Sequential nonparametric estimation via Hermite series estimators

Algorithms for estimating the statistical properties of streams of data in real time, as well as for the efficient analysis of massive data sets, are becoming particularly pertinent given the increasing ubiquity of such data. In this thesis we introduce novel approaches to sequential (online) estimation in both stationary and non-stationary settings based on Hermite series density estimators. In the univariate context we apply Hermite series based distribution function estimators to sequential cumulative distribution function estimation. These distribution function estimators are particularly useful because they allow the sequential estimation of the full cumulative distribution function. This is in contrast to the empirical distribution function estimator and smooth kernel distribution function estimator which only allow sequential cumulative probability estimation at predefined values on the support of the associated density function. We explore the asymptotic consistency and robustness properties of the Hermite series based cumulative distribution function estimator thereby redressing a gap in the literature. Given the sequential Hermite series based distribution function estimator, we obtain sequential quantile estimates numerically. Our algorithms go beyond existing sequential quantile estimation algorithms in that they allow arbitrary quantiles (as opposed to pre-specified quantiles) to be estimated at any point in time, in both the static and dynamic quantile estimation settings. In the bivariate context we introduce a Hermite series based sequential estimator for the Spearman's rank correlation coefficient and provide algorithms applicable in both the stationary and non-stationary settings. To treat the the non-stationary setting, we introduce a novel, exponentially weighted estimator for the Spearman's rank correlation, which allows the local nonparametric correlation of a bivariate data stream to be tracked. To the best of our knowledge this is the first algorithm to be proposed for estimating a time-varying Spearman's rank correlation that does not rely on a moving window approach. We explore the practical effectiveness of the Hermite series based estimators through real data and simulation studies, demonstrating competitive performance compared to leading existing algorithms. The potential applications of this work are manifold. Our sequential distribution function and quantile estimation algorithms can be applied to real time anomaly and outlier detection, real time provisioning for future demand as well as real time risk estimation for example. The Hermite series based Spearman's rank correlation estimator can be applied to fast and robust online calculation of correlation which may vary over time. Possible machine learning applications include fast feature selection and hierarchical clustering on massive data sets amongst others.

Grantor

Department of Statistical Sciences

Year dc:date.issued

2020

Author dc:creator

• Stephanou, Michael Jared

Advisor dc:contributor.advisor

• Varughese, Melvin

• statistical sciences

Handle dc:identifier.uri

http://hdl.handle.net/11427/32998

OAI identifier oai:identifier

oai:open.uct.ac.za:11427/32998