A Non-iterative Method for Fitting the Single Index Quantile Regression Model with Uncensored and Censored Data PDF Download

Author: Eliana Christou
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Languages : en
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Book Description
Quantile regression (QR) is becoming increasingly popular due to its relevance in many scientific investigations. Linear and nonlinear QR models have been studied extensively, while recent research focuses on the single index quantile regression (SIQR) model. Compared to the single index mean regression (SIMR) problem, the fitting and the asymptotic theory of the SIQR model are more complicated due to the lack of closed form expressions for estimators of conditional quantiles. Consequently, existing methods are necessarily iterative. We propose a non-iterative estimation algorithm, and derive the asymptotic distribution of the proposed estimator under heteroscedasticity. For identifiability, we use a parametrization that sets the first coefficient to 1 instead of the typical condition which restricts the norm of the parametric component. This distinction is more than simply cosmetic as it affects, in a critical way, the correspondence between the estimator derived and the asymptotic theory. The ubiquity of high dimensional data has led to a number of variable selection methods for linear/nonlinear QR models and, recently, for the SIQR model. We propose a new algorithm for simultaneous variable selection and parameter estimation applicable also for heteroscedastic data. The proposed algorithm, which is non-iterative, consists of two steps. Step 1 performs an initial variable selection method. Step 2 uses the results of Step 1 to obtain better estimation of the conditional quantiles and, using them, to perform simultaneous variable selection and estimation of the parametric component of the SIQR model. It is shown that the initial variable selection method of Step 1 consistently estimates the relevant variables, and that the estimated parametric component derived in Step 2 satisfies the oracle property. Furthermore, QR is particularly relevant for the analysis of censored survival data as an alternative to proportional hazards and the accelerated failure time models. Such data occur frequently in biostatistics, environmental sciences, social sciences and econometrics. There is a large body of work for linear/nonlinear QR models for censored data, but it is only recently that the SIQR model has received some attention. However, the only existing method for fitting the SIQR model uses an iterative algorithm and no asymptotic theory for the resulting estimator of the Euclidean parameter is given. We propose a new non-iterative estimation algorithm, and derive the asymptotic distribution of the proposed estimator under heteroscedasticity.