Estimating High Quantiles Based on Dependent Circular Data

Abstract

This paper gives an overview of the existing approaches for modelling high quantiles of dependent spatial data and apply the methods to the bivariate circular case. We also adapt the bootstrap to the situation at hand.

Since any data set one might use is finite, the interest lies in estimating a continuous curve as its upper limit (or quantile function). This can be obtained by either a kernel type regression or a fitted parametric model. We also introduce a new, more realistic correction formula for a nonparametric method for estimating the pointwise maximum (called frontier in this setup [8]).

An additional common challenge in real-life applications is the dependence among subsequent observations. The theoretical results about the GPD limit of the exceedances beyond high threshold remain valid under mixing-type conditions, called D(u_n) in the extreme-value literature. However, if one intends to use the bootstrap-based reliability estimators, then they need to be adjusted — e.g., by the block-bootstrap approach in [15]. We estimate the reliability of the estimators by a suitable application of the m out of n bootstrap, which turned out to be suitable for high-quantile estimation.

We illustrate the introduced methods for simulated as well as real enzymes data.