Non-Gaussian semi-stable distributions and their statistical applications Public Deposited

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  • March 22, 2019
Creator
  • Chaudhuri, Ritwik
    • Affiliation: College of Arts and Sciences, Department of Statistics and Operations Research
Abstract
  • The dissertation is motivated by problems arising in modern communication networks such as the Internet. Over these networks, information is sent in the form of data packets which are further grouped into flows. For example, a flow can be associated with a certain (document, music, movie or other) file. Knowing the structure of flows is of great interest to network operators and networking researchers. One quantity of particular interest is the distribution of flow sizes (the number of packets in a flow). Each packet carries information on the flow it belongs to. Hence, examining all packets allows reconstructing and studying the associated flows. Examining all packets, however, is becoming cumbersome due to the ever increasing amount of data and processing costs. To overcome these issues, packet sampling has become prevalent. One common sampling scheme is probabilistic sampling wherein each packet is sampled independently and with the same probability. The basic problem then becomes inference of the characteristics of original flows (e.g.\\ the flow size distribution) from sampled packets (forming sampled flows). This problem, known as an inversion problem, has attracted much attention in the networking community. In particular, a well-known nonparametric estimator of the flow size distribution is available under probabilistic sampling, based on sampled packets and sampled flows. From the application perspective, the focus of the dissertation is on some statistical properties of this nonparametric estimator. Under suitable and restrictive assumptions, the estimator has been known to be asymptotically normal. Going beyond these assumptions, it is shown in the dissertation that the estimator can be asymptotically semi-stable. To achieve this goal, the domains of attraction of semi-stable distributions are reexamined here. As a main theoretical contribution, general, sufficient and practical conditions are provided for a distribution to be in the domain of attraction of a semi-stable distribution. They lead to practical conditions for the aforementioned nonparametric estimator to be attracted to a semi-stable law. Examples of probability distributions and illustrations of the main results are provided throughout the dissertation. One practical consequence of the results is a confidence interval for the distribution of flow sizes, based on critical values of semi-stable distributions. Semi-stable distributions do not have closed forms in general. In order to compute their critical values, numerical calculation of semi-stable densities is also considered in the dissertation. This is carried out by using a celebrated method of Joseph Abate and Ward Whitt, allowing numerical calculation of a density given its characteristic function (Laplace transform). The code implementing the method for semi-stable densities is included. The numerically calculated densities are used to assess the goodness of approximations involving semi-stable distributions and, as indicated above, the computation of confidence intervals. These points are explored in numerical simulations throughout the dissertation. Finally, some multivariate extensions of the results and further directions are also discussed.
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  • In Copyright
Advisor
  • Pipiras, Vladas
Degree
  • Doctor of Philosophy
Graduation year
  • 2014
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