arXiv subjects

Jana Jurečková

Publications and source records attributed to Jana Jurečková.

10 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 $\boldgreekβ$ that is insensitive to a possible nuisance autoregression in the model errors. The main tool for estimating $\boldgreekβ$ 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↗

arXiv2022

Estimation of Expected Shortfall under Various Experimental Conditions

Our primary aim is to find an estimate of the expected shortfall in various situations: (1) Nonparametric situation, when the probability distribution of the incurred loss is unknown, only satisfying some general conditions. Then, following [3], the expected shortfall can be expressed through a minimization of a well known quantile criterion and its numerical estimate is based on the empirical quantile functionof the loss. (2) The distribution function of the loss is known, but the loss can be contaminated by an additive measurement error: Estimating the expected shortfallin such a case exploits the concept of pseudo-capacities elaborated in [11] and [6] and its numerical value is based on the empirical quantile function of the suitable capacity. (3) The loss distribution can be contaminated by the heavy right tail with Pareto index > 1. The problem of interest is in this case to evaluate the effect of the Pareto index on the resulting expected shortfall.

stat.ME↗

arXiv2021

Process of the slope components of $α$-regression quantile

We consider the linear regression model along with the process of its $α$-regression quantile, $0<α<1$. We are interested mainly in the slope components of $α$-regression quantile and in their dependence on the choice of $α.$ While they are invariant to the location, and only the intercept part of the $α$-regression quantile estimates the quantile $F^{-1}(α)$ of the model errors, their dispersion depends on $α$ and is infinitely increasing as $α\rightarrow 0,1$, in the same rate as for the ordinary quantiles. We study the process of $R$-estimators of the slope parameters over $α\in[0,1]$, generated by the Hájek rank scores. We show that this process, standardized by $f(F ^{-1}(α))$ under exponentially tailed $F$, converges to the vector of independent Brownian bridges. The same course is true for the process of the slope components of $α$-regression quantile.

math.ST↗

arXiv2019

Uniform minimum risk equivariant estimates for moment condition models

We consider semiparametric moment condition models invariant to transformation groups. The parameter of interest is estimated by minimum empirical divergence approach, introduced by Broniatowski and Keziou (2012). It is shown that the minimum empirical divergence estimates, including the empirical likelihood one, are equivariants. The minimum risk equivariant estimate is then identied to be any one of the minimum empirical divergence estimates minus its expectation conditionally to maximal invariant statistic of the considered group of transformations. An asymptotic approximation to the conditional expectation, is obtained, using the result of Jureckov{á} and Picek (2009).

math.ST↗

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↗

arXiv2012

Nonparametric multivariate rank tests and their unbiasedness

Although unbiasedness is a basic property of a good test, many tests on vector parameters or scalar parameters against two-sided alternatives are not finite-sample unbiased. This was already noticed by Sugiura [Ann. Inst. Statist. Math. 17 (1965) 261--263]; he found an alternative against which the Wilcoxon test is not unbiased. The problem is even more serious in multivariate models. When testing the hypothesis against an alternative which fits well with the experiment, it should be verified whether the power of the test under this alternative cannot be smaller than the significance level. Surprisingly, this serious problem is not frequently considered in the literature. The present paper considers the two-sample multivariate testing problem. We construct several rank tests which are finite-sample unbiased against a broad class of location/scale alternatives and are finite-sample distribution-free under the hypothesis and alternatives. Each of them is locally most powerful against a specific alternative of the Lehmann type. Their powers against some alternatives are numerically compared with each other and with other rank and classical tests. The question of affine invariance of two-sample multivariate tests is also discussed.

math.ST↗

arXiv2008

Regression rank scores in nonlinear models

Consider the nonlinear regression model $Y_i=g({\bf x}_i,\boldmath $θ$)+e_i,\quad i=1,...,n$(1) with ${\bf x}_i\in \mathbb{R}^k,$ $\boldmathθ=(θ_0,θ_1,...,θ_p)^{\prime}\in \boldmath $Θ$$ (compact in $\mathbb{R}^{p+1}$), where $g({\bf x},\boldmath $θ$)=θ_0+\tilde{g}({\bf x},θ_1,...,θ_p)$ is continuous, twice differentiable in $\boldmath $θ$$ and monotone in components of $\boldmath $θ$$. Following Gutenbrunner and Jurečková (1992) and Jurečková and Procházka (1994), we introduce regression rank scores for model (1), and prove their asymptotic properties under some regularity conditions. As an application, we propose some tests in nonlinear regression models with nuisance parameters.

math.ST↗