arXiv subjects

Saïd Maanan

Publications and source records attributed to Saïd Maanan.

4 recordsLinked to original sources

arXiv2026

Robust Interpolated Quantile Estimators: Asymptotic Theory and Efficiency

This paper introduces a unified family of interpolated quantile estimators obtained by augmenting the check loss with quadratic, Huber, or Tukey's bisquare regularization. The estimators are indexed by the quantile level $τ$ and an interpolation parameter $h$. They reduce to the classical empirical quantile when $h=0$, while increasing $h$ continuously shifts the effective probability level toward the center of the distribution. A complete asymptotic theory is developed. For the quadratic interpolation, the effective quantile level is characterized by an interpolation equation yielding a closed-form parametrization of neighboring quantiles. Asymptotic normality is established for all three interpolated estimators via M-estimation, and a decomposition of the asymptotic variance explains how efficiency depends on the underlying distribution. Numerical experiments show that the quadratic interpolated estimator can reduce asymptotic variance by up to 36\% for light-tailed distributions and up to 57\% for heavy-tailed or asymmetric distributions for suitable interpolation strength. The framework is extended to linear quantile regression, where Monte Carlo experiments show that Huber interpolation is beneficial only in a narrow neighborhood of the median, while ordinary quantile regression remains preferable elsewhere. An application to daily log-returns illustrates the practical relevance of the proposed methodology for tail estimation under heavy tails and asymmetry.

stat.ME↗

arXiv2026

Interpolated Quantile Estimation: A Unified Framework Bridging Quantiles and the Mean

This paper develops and analyzes three families of estimators that continuously interpolate between classical quantiles and the sample mean. The construction begins with a smoothed version of the $L_1$ loss, indexed by a location parameter $z$ and a smoothing parameter $h \ge 0$, whose minimizer $\hat q(z,h)$ yields a unified $M$-estimation framework. Depending on how $(z, h)$ is specified, this framework generates three distinct classes of estimators: fixed-parameter smoothed quantile estimators, plug -- in estimators of fixed quantiles, and a new continuum of mean -- estimating procedures. For all three families we establish consistency and asymptotic normality via a uniform asymptotic equicontinuity argument. The limiting variances admit closed forms, allowing a transparent comparison of efficiency across families and smoothing levels. A geometric decomposition of the parameter space shows that, for fixed quantile level $τ$, admissible pairs $(z, h)$ lie on straight lines along which the estimator targets the same population quantile while its asymptotic variance evolves. The theoretical analysis reveals two efficiency regimes. Under light-tailed distributions (e.g., Gaussian), smoothing yields a monotone variance reduction. Under heavy-tailed distributions (e.g., Laplace), a finite smoothing parameter $h^{*}(τ) > 0$ strictly improves efficiency for quantile estimation. Numerical experiments -- based on simulated data and real financial returns -- validate these conclusions and show that, both asymptotically and in finite samples, the mean-estimating family does not improve upon the sample mean.

stat.ME↗

arXiv2025

Smoothed Quantile Estimation via Interpolation to the Mean

This paper introduces a unified family of smoothed quantile estimators that continuously interpolate between classical empirical quantiles and the sample mean. The estimators q(z, h) are defined as minimizers of a regularized objective function depending on two parameters: a smoothing parameter h $\ge$ 0 and a location parameter z $\in$ R. When h = 0 and z $\in$ (-1, 1), the estimator reduces to the empirical quantile of order $τ$ = (1z)/2; as h $\rightarrow$ $\infty$, it converges to the sample mean for any fixed z. We establish consistency, asymptotic normality, and an explicit variance expression characterizing the efficiency-robustness trade-off induced by h. A key geometric insight shows that for each fixed quantile level $τ$ , the admissible parameter pairs (z, h) lie on a straight line in the parameter space, along which the population quantile remains constant while asymptotic efficiency varies. The analysis reveals two regimes: under light-tailed distributions (e.g., Gaussian), smoothing yields a monotonic but asymptotic variance reduction with no finite optimum; under heavy-tailed distributions (e.g., Laplace), a finite smoothing level h * ($τ$ ) > 0 achieves strict efficiency improvement over the classical empirical quantile. Numerical illustrations confirm these theoretical predictions and highlight how smoothing balances robustness and efficiency across quantile levels.

stat.ME↗

arXiv2025

Multiscale Asymptotic Normality in Quantile Regression: Hilbert Matrices and Polynomial Designs

This paper investigates the asymptotic properties of quantile regression estimators in linear models, with a particular focus on polynomial regressors and robustness to heavy-tailed noise. Under independent and identically distributed (i.i.d.) errors with continuous density around the quantile of interest, we establish a general Central Limit Theorem (CLT) for the quantile regression estimator under normalization using $Δ_n^{-1}$, yielding asymptotic normality with variance $τ(1-τ)/f^2(0) \cdot D_0^{-1}$. In the specific case of polynomial regressors, we show that the design structure induces a Hilbert matrix in the asymptotic covariance, and we derive explicit scaling rates for each coefficient. This generalizes Pollard's and Koenker's earlier results on LAD regression to arbitrary quantile levels $τ\in (0, 1)$. We also examine the convergence behavior of the estimators and propose a relaxation of the standard CLT-based confidence intervals, motivated by a theoretical inclusion principle. This relaxation replaces the usual $T^{j+1/2}$ scaling with $T^α$, for $α< j + 1/2$, to improve finite-sample coverage. Through extensive simulations under Laplace, Gaussian, and Cauchy noise, we validate this approach and highlight the improved robustness and empirical accuracy of relaxed confidence intervals. This study provides both a unifying theoretical framework and practical inference tools for quantile regression under structured regressors and heavy-tailed disturbances.

math.ST↗