimation of the correlation dimension v and inh~rmatiou dmlension
er. It has been recognized for some time that the error estimates accompanying
the usual procedures (which generally involve least squares methods and nearest
neighbor calculationsj grossly underestimate the true statistical error involved.
In least squares analyses of v and a we identify sources of error not previously
discussed in the literature and address the problem of obtaining accurate error
estimates. We then develop an estimation procedure for a which corrects for an
important bias term (the local measure density) and provides confidence inter-
vals for a. The general applicability of this method is illustrated with various
numerical examples.