Translation-invariant estimator in a quadratic measurement
error model
Alexander Kukush,
Kiev National Taras Shevchenko University
E-mail: alexander_kukush@univ.kiev.ua
An adjusted least squares (ALS) estimator is derived that
yields a translation-invariant and consistent estimate of
the parameters of an implicit quadratic measurement error
model (IQMEM). Consistency means that the estimate
converges to the true value of the parameter, as the
sample size tends to infinity. In addition, a consistent
estimator for the measurement error noise variance is
proposed. Important assumptions are: (1) all errors are
i.i.d. and (2) the error distribution is normal. The
estimators for the quadratic measurement error model are
used to estimate consistently conic sections and
ellipsoids. In the IQMEM, the ordinary least squares (OLS)
estimator is inconsistent, and due to the non-linearity of
the model, the orthogonal regression (OR) estimator is
inconsistent as well. Simulation examples, comparing the
ALS estimator with the OLS method and the OR method, are
discussed for the ellipsoid fitting problem.
The results are joint with Prof. S. Van Huffel and I.
Markovsky (Belgium). The consistency is shown in [1], and
the numerical algorithm is proposed in [2].
References
[1] A.Kukush, I.Markovsky, and S.Van Huffel, Consistent
estimation in an implicit quadratic measurement error
model, Tech. Rep. 02-115, SCD division, Dept. EE,
K.U.Leuven, 2002. To appear in Computational Statistics
and Data Analysis, 2004. Available at:
ftp://ftp.esat.kuleuven.ac.be/pub/SISTA/markovsky/reports/02-115.ps.gz
[2] A.Kukush, I.Markovsky, and S.Van Huffel, Adjusted
least squares fitting in quadratic measurement error
models, Tech. Rep. 02-116, SCD division, Dept. EE,
K.U.Leuven, 2002. To appear in Numerische Mathematik,
2004. Available at:
ftp://ftp.esat.kuleuven.ac.be/pub/SISTA/markovsky/reports/02-116.ps.gz