Regresyon ve dağılım tahmininde geometrik modelleme

The contribution of this thesis to inferential statistics by using geometric modeling methods is comprised of three parts. First, we propose a new nonparametric regression model that uses the rational Bézier curves. The main advantages of the proposed regression model are the flexibility over parametric models and having no restriction on the assumptions in regression function estimation. Moreover, rational Bézier curves provide many conveniences such as more control over the shape of a curve and projective invariance. We provide numerical examples to demonstrate superiority of the proposed model and verify it. Secondly, we introduce a new distribution function estimation method using the rational Bernstein polynomials (RBP). The performance of the proposed new method is compared with Bernstein polynomials and empirical distribution function methods using simulation. The new method guarantees monotone nondecreasing function by applying linear constraints on the coefficients of the rational Bernstein basis functions and smooth the empirical distribution function. Furthermore, as a special case, it reduces to Bernstein polynomial estimator method. Some theoretical properties of the new estimator are investigated. Simulation study shows that the proposed estimator is preferable to the Bernstein polynomials and empirical distribution function estimator methods. Thirdly, we introduce a unimodal density estimator based on the Schoenberg's spline operator and thus generalize that of Bernstein polynomials and the beta density. The advantage of this method is the local property. That is, refining the knots while keeping the degree fixed of B-splines yields better estimates. We also give a numerical example to verify our results.