## Convex Density Function EstimationThe 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 |

### Common terms and phrases

an_i an-i B(si bounded interval Chapter column vector consistent constraint set CONVEX DENSITY ESTIMATION convex density functions convex function convex piecewise linear convex probability density cut points density estimation problem density f(x distribution function domain of definition exists a convex f(xi following result function g GA)z given in Figure graph Grenander Guido Van Rossum ideal sample infinite dimensional inverse isotonic regression Let f Let g likelihood function linear constraints linear density function linear equations linear probability density maximization problem maximize L(f maximum likelihood estimator monotonically increasing non-negativity constraints nonparametric numbers objective function order statistics piecewise linear convex piecewise linear density piecewise linear function piecewise linear probability probability density function Proof random sample Sedge Warblers slopes solution given solution set solve system of equations system of linear Un-l unknown density Vn-l Wellner x-axis Figure zn-i Zn-l