  We discuss properties of two methods for ascribing probabilities to the shape
of a probability distribution. One is based on the idea of counting the number
of modes of a bootstrap version of a standard kernel density estimator. We
argue that the simplest form of that method suffers from the same difficulties
that inhibit level accuracy of Silverman's bandwidth-based test for modality:
the conditional distribution of the bootstrap form of a density estimator is
not a good approximation to the actual distribution of the estimator. This
difficulty is less pronounced if the density estimator is oversmoothed, but the
problem of selecting the extent of oversmoothing is inherently difficult. It is
shown that the optimal bandwidth, in the sense of producing optimally high
sensitivity, depends on the widths of putative bumps in the unknown density and
is exactly as difficult to determine as those bumps are to detect. We also
develop a second approach to ascribing a probability to shape, using Muller and
Sawitzki's notion of excess mass. In contrast to the context just discussed, it
is shown that the bootstrap distribution of empirical excess mass is a
relatively good approximation to its true distribution. This leads to empirical
approximations to the likelihoods of different levels of ``modal sharpness,''
or ``delineation,'' of modes of a density. The technique is illustrated
numerically.
