Mini-course "Topics in Applied Probability:Tempered Stable Distributions, Ad Placement, and Turing's Formula"
6-17 October 2014 Assistand Proffesor UNC Charllote Michael Grabchak gave a lectures on
"Tempered Stable Distributions, Ad Placement, and Turing's Formula".
Lecture 1: Do financial returns have a finite or infinite variance? A paradox and an explanation.
Abstract: One of the major points of contention in studying and modeling financial returns is whether or not the variance of the returns is finite or infinite. The available empirical evidence can be, and has been, interpreted in more than one way. The apparent paradox, which has puzzled many a researcher, is that the tails appear to become less heavy for less frequent (e.g. monthly) returns than for more frequent (e.g. daily) returns, a phenomenon not easily explainable by the standard models. We provide an explanation of this paradox and show that, for financial returns, natural families of models are those with tempered heavy tails. These models can generate observations that appear heavy tailed for a wide range of aggregation levels before becoming clearly light tailed at even larger aggregation scales.
Lecture 2: Tempered stable distributions and processes.
Lecture 3, Part 1: Does value-at-risk encourage diversification when losses follow a tempered stable or more general Levy processes?
Lecture 3, Part 2: How do tempered stable distributions appear in applications?
Abstract: In this lecture we discuss limits theorems for two models of summation of iid random variables with a parameter. The parameter determines how heavy the tails of the distribution are. The first model explains how tempered stable distributions arise in applications. Our discussion is motivated by the problem of modeling mobility, which is important in applications to computer science, anthropology, and ecology.
Lecture 4: Applications of stochastic knapsack and multi-armed bandit problems to internet ad placement.
Lecture 5: Turing's formula and its statistical implications.
Abstract: Turing's formula is a nonparametric estimator for seeing an observation of a previously unobserved type. It can be thought of as an estimator of the tail. We will discuss this formula and show how the ideas behind it can be used for the estimation of entropy and related quantities. Applications to ecology and linguistics will be stressed.
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