Problems in which a density or regression function is known to be smooth and to satisfy shape restrictions, like monotonicity or convexity, are investigated. Shape restricted regression and smoothing splines offer one promising approach to the problem, and the combination of shape restrictions with local polynomial methods another. In large samples the emphasis is on asymptotic distributions. With a lot of smoothing the contribution of the shape restrictions to the asymptotic distribution is negligible; with only a little smoothing, the shape restrictions dominate, leading to non-normal asymptotic distributions. The nature of the transition between a little smoothing a lot is investigated. In many cases, especially with smoothing splines, an estimator is the solution to a differential equation. In such cases it is possible to use the Green's function to construct an asymptotically equivalent kernel estimator from which the asymptotic distribution may be found. In moderate samples, the shape restrictions do affect the distribution of estimators, even in the presence of substantial smoothing. The size of this effect is investigated both analytically, using techniques of shrinkage estimation, and by simulation. Problems with known inequalities for parameters also arise in models with only a few parameters, like variance components, and classical confidence intervals can be empty, or degenerate, in such problems. In current work interest centers on finding Bayesian credible intervals with good frequentist properties.
The work on shape restricted density estimation and regression is motivated by a project to map the distribution of dark matter in nearby dwarf spheroidal galaxies, like Ursa Minor. Dark matter is matter that cannot be seen. Its existence is inferred from gravitational effects, but what it is comprised of is an open question. It is expected that knowing where the dark matter is may shed some light on what it is. The work on models with few parameters is motivated by the problem of disentangling signal and background events in both physics and astronomy. The confidence interval problem is an important part of the search for an elementary particle, the Higgs, in particular, and has been a featured topic at several recent meetings of high-energy physicists. The investigator is using these two examples and others in an interdisciplinary seminar on Statistics in the physical sciences.
