Note On Two
Estimators For The Polynomial Regression With Errors In The Variables
MOHAMMED I. AGEEL
Department
of Mathematics, College of Science,
King
Khalid University,
Abha,
Saudi Arabia
دراسة عن تقدرين لدالة
الانحدار متعددة الحدود ذات الأخطاء
في المتغيرات
مقدري معالم دالة بعلاقة انحدار تربيعية ذات الأخطاء في المتغيرات لحالتين حين وجود الخطأ، أو عدمه في المعادلة، ثم تعميمه في أ نموذج بدالة ذات علاقة متعددة الحدود بأي درجة، وبإرتباط أخطاء الاستجابة، ومتغيرات دالة الانحدار، وبأخطاء ليست بالضرورة طبيعية، كماتم اشتقاق مصفوفة التغاير لهذه المقدرات.
The estimators for the parameters of the quadratic regression relationship with errors in the variables for two different cases regarding the presence or absence of errors in the equation are generalized to the model of a polynomial functional relationship of any degree with correlated errors of response and regressor variables which is not necessarily normal error. The asymptotic covariance matrix of these estimators is also derived.
Keywords: Polynomial regression, functional relationship, errors in variables, adjusted least squares.
INTRODUCTION
Many investigators (Fuller
1987; Wolter & Fuller 1982) have considered the quadratic functional relationships
with errors in the variables for two different cases regarding the presence
(case 1) or absence (case 2) of errors in the equation. They developed
estimators for the parameters of the quadratic relationship in both cases,
assuming that, in case 1, the error variance of the regressor variable or, in
case 2, the error variances of dependent and regressor variables be known. In
case 1, the errors of dependent and regressor variables were assumed to be
uncorrelated. In both cases the errors were taken to be normally distributed.
Here, both these
estimators will be generalized to the model of a polynomial functional
relationship of any degree and with correlated errors of dependent and
regressor variables and with not necessarily normal errors. In case 1, the
resulting estimators seem to be
the same as the one developed by Cheng & Schneeweiss (1996). They
derived the asymptotic covariance
matrix of the estimators of case 1.
The same will be done here
for the estimator of case 2. For a discussion of the practical relevance of the
two cases (see Cheng & Schneeweiss (1996)) and the literature cited e.g. in Carroll & Ruppert (1996)
and Cheng & Van Ness (1994).
CASE
1: ERRORS IN THE EQUATION
Consider a polynomial
functional relationship with errors in the equation:
(1)
where 
are i.i.d. random
errors with expectation 0 and covariance matrix
The 
are unobservable
(latent) nonstochastic variables. The regressor error variance 
and the covariance
of regressor
error 
and dependent
variable error 
are assumed to
be known. The variance 
, which contains the error-in-the-equation variance component,
is unknown. If the error variables are jointly normally distributed we have
(N)
It is well-known that replacing
the latent variable 
by its
observable counterpart x in the polynomial relationship and estimating the
parameters 
in the resulting
polynomial regression by OLS (Ordinary Least Squares) yields inconsistent
estimates (Grilliches & Ringstad 1970). As a first step to remove this
inconsistency, Fuller (1987) suggests to view the powers of 
as k+1 different latent regressor variables, for which their
observable counterparts unbiased estimates 
computable from
the data are available, so that a linear functional relationship results:
where the 
are the new
measurement errors, with 
. Let 
, then the model can be written as
(2)
For this linear functional
relationship a consistent estimator of 
can now be constructed if unbiased estimates 
and 
of, respectively,
the covariance matrix 
and the
covariance vector 
are available.
The error adjusted least squares normal equations are given by (all summations
are for 
):
,
or
,
(3)
where the - denotes averages over i = 1, …, n; for the quadratic
relationship (Fuller 1987). Following the idea of Chan & Mak (1985), the
estimates 
of 
are easily
constructed as certain polynomials in 
of degree r.
Their coefficients depend on higher moments of 
up to the order
of 2r, which are assumed to be known. In case of (N) only 
needs to be
known, and the 
can be computed
by the recursive relation 
with 
. For non-normal errors 
the variables 
, 
can be computed
by solving the triangular linear system
, 
, 
, 
where the higher moments
of 
are assumed to be known.
The covariance matrix 
is given by
(4)
The elements of 
are powers of 
and can
therefore be estimated by the variables 
. Let
,
then 
. Thus, an unbiased estimate of 
is given by
.
Similarly,
.
Cheng & Schneeweiss
(1996) derived an unbiased estimate of 
in terms of a
linear combination of the 
, the coefficients of which depend on 
and 
only, 
, and which they denoted by 
. Thus 
. In case of (N), 
. The normal equation (1) can now be written as
with
, 
, and 
.
This is exactly the normal
equations system for the ALS estimator of Cheng & Schneeweiss (1996). Its
solution 
is consistent
for 
and under
general conditions asymptotically normally distributed. The asymptotic
covariance matrix of 
can be estimated
by Cheng & Schneeweiss (1996).
,
where
CASE
2: NO ERRORS IN THE EQUATION
Wolter & Fuller (1982)
construct an estimate of the quadratic functional relationship when the whole
of 
is known. This
corresponds to the case where there is no error in the equation but an only
measurement error is known. The estimator can be computed without any
iterations. It can be generalized to the case of a polynomial functional
relationship. Let 
and let 
be the error covariance
matrix of 
and 
its estimate, as
given below:
Furthermore, let 
, 
, 
, 
, so that
Then a generalization of
Wolter and Fuller’s estimator is given by
,
(5)
where 
is the smallest positive
root (eigenvalue) of
;
see also Moon & Gunst
(1995) for the special case (N).
THE ASYMPTOTIC COVARIANCE
MATRIX OF 
IN (CASE 2)
Under general conditions 
is
asymptotically normally distributed with an asymptotic covariance matrix 
which can be
computed as follows. First note that with 
and 
the estimating
equation (5) for 
can be written
as:
,
(6)
where 
is the smallest
positive eigenvalue and 
the
corresponding eigenvector.
Let 
, where
,
(7)
, 
, and 
. For large n all these differences will be small in
probability and the estimating equation (6) can be expanded as
This can be simplified
with the help of equation (7) and
using the fact that 
to:
Deleting the first
equation of this system we get the following system for 
:
(8)
Now by equation (6):
,
and taking differences
Thus
Substituting this
expression for 
in (8) we get:
(9)
with 
and
(10)
Obviously 
, and by the central limit theorem 
converges in
distribution to a normal distribution with covariance matrix
Thus by equation (9), 
also converges
to a normal distribution with covariance matrix
An estimate of the
asymptotic covariance matrix of 
is given by
,
with
AKNOWLEDGEMENT
The author is greatly
indebted to the referees and editorial board for their useful comments and
suggestions.
REFERENCES
Carroll, R.J., &
Ruppert, D. 1996. The use and misuse
of orthogonal regression in linear errors-in-variables models. The American
Statistician 50: 1-6.
Chan, L.K., & Mak,
M.K. 1985. On the polynomial
functional relationship. Journal of the Royal Statistical Society 47: 510-518.
Cheng, C.L., &
Schneeweiss, H. 1996. The polynomial
Regression with errors in the variables. Research report 386, University of
Munich, Munich, Germany.
Cheng, C.L., & Van
Ness, J.W. 1994. On estimating Linear
Relationship when both variables are subject to error. Journal of the Royal
Statistical Society 56 (1): 167-183.
Fuller, W.A. 1987. Measurement error models, John Wiley, New York.
Grilliches, Z. &
Ringstad, V. 1970.
Errors-in-variables bias in nonlinear context. Econometrica 38: 368-370.
Moon, M.S. & Gunst,
R. 1995. Polynomial measurement error
modeling. Computational Statistics and Data Analysis 19: 1-21.
Wolter, K.H., &
Fuller, W.A. 1982. Estimation of the
quadratic errors-in-variables model. Biometrica 69: 175-182.
(Received 1/10/1419; 18th January 1999, accepted
12/8/1420; 20th November 1999)