NONPARAMETRIC WEIGHTED AVERAGE QUANTILE DERIVATIVE

The weighted average quantile derivative (AQD) is the expected value of the partial derivative of the conditional quantile function (CQF) weighted by a function of the covariates. We consider two weighting functions: a known function chosen by researchers and the density function of the covariates that is parallel to the average mean derivative in Powell, Stock, and Stoker (1989, Econometrica 57, 1403–1430). The AQD summarizes the marginal response of the covariates on the CQF and defines a nonparametric quantile regression coefficient. In semiparametric single-index and partially linear models, the AQD identifies the coefficients up to scale. In nonparametric nonseparable structural models, the AQD conveys an average structural effect under certain independence assumptions. Including a stochastic trimming function, the proposed two-step estimator is root-n-consistent for the AQD defined by the entire support of the covariates. To facilitate tractable asymptotic analysis, a key preliminary result is a new Bahadur-type linear representation of the generalized inverse kernel-based CQF estimator uniformly over the covariates in an expanding compact set and over the quantile levels. The weak convergence to Gaussian processes applies to the differentiable nonlinear functionals of the quantile processes.

Some asymptotic properties between smooth empirical and quantile processes for dependent random variables

Пусть $\widehat F_n$ - гладкая эмпирическая оценка, полученная интегрированием оценки плотности ядерного типа, построенной по случайной выборке размера $n$ из распределения с непрерывной функцией распределения $F$. В статье изучается отклонение почти наверное между гладким эмпирическим и гладким квантильным процессами при условии $\phi$-перемешивания и при условии сильного перемешивания. Для гладких квантилей в случае $\phi$-перемешивания и в случае сильного перемешивания выводится представление Бахадура-Кифера, как поточечное, так и равномерное. Эти результаты являются распространением результатов Бабу-Сингха (1978) и Ралеску (1992).

SMOOTHED QUANTILE REGRESSION PROCESSES FOR BINARY RESPONSE MODELS

In this article, we consider binary response models with linear quantile restrictions. Considerably generalizing previous research on this topic, our analysis focuses on an infinite collection of quantile estimators. We derive a uniform linearization for the properly standardized empirical quantile process and discover some surprising differences with the setting of continuously observed responses. Moreover, we show that considering quantile processes provides an effective way of estimating binary choice probabilities without restrictive assumptions on the form of the link function, heteroskedasticity, or the need for high dimensional nonparametric smoothing necessary for approaches available so far. A uniform linear representation and results on asymptotic normality are provided, and the connection to rearrangements is discussed.