Shape-restricted Density Estimation for Financial Data

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dc.contributor.advisor Wang, Y en
dc.contributor.advisor Ziedins, IB en Liu, Yu en 2016-09-06T22:04:38Z en 2016 en
dc.identifier.uri en
dc.description.abstract The motivation of the study in this thesis is about how to estimate an asset return distribution in finance that is often skewed, high-peaked and heavy-tailed. To avoid misspecification which is possible for a parametric model, we turn to nonparametric methods to estimating a density function. There are many nonparametric approaches, such as kernel-based and penalty methods, to solving estimation problems, but they may easily fail to satisfy some practically known properties or have difficulty in choosing the value of the bandwidth or tuning parameter. By contrast, one can avoid these issues by imposing certain shape constraints on the density function, that appear very reasonable from a practical point of view. Nonparametric density estimation under shape restrictions offer many advantages, such as having the required shapes, easily described fitted models and possibly a higher estimation efficiency. Specifically, we are interested in estimating a density function that is log-concave, or unimodal with heavy tails. Three main objectives are addressed in this thesis. Firstly, nonparametric maximum likelihood estimation of a log-concave density function is investigated. In particular, a new fast algorithm is proposed and studied for computing the nonparametric maximum likelihood estimate of a logconcave density. Theoretically, the characterization of the nonparametric maximum likelihood estimate is studied and the algorithm is guaranteed to converge to the unique maximum likelihood estimate under log-concavity constraints. Numerical studies show that it outperforms other algorithms that are available in the literature. Tests for log-concavity based on the new algorithm are also developed. Secondly, nonparametric estimation under smoothness and log-concavity shape assumptions is studied. We propose several new smooth estimators based on the maximum likelihood approach by employing piecewise quadratic functions for the log-density function. This leads us to define a log-concave distribution family that allows the second derivative of the log-density to change the direction of monotonicity at most once. Algorithms for these likelihood maximization problems are developed. Numerical studies of simulated and real-world data show that the new smooth estimator has the best performance of all nonparametric estimators studied. We also apply our smooth estimator to the receiver operating characteristic curve estimation, with good results obtained. Finally, we study the problem of estimating a unimodal, highly heavy-tailed distribution, as normally seen in financial data. A novel idea is proposed that it imposes log-concavity on the main body, and log-convexity on the tails. With the corresponding algorithm developed, the new shape-restricted estimator very much dominates the other ones for both simulated and real-world financial data, by providing excellent, nonparametric fits to the data in both the center and tails of the distribution. Bootstrap testing for identifying the function form implied by the new estimator has been developed. Tail performance is further studied in great detail and an application to Value-at-risk estimation is investigated. As a matter of fact, the study provides a very general approach to nonparametric density estimation under shape restrictions. Different pieces of shape restrictions can be combined easily in a seamless way, with fast computing algorithms available. Shape-restricted estimation is able to provide more accurate estimates compared with unconstrained estimates, and the work reported in this thesis lies a promising foundation for many more shape-restricted estimation methods to be developed and applied in the future. en
dc.publisher ResearchSpace@Auckland en
dc.relation.ispartof PhD Thesis - University of Auckland en
dc.relation.isreferencedby UoA99264880511802091 en
dc.rights Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated. Previously published items are made available in accordance with the copyright policy of the publisher. en
dc.rights.uri en
dc.rights.uri en
dc.title Shape-restricted Density Estimation for Financial Data en
dc.type Thesis en Statistics en The University of Auckland en Doctoral en PhD en
dc.rights.holder Copyright: The author en
dc.rights.accessrights en
pubs.elements-id 540827 en
pubs.record-created-at-source-date 2016-09-07 en

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