Bias correction and uniform inference
for the quantile density function

The derivative of the quantile function, the quantile density (QD), has been long recognized as an important object in statistical inference.²²2This function is sometimes also called the sparsity function (Tukey, 1965). In particular, it arises as a factor in the asymptotically linear expansion for the quantile function (Bahadur, 1966; Kiefer, 1967), and hence may be used for asymptotically valid inference on quantiles (Csörgő and Révész, 1981a, b; Koenker, 2005).

Given its importance, several estimators of the QD have been proposed in the literature. The most widely used estimator is the kernel quantile density (KQD), originally developed by Siddiqui (1960) and Bloch and Gastwirth (1968) for the case of rectangular kernel, and generalized to arbitrary kernels by Falk (1986), Welsh (1988), Csörgő et al. (1991), and Jones (1992). This estimator is simply a smoothed derivative of the empirical quantile function, where smoothing is performed via convolution with a kernel function.

Similarly to the classical case of kernel density estimation, the KQD suffers from bias close to the boundary points $\{0,1\}$ of its domain $[0,1]$, rendering the estimator inconsistent. To the best of my knowledge, no bias correction procedures have been developed for the QD.

In this paper, I show how to perform correction for the boundary bias, recovering strong uniform consistency for the resulting bias-corrected KQD (BC-KQD) estimator. The bias correction is computationally cheap and is based on the fact that the bias of the KQD is approximately equal to the integral of the localized kernel function, a quantity that only depends on the chosen kernel and bandwidth. I also develop an algorithm for construction of the uniform confidence bands around the QD on its entire domain $[0,1]$. This procedure relies on the fact that the studentized BC-KQD exhibits an influence function that is pivotal. This makes it possible to calculate the critical values by simulating from either the known influence function or the studentized BC-KQD under an alternative (pseudo) distribution of the data.

The rest of the paper is organized as follows. Section 2 outlines the framework and defines the KQD estimator. Section 3 introduces the BC-KQD estimator and establishes its Bahadur-Kiefer expansion. Section 4 develops the uniform confidence bands based on the BC-KQD. Section 5 illustrates the performance of the confidence bands in a set of Monte Carlo simulations. Section 6 concludes. Proofs of theoretical results are given in the Appendix.

The data consist of independent identically distributed draws $X_{1},\dots,X_{n}$ from a distribution on $\mathbb{R}$ with a cumulative distribution function (CDF) $F$ satisfying the following assumption.

Assumption 1 implies that the quantile density

is continuously differentiable and bounded away from zero and infinity on the support $[\underline{x},\bar{x}]$.

Let $X_{(1)}\leq\cdots\leq X_{(n)}$ be the order statistics of the sample $X_{1},\dots,X_{n}$, and let $\hat{Q}$ denote the empirical quantile function,

The KQD estimator is defined as

where $K$ is a kernel function, $K_{h}(z)\coloneqq h^{-1}K\left(h^{-1}z\right)$, and $h>0$ is bandwidth (see, e.g., Csörgő et al., 1991). We impose the following assumptions on the kernel and bandwidth.

Assumption 2 is standard; boundedness of the total variation of $K$ ensures that the class

is a bounded VC class of measurable functions, see, e.g., Nolan and Pollard (1987).

Assumptions 3 and 4 are essentially the same, up to the log terms in the bandwidth rates, with Assumption 3 being slightly weaker. Assumption 4.1 states that the bandwidth rate is large enough (slightly larger than $n^{-1/2}$) to guarantee that the smoothed remainder of the classical Bahadur-Kiefer expansion vanishes asymptotically, see the proof of 1 below. Assumption 4.2 imposes the undersmoothing bandwidth rate (slightly smaller than $n^{-1/3}$), which ensures that the smoothing bias disappears fast enough for the confidence bands to be valid, see the proof of Theorem 2 below.

In this section, I introduce the bias-corrected estimator and develop its asymptotically linear expansion with an explicit a.s. uniform rate of the remainder (the Bahadur-Kiefer expansion).

To see the necessity of bias correction, note that, for $u$ close to the boundary, the kernel weights $K_{h}(u-i/n)$, $i=1,\dots,n-1$, do not approximately sum up to one, rendering the KQD $\hat{q}_{h}(u)$ inconsistent. Therefore, dividing the KQD by the sum of the kernel weights (or the corresponding integral of the kernel function) may eliminate the boundary bias. To this end, define

For computational purposes, note that $\psi_{h}$ is symmetric around $1/2$ (i.e. $\psi_{h}(u)=\psi_{h}(1-u)$ for all $u\in[0,1]$), $\psi_{h}\in[1/2,1]$ and $\psi_{h}(u)=1$ for $u\in[h/2,1-h/2]$. The bias-corrected KQD (BC-KQD) is then defined as

The following theorem establishes that the studentized BC-KQD is approximately equal to the centered kernel density estimator with an approximation error that converges to zero a.s. at an explicit uniform rate. Since this result resembles (and relies on) the classical asymptotically linear expansion for the quantile function (Bahadur, 1966; Kiefer, 1967), we call it the Bahadur-Kiefer expansion for the BC-KQD. Denote $U_{i}=F(X_{i})$, $i=1,\dots,n.$

This representation allows us to establish the exact rate of strong uniform consistency of the BC-KQD under a bandwidth that achieves undersmoothing (Assumption 3.2).

One of the convenient features of the KQD (and BC-KQD) estimator is that its bandwidth has a natural scale $[0,1]$ which is independent of the data generating process. Hence, I put aside the choice of constant $c$ in the bandwidth $h=cn^{-\eta}$ and suggest setting $c=1$.

Regarding the choice of the rate $\eta$, ignoring the log terms, it is easy to establish the rate-optimal bandwidth, which is achieved whenever the rate of the smoothing bias $n^{1/2}h^{3/2}$ matches that of the remainder in the original Bahadur-Kiefer expansion $n^{-1/4}h^{-1/2}$. It follows that the nearly-optimal bandwidth is

Under this bandwidth, the exact rate of strong uniform convergence is

which is just slightly worse than the familiar “cube-root” rate (Kim and Pollard, 1990).

Suppose we had access to valid approximations $c_{n,\tau}$, $c_{n,\tau}^{abs}$ to the $\tau$-quantiles of the random variables

respectively, in the sense that

Then the following confidence bands for $q(\cdot)$ would be asymptotically valid at the confidence level $1-\alpha$:

1. 1.

   the one-sided CB

   $\displaystyle\left[\frac{\hat{q}_{h}^{bc}(u)}{1+\frac{c_{n,1-\alpha}}{\psi_{h}(u)\sqrt{nh}}},\,+\infty\right),\quad u\in[0,1],$

   (19)

2. 2.

   the one-sided CB

   $\displaystyle\left(-\infty,\,\frac{\hat{q}_{h}^{bc}(u)}{1-\frac{c_{n,1-\alpha}}{\psi_{h}(u)\sqrt{nh}}}\right],\quad u\in[0,1],$

   (20)

3. 3.

   the two-sided CB

   $\displaystyle q(u)\in\left[\frac{\hat{q}_{h}^{bc}(u)}{1+\frac{c_{n,1-\alpha/2}^{abs}}{\psi_{h}(u)\sqrt{nh}}},\,\,\frac{\hat{q}_{h}^{bc}(u)}{1-\frac{c_{n,1-\alpha/2}^{abs}}{\psi_{h}(u)\sqrt{nh}}}\right],\quad u\in[0,1].$

   (21)

I propose two ways of obtaining such approximate critical values, both making use of the pivotality of the studentized bias-corrected KQD $Z_{n}^{bc}(u)$, see Theorem 1. I focus on the one-sided critical value $c_{n,\tau}$ for simplicity; the proofs for the two-sided critical value are analogous.

The first approach is to let $c_{n,\tau}$ be the $\tau$-quantile of the random variable

Since $\mathbb{G}_{n}$ is a known process, $c_{n,\tau}$ can be obtained easily by simulation. In principle, $c_{n,\tau}$ can be tabulated for different choices of the kernel $K$ and values of the sample size $n$ and the bandwidth $h$.

The other approach is to let $c_{n,\tau}$ be the $\tau$-quantile of the random variable

where $Z_{n}^{bc,U[0,1]}(u)$ is equal to $Z_{n}^{bc}(u)$ evaluated at a pseudo-sample $\tilde{X}_{1},\dots,\tilde{X}_{n}\sim U[0,1]$ in place of the original sample. For the uniform distribution, $q\equiv 1$, and hence

where $\tilde{q}_{n}(u)$ is the (non-bias-corrected) KQD calculated using the pseudo-sample, i.e.

The following theorem establishes that the two aforementioned approximations to the critical values are valid, implying the asymptotic validity of the confidence bands. These confidence bands are centered at an AMSE-suboptimal estimator $\hat{q}_{h}^{bc}$ and are expected to shrink at a rate slightly slower than the minimax optimal rate, as noted by Chernozhukov et al. (2014a, p.1795). This is compensated for by the confidence bands exhibiting the coverage that is asymptotically exact.

In this section I study the finite-sample behavior of the proposed confidence bands in a set of Monte Carlo simulations.

I consider the following distributions of the data, all supported on the interval $[0,1]$: (i) uniform[0,1] distribution (ii) the distribution $N(1/2,1)$ truncated to $[0,1]$ (iii) the linear distribution with the PDF $f(x)=x+1/2$, $x\in[0,1]$. I set the nominal confidence level to be $1-\alpha\in\{0.8,0.9,0.95,0.99\}$ and the sample size $n\in\{100,500,1000,5000\}$. The critical values are obtained by simulating $\mathbb{G}_{n}(u)$ and calculating the quantiles of its supremum on the grid $u\in\{0.005,0.015,0.02,\dots,0.995\}$, with the number of simulations set to $20000$ (simulation results for the critical values based on $Z_{n}^{bc,U[0,1]}(u)$ are very similar, so I do not report them here). I use the kernel corresponding to the standard normal distribution truncated to $[-1/2,1/2]$ and the nearly-optimal bandwidth $h=cn^{-3/8}$, where I set $c=1$ since the scale of the bandwidth is $[0,1]$, see Section 3.

In Figure 1, included for illustration, I plot 100 independent realizations of the $90\%$ confidence bands for the linear distribution, along with the true quantile density (in blue). Table 1 contains simulated coverage values for the two-sided confidence bands. The coverage is almost invariant to the distribution of the data, but the size distortion tends to be smaller for higher nominal confidence levels.

To the best of my knowledge, no boundary bias correction or uniform inference procedures have been developed for the quantile density (sparsity) function. In this paper, I develop such procedures, establish their validity and show in a set of Monte Carlo simulations that they perform reasonably well in finite samples. I hope that, even when the quantile density itself is not the main inference target, these results may be employed for improving the quality of inference for other statistical objects, including the quantile function.