On Statistical Diagnostics For Nonlinear Quantile Regression

Quantile regression models are important statistical models, which are more powerful than least square (LS) regression and least absolute deviation (LAD) regression in statistical analysis, and can supply much more attractive inference results for the datasets. Regression quantiles are robust against the influence of outliers and, taken several at a time, they give a more complete picture of the conditional distribution than a single estimate of the center, and they are more efficient than the least square estimate when the data comes from heavy-tailed distribution or the mixture of several different distributions. So quantile regression models and regression quantiles have been noticed and favoured by the statistists and economists while they appeared. This thesis is devoted to investigate the statistical diagnostics, particularly the influence analysis, for nonlinear quantile regression.In chapter 2, the essential concepts and main properties of nonlinear quantile regression model and linear regression quantiles are introducted, and the MM algorithm is given to compute nonlinear regression quantiles.In chapter 3, under quite common regular assumputions, we show the equivalence theorem of case deletion model (CDM) and mean shift outlier model (MSOM) for a wide class of statistical models which adopt the M-estimate and include quantile regression and LS regression. Based on the objective function of the MM algorithm, the same theorem is proved for quantile regression, which is valuable in practice. γ|^ = e_i|^(i) is a byproduct of the equivalence theorem, which is used to demonstrate the relationship between CDM and MSOM. According to Bayesian method, it is proved that the estimates of CDM and MSOM are not equal in a wide class of statistical models when 7 has informative prior.In chapter 4, firstly, we obtain the likelihood displacement LD_τi(β,σ), LD_τi(β,σ) and LD_τi(σ|β) for nonlinear quantile regression models when the random errors are independently distributed asymmetric Laplace density, i.e. ALA_τ(0, σ). Secondly, Cook distance and quasi-likelihood displacement are proposed from the view of the confidence region of large sample, and a new influence measure, i.e. the MM distance, is also proposed based on the new objective function Q_τ^ε(β|β_τ|^) Thirdly, the approximate equivalent formula between the three likelihood displacements is obtained in a wide class of statistical models which include nonlinear quantile regression model. Finally, several other diagnostic measures are also considered for nonlinear quantile regression.In chapter 3 and chapter 4, many datasets are analyzed to illustrate the accuracy and practice of the theories and diagnostic methods proposed in this thesis.