A Topologically Valid Definition of Depth for Functional Data

The main focus of this work is on providing a formal definition of statistical depth for functional data on the basis of six properties, recognising topological features such as continuity, smoothness and contiguity. Amongst our depth defining properties is one that addresses the delicate challenge of inherent partial observability of functional data, with fulfillment giving rise to a minimal guarantee on the performance of the empirical depth beyond the idealised and practically infeasible case of full observability. As an incidental product, functional depths satisfying our definition achieve a robustness that is commonly ascribed to depth, despite the absence of a formal guarantee in the multivariate definition of depth. We demonstrate the fulfillment or otherwise of our properties for six widely used functional depth proposals, thereby providing a systematic basis for selection of a depth function

Distributed Estimation and Inference with Statistical Guarantees

This paper studies hypothesis testing and parameter estimation in the context of the divide and conquer algorithm. In a unified likelihood based framework, we propose new test statistics and point estimators obtained by aggregating various statistics from $k$ subsamples of size $n/k$, where $n$ is the sample size. In both low dimensional and high dimensional settings, we address the important question of how to choose $k$ as $n$ grows large, providing a theoretical upper bound on $k$ such that the information loss due to the divide and conquer algorithm is negligible. In other words, the resulting estimators have the same inferential efficiencies and estimation rates as a practically infeasible oracle with access to the full sample. Thorough numerical results are provided to back up the theory

Improving confidence set estimation when parameters are weakly identified

We consider inference in weakly identified moment condition models when additional partially identifying moment inequality constraints are available. We detail the limiting distribution of the estimation criterion function and consequently propose a confidence set estimator for the true parameter.National Science Foundation of China (Grant ID: 71301026)This is the author accepted manuscript. The final version is available from Elsevier via http://dx.doi.org/10.1016/j.spl.2016.06.01

Dynamics of value-tracking in financial markets

The effciency of a modern economy depends on value-tracking: that market prices of key assets broadly track some underlying value. This can be expected if a suffcient weight of market participants are valuation- based traders, buying and selling an asset when its price is, respectively, below and above their well-informed private valuations. Such tracking will never be perfect, and we propose a natural unit of tracking error, the 'deciblack' . We then use a simple discrete-time model to show how large tracking errors can arise if enough market participants are not valuation-based traders, regardless of how much information the valuation-based traders have. Similarly to Lux [17] and others who study subtly different models, we find a threshold above which value-tracking breaks down without any changes in the underlying value of the asset. We propose an estimator of the tracking error and establish its statistical properties. Because financial markets are increasingly dominated by non-valuation-based traders, assessing how much valuation-based investing is required for reasonable value tracking is of urgent practical interest

Functional Symmetry and Statistical Depth for the Analysis of Movement Patterns in Alzheimer’s Patients

Black-box techniques have been applied with outstanding results to classify, in a supervised manner, the movement patterns of Alzheimer’s patients according to their stage of the disease. However, these techniques do not provide information on the difference of the patterns among the stages. We make use of functional data analysis to provide insight on the nature of these differences. In particular, we calculate the center of symmetry of the underlying distribution at each stage and use it to compute the functional depth of the movements of each patient. This results in an ordering of the data to which we apply nonparametric permutation tests to check on the differences in the distribution, median and deviance from the median. We consistently obtain that the movement pattern at each stage is significantly different to that of the prior and posterior stage in terms of the deviance from the median applied to the depth. The approach is validated by simulation