as tr o - ph / 9 51 01 50 30 O ct 1 99 5 DYNAMICAL MODELLING

Estimation of the distribution function f and potential of hot stellar systems from kinematical data is discussed. When the functional forms of f and are not speciied a priori, accurate estimation of either function requires very high quality data : either accurate \line prooles" at radii extending well beyond an eeective radius, or large samples (N > 10 3) of discrete radial velocities. Estimates of (r) based on much smaller data sets can be very strongly innuenced by assumptions, explicit or implicit, about the form of f. The importance of casting the estimation problem into a mathematically determined form is stressed. Some techniques for nonparametric estimation are presented, with some preliminary results of their application to real stellar systems. The problem of inferring the dynamical state of a hot stellar system like an elliptical galaxy from kinematical observations is an example of what statisticians call \inverse problems." The simplest sort of inverse problem consists of nding a function f that appears inside of an integral : g(x) = Z b a k(x; y)f(y) dy; (1) with g(x) and k(x; y) known functions. For instance, f might represent the intrinsic luminosity distribution of a star on the sky, k the smearing eeect of the earth's atmosphere, and g the observed image. However statisticians deene inverse problems more broadly, to include any problem that requires \making inferences about a phenomenon from partial or incomplete information" (O'Sullivan 1986). This deenition includes statistical estimation { that is, estimating the parameters of a function given a sample drawn from that function { as well as model building. They further divide the class of inverse problems into two types, \well-posed" and \ill-posed." Well-posed problems have { at least mathematically { unique solutions, and those solutions are possible to nd, in an approximate sense, even when the information is incomplete or imperfect. Much more common are ill-posed inverse problems : problems that have no unique solution, even given perfect or near-perfect data. The most common ill-posed problems are underdetermined ones; examples are nding the axis ratio distribution of triaxial galaxies from the distribution of Hubble types, calculating the distribution function of a spherical galaxy from its density proole, and inferring the 3-D velocity eld of a triaxial galaxy from line-of-sight velocities. To the extent that we see galaxies in projection, almost every attempt at model building begins as an underdetermined problem. However, even problems that are mathematically …