On average derivative quantile regression. (English) Zbl 0898.62082

Chaudhuri, Probal; Doksum, Kjell; Samarov, Alexander

Ann. Stat. 25, No. 2, 715-744 (1997).

Summary: For fixed \(\alpha\in (0,1)\), the quantile regression function gives the \(\alpha\) th quantile \(\theta_\alpha ({\mathbf x})\) in the conditional distribution of a response variable \(Y\) given the value \({\mathbf X}= {\mathbf x}\) of a vector of covariates. It can be used to measure the effect of covariates not only in the center of a population, but also in the upper and lower tails. A functional that summarizes key features of the quantile specific relationship between \({\mathbf X}\) and \(Y\) is the vector \(\beta_\alpha\) of weighted expected values of the vector of partial derivatives of the quantile function \(\theta_\alpha ({\mathbf x})\). In a nonparametric setting, \(\beta_\alpha\) can be regarded as a vector of quantile specific nonparametric regression coefficients. In survival analysis models (e.g., Cox’s proportional hazard model, proportional odds rate model, accelerated failure time model) and in monotone transformation models used in regression analysis, \(\beta_\alpha\) gives the direction of the parameter vector in the parametric part of the model. \(\beta_\alpha\) can also be used to estimate the direction of the parameter vector in semiparametric single index models popular in econometrics.
We show that, under suitable regularity conditions, the estimate of \(\beta_\alpha\) obtained by using the locally polynomial quantile estimate of P. Chaudhuri [ibid. 19, No. 2, 760-777 (1991; Zbl 0728.62042)] is \(n^{1/2}\)-consistent and asymptotically normal with asymptotic variance equal to the variance of the influence function of the functional \(\beta_\alpha\). We discuss how the estimate of \(\beta_\alpha\) can be used for model diagnostics and in the construction of a link function estimate in general single index models.

MSC:

62J02	General nonlinear regression
62G20	Asymptotic properties of nonparametric inference
62G99	Nonparametric inference
62G05	Nonparametric estimation

Keywords:

average derivative estimate; transformation model; projection pursuit model; survival analysis; heteroscedasticity; reduction of dimensionality; quantile specific nonparametric regression coefficients; index models

Citations:

Zbl 0728.62042

References:

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