arXiv subjects

Alexander Kukush

Publications and source records attributed to Alexander Kukush.

9 recordsLinked to original sources

arXiv2026

Polynomial regression under a mixture of classical and Berkson errors

A polynomial structural regression model is studied, where the covariate is observed with a mixture of the classical and Berkson measurement errors. Both variances of the classical and Berkson errors, as well as some of their higher moments are assumed to be known. Without normality assumptions, consistent estimators of model parameters are constructed using the Corrected Score method, and conditions for their asymptotic normality are given. Under mild conditions, we found pairs of asymptotically independent estimators. A simulation study illustrates the results.

math.ST↗

arXiv2026

Integrability of a composition of functions

The paper deals with the Riemann and Lebesgue integrability of functions. Sufficient conditions and criteria are stated for the integrability of a composition of functions. Relevant counterexamples are constructed based on Cantor's perfect sets and the Cantor staircase function. The main tool of research is Lebesgue's criterion for Riemann integrability.

math.CA↗

arXiv2020

Prediction in polynomial errors-in-variables models

A multivariate errors-in-variables (EIV) model with an intercept term, and a polynomial EIV model are considered. Focus is made on a structural homoskedastic case, where vectors of covariates are i.i.d. and measurement errors are i.i.d. as well. The covariates contaminated with errors are normally distributed and the corresponding classical errors are also assumed normal. In both models, it is shown that (inconsistent) ordinary least squares estimators of regression parameters yield an a.s. approximation to the best prediction of response given the values of observable covariates. Thus, not only in the linear EIV, but in the polynomial EIV models as well, consistent estimators of regression parameters are useless in the prediction problem, provided the size and covariance structure of observation errors for the predicted subject do not differ from those in the data used for the model fitting.

math.ST↗

arXiv2019

Does Regression Approximate the Influence of the Covariates or Just Measurement Errors? A Model Validity Test

A criterion is proposed for testing hypothesis about the nature of the error variance in the dependent variable in linear model, which separates correctly and incorrectly specified models. In the former only measurement errors determine the variance (i.e., dependent variable is correctly explained by independent ones, up to measurement errors), while in the latter the model lacks some independent covariates (or has nonlinear structure). The proposed MEM-V (Measurement Error Model Validity) test checks the validity of the model when both dependent and independent covariates are measured with errors. The criterion has asymptotic character, but numerical simulations outlined approximate boundaries where estimates make sense. A practical example of the implementation of the test is discussed in detail; it shows ability of the test to detect wrong specification even in seemingly perfect models. This type of relation between measurement errors and model specification has not been studied before, and the proposed criterion may stimulate future research in this important area.

stat.ME↗

arXiv2018

Confidence regions in Cox proportional hazards model with measurement errors and unbounded parameter set

Cox proportional hazards model with measurement errors is considered. In Kukush and Chernova (2017), we elaborated a simultaneous estimator of the baseline hazard rate $λ(\cdot)$ and the regression parameter $β$, with the unbounded parameter set $\varTheta=\varTheta_λ\times\varTheta_β$, where $\varTheta_λ$ is a closed convex subset of $C[0,τ]$ and $\varTheta_β$ is a compact set in $\mathbb{R}^m$. The estimator is consistent and asymptotically normal. In the present paper, we construct confidence intervals for integral functionals of $λ(\cdot)$ and a confidence region for $β$ under restrictions on the error distribution. In particular, we handle the following cases: (a) the measurement error is bounded, (b) it is a normally distributed random vector, and (c) it has independent components which are shifted Poisson random variables.

math.PR↗

arXiv2017

Consistent estimation in Cox proportional hazards model with measurement errors and unbounded parameter set

Cox proportional hazards model with measurement error is investigated. In Kukush et al. (2011) [Journal of Statistical Research 45, 77-94] and Chimisov and Kukush (2014) [Modern Stochastics: Theory and Applications 1, 13-32] asymptotic properties of simultaneous estimator $λ_n(\cdot)$, $β_n$ were studied for baseline hazard rate $λ(\cdot)$ and regression parameter $β$, at that the parameter set $Θ=Θ_λ\times Θ_β$ was assumed bounded. In the present paper, the set $Θ_λ$ is unbounded from above and not separated away from $0$. We construct the estimator in two steps: first we derive a strongly consistent estimator and then modify it to provide its asymptotic normality.

math.ST↗

arXiv2017

Goodness-of-fit test in a multivariate errors-in-variables model $AX=B$

We consider a multivariable functional errors-in-variables model $AX\approx B$, where the data matrices $A$ and $B$ are observed with errors, and a matrix parameter $X$ is to be estimated. A goodness-of-fit test is constructed based on the total least squares estimator. The proposed test is asymptotically chi-squared under null hypothesis. The power of the test under local alternatives is discussed.

math.ST↗

arXiv2016

Asymptotic normality of total least squares estimator in a multivariate errors-in-variables model $AX=B$

We consider a multivariate functional measurement error model $AX\approx B$. The errors in $[A,B]$ are uncorrelated, row-wise independent, and have equal (unknown) variances. We study the total least squares estimator of $X$, which, in the case of normal errors, coincides with the maximum likelihood one. We give conditions for asymptotic normality of the estimator when the number of rows in $A$ is increasing. Under mild assumptions, the covariance structure of the limit Gaussian random matrix is nonsingular. For normal errors, the results can be used to construct an asymptotic confidence interval for a linear functional of $X$.

math.PR↗

arXiv2016

Hypothesis testing of the drift parameter sign for fractional Ornstein-Uhlenbeck process

We consider the fractional Ornstein-Uhlenbeck process with an unknown drift parameter and known Hurst parameter $H$. We propose a new method to test the hypothesis of the sign of the parameter and prove the consistency of the test. Contrary to the previous works, our approach is applicable for all $H\in(0,1)$. We also study the estimators for drift parameter for continuous and discrete observations and prove their strong consistency for all $H\in(0,1)$.

math.PR↗