Order restricted inference over countable preordered sets. Statistical aspects of neutron detection

Sammanfattning: This thesis consists of four papers. In the first paper, we study the isotonic regression estimator over a general countable preordered set. We obtain the limiting distribution of the estimator and study its properties. Also, it is shown that the isotonisation preserves the rate of convergence of the underlying estimator. We apply these results to the problems of estimation of a bimonotone regression function and estimation of a bimonotone probability mass function.In the second paper, we propose a new method of estimating a discrete monotone probability mass function. We introduce a two-step procedure. First, we perform a model selection introducing the Akaike-type information criterion (CMAIC). Second, using the selected class of models we construct a modified Grenander estimator by grouping the parameters in the constant regions and then projecting the grouped empirical estimator onto the isotonic cone. It is shown that the post-model-selection estimator performs asymptotically better, in $l_{2}$-sense, than the regular Grenander estimator. In the third paper, we use a stochastic process approach to determine the neutron energy in a novel detector. The data from a multi-layer detector consists of counts of the number of absorbed neutrons along the sequence of the detector's layers, in which the neutron absorption probability is unknown. These results are combined with known results on the relation between the absorption probability and the wavelength to derive an estimator of the wavelength and to show consistency and asymptotic normality. In the forth paper, the results of the third paper are generalised to the case of a multimode Poisson beam. We study the asymptotic properties of the maximum likelihood estimator of the spectrum and thinning parameters for the spectrum's components.