# Convex Density Function Estimation

ProQuest, 2008 - 104 pages
The main goal of the thesis is density estimation. The basic framework is as follows. Let X be a random variable with an unknown convex probability density function. Let C be the collection of all convex probability density functions on the real line. The set C is infinite dimensional. For any given data drawn on X, one can write down the likelihood of the data as a function on C . Maximizing the likelihood over C is an infinite dimensional problem. In the dissertation, we show that the maximum of the likelihood is attained at a piecewise linear convex function on a bounded domain. The maximization problem simplifies considerably and it involves linear constraints. Exploiting the theory of generalized inverses of matrices and isotonic regression, we solve the simplified maximization problem. Consistency of the of the solution has been established. The case of ties in the data has been tackled. A computer code using R and Python is developed.

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### Contents

 INTRODUCTION 1 is the i th catching time of a bird and X is the resting time of a bird on the island 4 Convexity property 7 Convex function discontinuous at the boundary points 8 Examples of convex density functions 9 Example of a piecewise linear function 11 Another example of a piecewise linear function 12 Area under the piecewise convex function 14
 Convex piecewise density function 56 Example of a fourpoint piecewise linear function which is nonconvex 57 Example of a fourpoint piecewise linear function solution which is non convex 63 Convex modification of g of g 64 An example of a fourpoint convex piecewise linear density function restored 65 TIES 67 Optimal solution of density estimation in the case of ties 77 CONSISTENCY OF THE ESTIMATOR 82

 DISTINCT POINTS 28 A convex piecewise linear density function 31 Constraint set of the 3data point optimization problem 35 3data point optimization problem 37
 Convex piecewise density function 83 SUMMARY AND FUTURE WORK 89 Copyright