Nonparametric inference with applications to dark matter estimation in astronomy and degradation modeling in reliability

This dissertation deals with nonparametric inference methods that arise in two major areas of application: astronomy and reliability engineering. In both of these areas, extensive amounts of data are now routinely being collected. Nonparametric methods are especially useful in such environments.; The first part of this dissertation focused on mapping the distribution of dark matter in galaxies close to the Milky Way. One interesting and difficult problem is an inverse problem in which one attempts to infer the mass distribution of the galaxy from the projected positions and line-of-sight velocities of the stars. Let X = (X1, X2, X3) and V = (V1, V2, V3) denote the 3-dimensional position and velocity of a star within a galaxy. We observe only a sample of (X 1, X2, V3) with a proper choice of coordinates. With the assumptions of spherical symmetry and velocity isotropy, estimating the mass distribution tends to determine the second derivative of function Psi, which is subject to certain shape restrictions: non-negative, decreasing and convex. The new procedure of estimation we have developed combines spline techniques with the isotonic methods to obtain a consistent estimate of mass distribution. Within the context of dark matter distributions, an important quantity to estimate is the velocity dispersion, E&parl0;∥V∥ 2&vbm0;X=x&parr0; . We extend the isotonic analysis in Groneboom and Jongbloed (Ann. of Stat., 1995) to estimate a regression function. The main result is a version of the Kiefer-Wolfowitz theorem comparing the empirical distribution to its least concave majorant, but with a faster rate, n -1 log n, of convergence than n -2/3 log n. The main result is useful in obtaining asymptotic distributions for different estimators such as isotonic or kernel estimators.; The second part of the dissertation is aimed at developing a class of degradation models based on non-homogeneous Levy processes. Let X (t) be a homogeneous Levy process with mean t and variance betat where beta is an unknown parameter. We consider degradation models obtained through time-transformations of the form Y(t) = X(Λ( t)) for some nondecreasing function Λ(t). This yields a very flexible class of models that accommodate a variety of degradation shapes including increasing, decreasing, and constant degradation rates. We consider nonhomogeneous Gaussian processes with X( t) given by a Brownian motion with linear drift and nonhomogeneous Gamma process as the special cases of Levy process. If we define time-to-failure as the first-passage time, this model leads to tractable time-to-failure distributions. This allows us to combine degradation and time-to-failure data for inference. Nonparameric methods are used to estimate the degradation function Λ( t) based on different types of degradation data (destructive and non-destructive), time-to-failure data, and different sample configurations (number of degradation pathes leading to infinity, number of sample points in a degradation path going to infinity, etc.). The asymptotic behavior is used for model selection.