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ABSTRACT: Under consideration is the problem of estimating the linear regression parameter in the case when the variances of observations
depend on the unknown parameter of the model, while the coefficients (independent variables) are measured with random errors.
We propose a new two-step procedure for constructing estimators which guarantees their consistency, find general necessary
and sufficient conditions for the asymptotic normality of these estimators, and discuss the case in which these estimators
have the minimal asymptotic variance.
Keywordslinear regression–errors in the independent variables–dependence of variance on a parameter–two-step estimation–asymptotically normal estimator
Siberian Mathematical Journal 05/2012; 52(1):113-126. · 0.37 Impact Factor
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ABSTRACT: We consider the linear regression model in the case when the independent variables are measured with errors, while the variances
of the main observations depend on an unknown parameter. In the case of normally distributed replicated regressors we propose
and study new classes of two-step estimates for the main unknown parameter. We find consistency and asymptotic normality conditions
for first-step estimates and an asymptotic normality condition for second-step estimates. We discuss conditions under which
these estimates have the minimal asymptotic variance.
Keywordslinear regression–errors in independent variables–replicated regressors–dependence of variances on a parameter–two-step estimates–consistent estimate–asymptotically normal estimate
Siberian Mathematical Journal 04/2012; 52(4):711-726. · 0.37 Impact Factor
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ABSTRACT: We consider the problem of estimating the unknown parameters of linear regression in the case when the variances of observations
depend on the unknown parameters of the model. A two-step method is suggested for constructing asymptotically linear estimators.
Some general sufficient conditions for the asymptotic normality of the estimators are found, and an explicit form is established
of the best asymptotically linear estimators. The behavior of the estimators is studied in detail in the case when the parameter
of the regression model is one-dimensional.
Siberian Mathematical Journal 04/2012; 50(2):302-315. · 0.37 Impact Factor
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ABSTRACT: We consider the problem of estimating the unknown parameter of the one-dimensional analog of the Michaelis-Menten equation
when the independent variables are measured with random errors. We study the behavior of the explicit estimates that we have
found earlier in the case of known independent variables and establish almost necessary conditions under which the presence
of the random errors does not affect the asymptotic normality of these explicit estimates.
Siberian Mathematical Journal 01/2008; 49(3):474-497. · 0.37 Impact Factor
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ABSTRACT: Considering the linear-fractional regression problem with errors in independent variables, we construct and study asymptotically
optimal estimators for unknown parameters in the case of violation of the classical regression assumptions (the variances
of the observations are different and depend on the unknown parameters).
Siberian Mathematical Journal 01/2006; 47(6):1128-1153. · 0.37 Impact Factor
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ABSTRACT: Under consideration is the problem of estimating unknown parameters in the Michaelis–Menten equation which is frequent in natural sciences. The authors suggest and study asymptotically normal explicit estimates of unknown parameters which often have a minimal covariance matrix.
Siberian Mathematical Journal 04/2001; 42(3):517-536. · 0.37 Impact Factor
