NEWS.md

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

qdfauction 0.3.0

New features

Procurement (lowest-bid-wins) auctions

• Procurement auctions are supported by reflection: proc_values(), proc_cost(), proc_optimal_ceiling(), proc_cost_boot(), proc_simulate() and proc_M(). Bids are reflected, b' = M - b, every existing estimator, selector, monotonisation, risk-aversion and copula adjustment applies unchanged to the reflected sample, and results are mapped back (costs c = M - v', ranks u = 1 - u'). Expected procurement cost with a price ceiling r and buyer outside option v0 is M - R'(M - r; M - v0), evaluated with the quantile-density functionals.
• proc_M() sets the reflection constant just above the largest bid, so the reflected sample starts near zero where the Q_n(0) = 0 anchor of the series estimators holds.
• Supported dependence families for procurement are "independence", "frank" (radially symmetric, so unchanged by reflection) and "nonparametric" (rank based, hence also unaffected). "clayton" and "gumbel" are refused with an explanatory error: reflection maps them to their 180-degree rotated (survival) copulas, which are not yet implemented.
• proc_values() defaults to monotone = "isotonic", because the natural-spline replacement extrapolates into the lower bid tail, which in procurement is the winners' tail. The default of fpa_values() is unchanged.

Bootstrap inference under dependence across auctions

• fpa_revenue_boot(), fpa_density_boot() and proc_cost_boot() gain resample = "blocks", a circular moving-block bootstrap over consecutive auctions, with block length L = "auto" or an integer. Whole auctions are still resampled together, so any within-auction copula structure is kept, while dependence across auctions is retained as well. The bandwidth is selected once on the full sample and held fixed in every refit, as for the existing schemes. Rows must be in time order, which the package cannot check, so this is never the default.
• block_length_auto() returns ceiling(2 * L_pw), with L_pw the Politis-White block length of the per-auction winning bid (winner = "max" for sales, "min" for procurement), capped at half the number of auctions. It warns when the number of auctions divided by L_pw falls below 15: the sample is then short relative to its dependence range and the interval is a lower bound on the sampling uncertainty. Requires the suggested package np.
• resample_blocks(), fpa_block_boot(), fpa_revenue_boot_blocks() and fpa_quantile_boot_blocks() expose the scheme for any statistic of an fpa_values() fit.

Other

• qdf() gains zero_at_origin for the Bernstein estimator, previously reachable only by calling qdf_bernstein() directly. The default TRUE reproduces earlier versions; FALSE anchors at the smallest observation instead, which is preferable when the data do not start near zero. The exact O(nK) leave-one-out path is used for either setting.
• proc_check_winning_bid() compares the model's expected winning bid with the sample mean of the auction minima: under the correct dependence specification the two agree up to sampling error, so a gap is a specification check.
• bid_icc() gives the between-group share of residual bid variance. Grouped by auction it measures within-auction dependence; grouped by firm it measures a permanent firm effect.
• fpa_values() documents that zero_at_origin, grid and the other arguments of qdf() can be passed through ..., and records the exact combination that reproduces the published revenue estimator. A new test checks that reproduction on the timber data.
• The package now ships inst/CITATION.

Bug fixes

• copula_psi(), copula_delta() and copula_delta_prime() now use the independence formulas when the fitted dependence is numerically negligible (Kendall's tau at or below 0.01). Near those limits the Frank and Clayton generators are ill-conditioned, and the reserve-price integrals built on them lost accuracy well before the multiplier itself did: with family = "frank" and five independent bidders the optimal reserve or ceiling was biased by about +0.2 while the pseudo-values were unaffected.
• fpa_revenue() accepts a vector v_s of the same length as r; previously this failed.
• resample_blocks() accepts a double number of auctions; previously it failed unless the count was an integer.

Documentation

• The help pages of fpa_revenue_boot() and fpa_density_boot() state that resampling auctions or bids independently understates uncertainty when the auctions have a time order and the same bidders recur, and point to resample = "blocks".
• ?fpa_revenue notes that its plug-in form rebuilds winners' bids from the pseudo-value distribution and assumes values independent within an auction; for procurement data, or dependent values, fpa_revenue_qd() is the right functional. The error for a pooled vector now also points to proc_values() and proc_cost().

qdfauction 0.2.0

This release combines the changes made for the revisions of Doosti (2026b, 2026c) with fixes from the wavelet branch.

Bug fixes (these change numerical results)

• Frank copula objects (copula_psi(), copula_delta(), copula_delta_prime()) are evaluated without the catastrophic cancellation in 1 + a^n / D^(n-1) at large theta (Kendall's tau above about 0.8); previously psi was exactly 0 near u = 1 at tau = 0.9.
• The tail integral in the revenue engine of fpa_revenue_qd() and fpa_optimal_reserve_qd() is accumulated from the top; the previous forward difference lost precision for Clayton at theta >= 2 (up to 0.05 in revenue at the 90th reserve percentile for tau = 0.75).
• fpa_optimal_reserve_model() searches from p_lo = 1e-4 (was 0.02), so interior optima close to the lower support are no longer reported as corners.
• select_bandwidth() and qdf() use exact leave-one-out by default for every selector, including "wbcv". The scaling approximation q (n-1)/n, invalid for estimators built from order statistics, is used only with loo = FALSE.
• bcv_criterion(), rlcv_criterion() and wbcv_criterion() can be called directly with their default u_scheme (previously an error), and wbcv_criterion() now uses exact leave-one-out by default (loo = TRUE), as select_bandwidth() does.
• qdf() with a wavelet method and no bandwidth argument now uses cross-validation (bandwidth = "cv") instead of failing.
• The print() method of fpa_values() no longer shows theta = NULL for the nonparametric copula multiplier.

Documentation and data

• The documentation of the timber dataset now states that the bids are homogenised, not raw: within each stratum the log bids were regressed on the log appraised per-unit value and the log timber volume and rescaled to the average lot (R-squared 0.671 for n = 2 and 0.554 for n = 3). The data are unchanged.
• data-raw/timber.R now rebuilds the dataset from the Journal of Applied Econometrics Data Archive files of Lu and Perrigne (2008) (doi:10.15456/jae.2022319.0720356598), applying that regression, instead of storing the bids as literal values.

New features

• fpa_virtual_value() and fpa_virtual_value_model(): the affiliated virtual value omega(u) = Q(u) - q(u) H(u).
• fpa_corner_test() (with a print() method): a crossing test for whether any binding reserve price is optimal (Doosti 2026c). The decision is the point rule on the full sample, `min over u in [0.10, 0.98] of omega_hat(u)
• v_s >= 0; a bootstrap over auctions gives the frequency of that decision. The copulafamily` is a required argument, because the result depends on it.
• qdf(bandwidth = "sqrt") (and fpa_values(), the revenue functions and fpa_revenue_boot() through it): the deterministic rule m = ceiling(sqrt(N)) for the Bernstein estimator, used in Doosti (2026b). It lies inside the range covered by that paper's asymptotic theory.
• qdf_bernstein() gains zero_at_origin; qdf_bernstein_loo() computes exact leave-one-out Bernstein estimates by an index shift in O(nK) and is used automatically by the cross-validation criteria.
• select_bandwidth(grid = "exhaustive") returns the global minimiser over all integer m in m_range (not the default).
• fpa_montecarlo() gains p0 and eta, passed to the simulator.
• Stage 3 of fpa_revenue_qd(), fpa_optimal_reserve_qd(), fpa_virtual_value() and fpa_corner_test(): if the data-driven bandwidth for the pseudo-values is minimised at an end of m_range, the quantile density is refitted with the published range m_range = c(10, 50) and stats::optimize() (Doosti 2026c), with a warning; if that fit is also at an end, the function stops and asks for a numeric bandwidth.
• Argument checks: select_bandwidth() and qdf() accept only grid = "exhaustive" or a non-negative integer (a mistyped value used to fall back silently to optimize()); qdf_bernstein_loo() checks that u has one increasing value per observation, that the data are positive and that floor(1/h) >= 2.
• Documentation: qdf() lists all ten estimators and gives the correct pages for Parzen (1979); the default grid of fpa_density_ci() is described correctly; qdf_wavelet_nl() explains when post-smoothing helps.
• inst/benchmarks/scaling.R also times the functions for affiliated values, the specification test and the density bootstrap.

Changed defaults

• fpa_revenue_boot() resamples whole auctions by default (resample = "auctions"; previously "bids"). The revenue estimator uses the highest pseudo-value in each auction, and resampling pooled bids breaks that structure: in the design of Zincenko (2024), with the Bernstein estimator and m = ceiling(sqrt(N)), pooled resampling gave pointwise coverage of about 0.65-0.80 at nominal 0.90, against 0.85-0.89 with auction resampling. "bids" is kept for comparison. fpa_corner_test() already resamples auctions; fpa_density_boot() keeps "bids", because the density estimators do not use the auction structure.
• bandwidth = "wbcv", select_bandwidth(selector = "wbcv") and wbcv_criterion() are kept only for backward compatibility, are not recommended, and now warn. The two terms of the WBCV criterion have different units, so the selected bandwidth depends on the units of the data, and on data of order one or larger the criterion tends to select the smoothest fit in m_range. The criterion is being revised (Doosti 2026b); it is kept so that existing code runs. Examples, the README and the vignette now use BCV.
• select_bandwidth() and qdf() keep stats::optimize() over m_range as the default search for every selector, as in the published simulations; grid = "exhaustive" (every integer m in m_range, the global minimiser) is available as an option. A user-supplied optimizer takes precedence over the exhaustive search.
• Exact leave-one-out is the default for "wbcv" in select_bandwidth() and qdf() (previously the scaling approximation); see Bug fixes.
• fpa_revenue_boot() and the bootstrap of fpa_corner_test() select the bandwidth once, on the full sample, and hold it fixed in every bootstrap replication, as the theory assumes (Doosti 2026b, Theorems 6-7); previously the bandwidth was re-selected in each replicate. Results with a data-driven bandwidth change; results with a numeric bandwidth do not. The selected m and whether it lies at a boundary are returned and printed.
• select_bandwidth() detects boundary solutions for every selector and search method: if the minimiser lies at either end of m_range, it warns and returns at_boundary = TRUE (and boundary = "lower" or "upper"), and print() of a qdf() fit reports it. A boundary minimum means that the criterion did not select a bandwidth in that range.

qdfauction 0.1.0

First CRAN release.

Quantile density estimation and auction methods

• Quantile density estimation: ten estimators (kernel, corrected kernel, Poisson, Bernstein, Jones, Soni, indirect Poisson, linear / hard- / block-thresholded wavelets) with BCV, RLCV, WBCV and cross-validated wavelet tuning (qdf()).
• Private-value recovery in first-price auctions for independent or Archimedean-copula affiliated values and risk-neutral or CRRA bidders (fpa_values()), with GPV, boundary-corrected and Zincenko kernel benchmarks and monotonicity options.
• Private-value density estimation with wavelet, Hickman-Hubbard, Marmer-Shneyerov and empirical-quantile second stages (fpa_density()), tuned by interior least-squares or log-likelihood cross-validation (select_fpa_density()).
• Inference for the GPV estimator of the private-value density: analytic standard errors, pointwise confidence intervals and bootstrap uniform confidence bands (Ma, Marmer and Shneyerov 2019) (fpa_density_ci()).
• Expected revenue, optimal reserve price and bootstrap inference (fpa_revenue(), fpa_optimal_reserve(), fpa_revenue_boot()), exact equilibrium bids and simulation (fpa_bid_function(), fpa_simulate()), Monte Carlo and evaluation tools (fpa_montecarlo(), fpa_evaluate_values(), fpa_evaluate_density()).
• Extensibility: user-defined kernels via make_kernel(), with check_kernel() reporting mass, symmetry, order and roughness; a user-supplied optimiser for bandwidth selection (optimizer in select_bandwidth() and qdf()).
• Bid matrices may be supplied as data frames; incomplete auctions, missing values and univariate/multivariate mix-ups give informative errors.
• inst/benchmarks/scaling.R times the main functions as the sample grows.
• USFS timber auction data (timber) and a vignette.

Further estimators and inference

• Expected revenue and optimal reserve under affiliated private values by the copula--quantile density functional (fpa_revenue_qd(), fpa_optimal_reserve_qd(); Doosti 2026c).
• Nonlinear wavelet estimators of the quantile density built on genuine wavelet coefficients, with the theoretical thresholds of Chesneau, Dewan and Doosti (2016, Theorems 3.2 and 3.4) and the block thresholding of Shirazi and Doosti (2022) (qdf_wavelet_nl()), on a periodised orthonormal basis (phi_per(), psi_per(), psi_jk()).
• The hard-thresholding estimator of Theorem 2 of Doosti, Dewan and Sbai (2026) with level-dependent thresholds (fpa_density(method = "wavelet_theory")).
• Flexible, nonparametric copula multiplier (copula_psi_np(), fpa_values(family = "nonparametric")) and a bootstrap specification test for parametric copulas (copula_spec_test()) (Doosti 2026a). The semiparametric estimator of Hubbard, Li and Paarsch (2012) is documented as fpa_values(method = "gpv", family = ...).
• Bootstrap pointwise intervals and uniform bands for any private-value density estimator, including Marmer and Shneyerov (2012) (fpa_density_boot()).
• qdf() is now scale-equivariant for every estimator: it estimates on x / sd(x) and rescales (rescale = TRUE). This matters for the indirect Poisson, Jones and Soni estimators, whose smoothing parameter lives on the data scale; rescale = FALSE reproduces the earlier behaviour. The default evaluation grid is now i/(n+1).
• select_wavelet() (and qdf(..., bandwidth = "cv") for the wavelet estimators) uses least-squares cross-validation by default (criterion = "lscv") and skips bandwidths below the lattice spacing; cross-validation arguments are no longer passed on to the estimator.

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