Convex Density Function Estimation

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ProQuest, 2008 - 104 pages
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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

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