arXiv subjects

Jan Picek

Publications and source records attributed to Jan Picek.

7 recordsLinked to original sources

arXiv2026

R-estimation in a Linear Model with Autoregressive Errors

In the linear regression model, we construct a nonparametric estimate of the regression parameter vector $\boldgreek\beta$ that is insensitive to a possible nuisance autoregression in the model errors. The main tool for estimating $\boldgreek\beta$ is based on the autoregression rank scores of the model. The resulting estimator is invariant to the autoregression parameters and thus remains insensitive to potential hidden linear trends or other structured disturbances, which frequently occur in economic, hydrological, and related applications.

stat.ME↗

arXiv2026

Estimation of the Risk Measure under a Nuisance Autoregression

The goal of an experiment is to evaluate the profit, loss, or the amount of a physical entity over a period. The measurements $X_t$ can be influenced by the values measured in the past; hence we describe the situation with an autoregression model, whose autoregression coefficients are generally unknown. The variable of interest is the error term $Z_t$ of the model, which is the increment of $X_t$ with respect to the past, but itself unobservable. The problem is to estimate various quantile functions of $Z$, as the risk measure of the loss or the related economic indicators. We construct an estimate of quantile functions of $Z$ in the situation that the inference is possible only by means of observations $X$. The proposed estimates are based on the R-estimators of autoregression coefficients, combined with the autoregression quantiles.

stat.ME↗

arXiv2024

Estimation of Quantile Functionals in Linear Model

Various indicators and measures of the real life procedures rise up as functionals of the quantile process of a parent random variable Z. However, Z can be observed only through a response in a linear model whose covariates are not under our control and the probability distribution of error terms is generally unknown. The problem is that of nonparametric estimation or other inference for such functionals. We propose an estimation procedure based on the averaged two-step regression quantile, recently developed by the authors, combined with an R-estimator of slopes of the linear model.

stat.ME↗

arXiv2017

Empirical regression quantile process with possible application to risk analysis

The processes of the averaged regression quantiles and of their modifications provide useful tools in the regression models when the covariates are not fully under our control. As an application we mention the probabilistic risk assessment in the situation when the return depends on some exogenous variables. The processes enable to evaluate the expected $α$-shortfall ($0\leqα\leq 1$) and other measures of the risk, recently generally accepted in the financial literature, but also help to measure the risk in environment analysis and elsewhere.

math.ST↗

arXiv2016

Behavior of R-estimators under measurement errors

As was shown recently, the measurement errors in regressors affect only the power of the rank test, but not its critical region. Noting that, we study the effect of measurement errors on R-estimators in linear model. It is demonstrated that while an R-estimator admits a local asymptotic bias, its bias surprisingly depends only on the precision of measurements and does neither depend on the chosen rank test score-generating function nor on the regression model error distribution. The R-estimators are numerically illustrated and compared with the LSE and $L_1$ estimators in this situation.

math.ST↗

arXiv2015

Affine equivariant rank-weighted L-estimation of multivariate location

In the multivariate one-sample location model, we propose a class of flexible robust, affine-equivariant L-estimators of location, for distributions invoking affine-invariance of Mahalanobis distances of individual observations. An involved iteration process for their computation is numerically illustrated.

math.ST↗

arXiv2015

Rank tests for corrupted linear models

For some variants of regression models, including partial, measurement error or error-in-variables, latent effects, semi-parametric and otherwise corrupted linear models, the classical parametric tests generally do not perform well. Various modifications and generalizations considered extensively in the literature rests on stringent regularity assumptions which are not likely to be tenable in many applications. However, in such non-standard cases, rank based tests can be adapted better, and further, incorporation of rank analysis of covariance tools enhance their power-efficiency. Numerical studies and a real data illustration show the superiority of rank based inference in such corrupted linear models.

math.ST↗