qdf_kernel: Direct kernel estimators of the quantile density

In qdfauction: Nonparametric Quantile Density Estimation and Structural Analysis of First-Price Auctions

Direct kernel estimators of the quantile density

Description

Kernel-type estimators that smooth the empirical quantile function directly.

* 'qdf_kernel()' is the Jones (1992) estimator

\hat q(u) = \sum_{i=1}^n X_{(i)} \{ k_h((i-1)/n - u) - k_h(i/n - u) \}

with a Gaussian kernel k_h(a) = k(a/h)/h. * 'qdf_kernel_corrected()' is the boundary-corrected version (estimator 2 of Doosti, Dewan and Talebian 2025), which renormalises the kernel weights by the mass falling inside [0,1] and adds a correction proportional to the kernel quantile estimator.

Usage

qdf_kernel(u, x, h, kernel = "gaussian")

qdf_kernel_corrected(u, x, h, kernel = "gaussian")

Arguments

u	numeric vector of evaluation points in [0, 1].
x	numeric sample (sorted internally).
h	bandwidth on the probability scale (0 < h < 1).
kernel	'"gaussian"' (default), '"triangular"' or '"epanechnikov"', or a user-defined kernel from [make_kernel()] (or a kernel density function). Soni, Dewan and Jain (2012) used the triangular and Epanechnikov kernels.

Value

numeric vector of quantile density estimates at 'u'.

References

Jones, M. C. (1992). Estimating densities, quantiles, quantile densities and density quantiles. *Annals of the Institute of Statistical Mathematics*, 44, 721–727.

Chaubey, Y. P., Dewan, I. and Li, J. (2024). On some non parametric estimators of the quantile density function for a stationary associated process. *Communications in Statistics – Theory and Methods*, 53, 5553–5573.

See Also

[qdf()] for automatic bandwidth selection.

Examples

x <- rgamma(100, 5, 1)
u <- seq(0.05, 0.95, by = 0.05)
qdf_kernel(u, x, h = 0.05)
qdf_kernel_corrected(u, x, h = 0.05)

qdfauction documentation built on Nov. 10, 2026, 5:07 p.m.