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Abstract
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 twostep procedure. First, we perform a model selection introducing the Akaiketype 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 postmodelselection 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 multilayer 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.
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 twostep procedure. First, we perform a model selection introducing the Akaiketype 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 postmodelselection 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 multilayer 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.
Original language  English 

Qualification  Doctor 
Awarding Institution 

Supervisors/Advisors 

Award date  2018 Oct 5 
Place of Publication  Lund 
Publisher  
Print ISBNs  9789177538080 
Electronic ISBNs  9789177538097 
Publication status  Published  2018 Sep 
Bibliographical note
Defence detailsDate: 20181005
Time: 13:15
Place: MH:G Matematikhuset, Sölvegatan 18, Lund
External reviewer(s)
Name: Dümbgen, Lutz
Title: Professor
Affiliation: Institute of Mathematical Statistics and Actuarial Science, University of Bern, Switzerland

Subject classification (UKÄ)
 Probability Theory and Statistics
Keywords
 Constrained inference
 Isotonic regression
 Density estimation
 Grenander estimator
 Limit distribution
 Neutron detection
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 1 Examination

Order restricted inference over countable preordered sets. Statistical aspects of neutron detection.
Krzysztof Nowicki (Examiner)
2018 Oct 5Activity: Examination and supervision › Examination