Adaptive quantile regressions for massive datasets

Abstract

Analysis of massive datasets is challenging owing to limitations of computer primary memory. Adaptive quantile regressions is a robust and efficient estimation method. For computational efficiency, we propose an adaptive smoothing quantile regressions (ASQR). The ASQR method is used to analyze massive datasets. The proposed approach significantly reduces the required amount of primary memory, and the resulting estimate will be as efficient as if the entire data set is analyzed simultaneously. Both simulations and data analysis are conducted to illustrate the finite sample performance of the proposed methods.

Appendix

Proof of Theorem 2.1

By the Theorem 4.3 in Chen et al. (2019) and the error term $\varepsilon$ is independent of $\mathbf{X}$, we have

$$\begin{aligned} {\hat{\beta }}_{\tau _{q},h}-{\beta }_0=\frac{1}{N}f^{-1}(b_{\tau _q})\mathbf{C}^{-1}\sum _{i=1}^N\mathbf{x}_i \left( I\{\varepsilon _i\ge 0\}+ \tau _{k}-1 \right) +r_N, \end{aligned}$$

where $\Vert r_N\Vert _2=O_p\left( (p/N)^{3/4}(\log N)^{1/2}\right)$. Thus, for a given quantile $\tau _{q}$,

$$\begin{aligned} E[{\hat{\beta }}_{\tau _{q},h}]&=\beta _0+r_N,\\ Var[{\hat{\beta }}_{\tau _{q},h}]&=E[({\hat{\beta }}_{\tau _{q},h}-\beta _0)^2|\tau _{q}] =N^{-1}{} \mathbf{C}^{-1}\tau _{q}(1-\tau _{q})/f^2(b_{\tau _{q}})+r_N^2. \end{aligned}$$

See, for example, Koenker (2005) for details about the above two results. It is also straightforward to verify that, for two independent quantile $\tau _{q}$ and $\tau _{q'}$,

$$\begin{aligned} Cov({\hat{\beta }}_{\tau _{q},h},{\hat{\beta }}_{\tau _{q'},h}) =N^{-1}{} \mathbf{C}^{-1}(\min (\tau _{q},\tau _{q'})-\tau _{q}\tau _{q'})/\{f(b_{\tau _{q}}) f(b_{\tau _{q'}})\}+r_N^2. \end{aligned}$$

In addition, the distributions of ${\hat{\beta }}_{\tau _{q},h}$, $q=1,\ldots ,Q$ and the joint distributions of ${\hat{\beta }}_{\tau _{q},h}$, ${\hat{\beta }}_{\tau _{q'},h}$, $q,q'=1,\ldots ,Q$ are all enjoy asymptotic normality. Thus, ${\hat{\beta }}=\sum _{q=1}^Qw_q{\hat{\beta }}_{\tau _{q},h}$ also asymptotically follows normal distribution. The mean and variance of ${\hat{\beta }}$ are established as follows,

$$\begin{aligned} E[{\hat{\beta }}]&=\sum _{q=1}^Qw_qE[{\hat{\beta }}_{\tau _{q},h}]=\beta _0+r_N,\\ Var[{\hat{\beta }}]&=\sum _{q=1}^Q\sum _{q'=1}^Qw_qw_{q'}Cov({\hat{\beta }}_{\tau _{q},h}, {\hat{\beta }}_{\tau _{q'},h})\\&=N^{-1}{} \mathbf{C}^{-1}\sum _{q=1}^Q\sum _{q'=1}^Qw_qw_{q'} \frac{\min (\tau _{q},\tau _{q'})-\tau _{q}\tau _{q'}}{f(b_{\tau _{q}})f(b_{\tau _{q'}})} +r_N^2. \end{aligned}$$

Then, the density function of ${\hat{\beta }}$ is as follows,

$$\begin{aligned} f({\hat{\beta }})\rightarrow \left\{ 2\pi N^{-1}\det \left( \varSigma (\mathbf{w})\right) \right\} ^{-N/2} \exp \left\{ -1/2N^{-1}(\hat{{\beta }}-{\beta }_0)^{\top }\varSigma ^{-1}(\mathbf{w})(\hat{{\beta }}-{\beta }_0) \right\} . \end{aligned}$$

Thus, under conditions in Theorem 2.1,

$$\begin{aligned} \sqrt{N}(\hat{{\beta }}-{\beta }_0)\xrightarrow {L} \left( 0,\varSigma (\mathbf{w})\right) . \end{aligned}$$

This completes the proof of Theorem 2.1. $\square$