Grant: Discovery Grants Program - Individual

The objective of this research is to develop novel inference methods in two related areas: 1) Inference for heavy tailed distributions; 2) nonparametric inference (for general distributions). 1. Inference for Heavy Tailed Distributions Extreme events occur in financial markets, natural disasters, disease control and industrial quality control, which affect human life. It is important but difficult to study these extreme events by using suitable mathematical models. Heavy tailed distributions comprise one class of the extreme value distributions which has been applied widely to risk analysis with applications in economics, industrial engineering, actuarial science, medical research and networks. There are theoretical difficulties in the inference of heavy tailed distributions. The proposed program will explore innovative methods to overcome these difficulties. (1) Heavy tailed data is often complicated. A single distribution may not fit data well. Two new cluster and sieve methods are proposed, which are found to e generally better fitting compared with existing methods. (2) Estimation of a high quantile (Value-at-Risk) of a heavy tailed distribution is an important but difficult problem. Existing inference methods have bias and deficiency problems. A geometric mean method is proposed. Preliminary work shows that the new method yields improvements. Also, I will study estimation for high conditional quantiles (quantile regression). A weighted loss function to produce more accurate predictions is proposed. (3) I will explore innovative approximation methods for heavy tailed distributions, e.g., truncation, level crossing, generalized hyperexponential algorithm, etc. 2. Nonparametric Inference (for general distributions) Nonparametric methods are superior to parametric methods on data sets having complex distributions, e.g. multiple modes. However, compared to parametric methods, nonparametric methods may lack efficiency on tails of the distribution or have technique difficulties. The proposed program will develop novel methods to overcome these problems. (4) A family of weighted empirical distribution functions will be studied. I will explore the optimal weights that minimize the estimation errors to improve tail estimation, by utilizing criteria for measurement of the errors, e.g., Lp-norm, exceedance measure and Hellinger distance. (5) The classical kernel methods have selection problems of bandwidth and kernel. I will develop some non-kernel methods by using Hermite orthogonal series, wavelet and L-statistics to avoid these difficulties and keep good properties. Measurements of the proposed methods (1) to (5) will be: a) Comparing the proposed methods with existing methods on theoretical properties: efficiency, consistency, rate of convergence and robustness; b) Studying Monte Carlo simulations to search for good methods and confirm theoretical results; c) Applying the proposed methods to real problems, to find a best model fitting the data; estimating value-at-risk, survival functions, waiting time in networks and performing goodness of fit tests. Scientific Approach: a) The theoretical approach includes probability, statistical theory, order statistics, stochastic processes, harmonic analysis and approximation theory. b) The computational approach includes: Monte Carlo, bootstrapping, computer programming. Expected Impact: The work will provide an alternative approach for statistical inference. The results are expected to overcome some difficulties in the field, to explore and suggest new ideas for future research, and to benefit Canadian research on extreme values field. This research will contribute to solving real world problems and obtaining accurate solutions.