On local linear estimation in nonparametric errors-in-variables models
Local linear methods are applied to a nonparametric regression model with normal errors in the variables and uniform distribution of the variables. The local neighborhood is determined with help of deconvolution kernels. Two different linear estimation method are used: the naive estimator and the to...
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irk-123456789-45002009-11-20T12:00:38Z On local linear estimation in nonparametric errors-in-variables models Zwanzig, S. Local linear methods are applied to a nonparametric regression model with normal errors in the variables and uniform distribution of the variables. The local neighborhood is determined with help of deconvolution kernels. Two different linear estimation method are used: the naive estimator and the total least squares estimator. Both local linear estimators are consistent. But only the local naive estimator delivers an estimation of the tangent. 2007 Article On local linear estimation in nonparametric errors-in-variables models / S. Zwanzig // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 316-327. — Бібліогр.: 5 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4500 en Інститут математики НАН України |
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Local linear methods are applied to a nonparametric regression model with normal errors in the variables and uniform distribution of the variables. The local neighborhood is determined with help of deconvolution kernels. Two different linear estimation method are used: the naive estimator and the total least squares estimator. Both local linear estimators are consistent. But only the local naive estimator delivers an estimation of the tangent. |
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Zwanzig, S. |
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Zwanzig, S. On local linear estimation in nonparametric errors-in-variables models |
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Zwanzig, S. |
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Zwanzig, S. |
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On local linear estimation in nonparametric errors-in-variables models |
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On local linear estimation in nonparametric errors-in-variables models |
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On local linear estimation in nonparametric errors-in-variables models |
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On local linear estimation in nonparametric errors-in-variables models |
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On local linear estimation in nonparametric errors-in-variables models |
| title_sort |
on local linear estimation in nonparametric errors-in-variables models |
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Інститут математики НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/4500 |
| citation_txt |
On local linear estimation in nonparametric errors-in-variables models / S. Zwanzig // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 316-327. — Бібліогр.: 5 назв.— англ. |
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AT zwanzigs onlocallinearestimationinnonparametricerrorsinvariablesmodels |
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2025-07-02T07:43:48Z |
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2025-07-02T07:43:48Z |
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1836520286788255744 |
| fulltext |
Theory of Stochastic Processes
Vol.13 (29), no.1-2, 2007, pp.316-327
SILVELYN ZWANZIG
ON LOCAL LINEAR ESTIMATION IN
NONPARAMETRIC ERRORS-IN-VARIABLES
MODELS
Local linear methods are applied to a nonparametric regression model
with normal errors in the variables and uniform distribution of the
variables. The local neighborhood is determined with help of decon-
volution kernels. Two different linear estimation method are used:
the naive estimator and the total least squares estimator. Both local
linear estimators are consistent. But only the local naive estimator
delivers an estimation of the tangent.
1. Introduction
Errors-in-variables models are essentially more complicated as ordinary
regression models. The design points are observed with errors only, such
that the models include an increasing number of nuisance parameters. Nev-
ertheless it is wanted to estimate a regression function belonging to some
smoothness class.
Fan and Truong (1993) applied the deconvolution technique of density
estimation to nonparametric regression with errors in variables. The main
idea was to use kernel estimators with deconvolution kernels.
In ordinary regression a method of bias reducing is the local polynomial
regression, see [1]. Local linear regression has two main aspects.
- Around the wanted x a local neighborhood is defined.
- The regression function is approximated by its tangent at x. This
tangent is estimated by usual methods with observations coming from the
neighborhood.
The definition of the neighborhood of x is not trivial in errors-in-variables
models, because observations of design points near the wanted x can come
from a design point lying far away.
Invited lecture.
2000 Mathematics Subject Classifications: 62G08, 62G05, 62G20
Key words and phrases: error in variable, measurement errors, nonparametric regres-
sion, deconvolution, kernel methods, local linear regression, naive estimation, total least
squares.
316
LOCAL LINEAR ESTIMATION 317
J. Staudenmayer and D. Ruppert (2004) applied the local polynomial
approach to errors-in-variables models. They derived asymptotic results for
decreasing error variances. The local neighborhood is described by weights
basing on ordinary kernels at the observed variables. The tangent (or the
best polynomial) was estimated by a weighted naive estimator.
In this paper we are interested in consistent local linear regression, where
the consistency is based on increasing sample size. Unfortunately the best
possible rates are slow. For normal errors of the variables the best possible
rate is OP
(
(log n)−
1
2
)
.
In this paper we describe the local environment by deconvolution kernels.
That is an adjustment of the errors in variables. As estimation methods for
the tangent two different estimators from the theory of the linear errors-in-
variables model are taken: a weighted linear naive estimator and a weighted
total least squares estimator. The unweighted naive estimator has a bias in
usual linear errors-in-variables models. The unweighted total least squares
estimator is consistent and is the maximum likelihood estimator in linear
errors-in-variables models with normal error distributions.
The main result is that both procedures deliver consistent estimators,
but only the weighted naive estimator is an estimator of the tangent. The
weighted total least squares estimator estimates something else. The limit
value of the slope is derived. A heuristic interpretation of this effect may
be, that the weights basing on the deconvolution kernels and the total least
squares estimation principle are two independent adjustments for the same
thing.
2. The model and methods
Let the observations (x1, y1), ...(xi, yi), ...(xn, yn) be independently dis-
tributed, generated by
yi = g (ξi) + εi (1)
xi = ξi + δi. (2)
The error variables εi, δi, i = 1, ..., n are mutually independent with
expectation zero and bounded fourth moments, with V ar (εi) = σ2 and
V ar (ε2
i ) ≤ μ4
ε. The errors in the errors-in-variables equation (2) are normal
distributed
δi ∼ N(0, σ2) i.i.d., σ2 > 0. (3)
The unobserved design points come from an uniform distribution
ξi ∈ [0, 1] , i.i.d. U [0, 1]. (4)
and ξi, εi, δi are mutually independent. We assume a smooth regression
function g ∈ C2
[0,1] with
318 SILVELYN ZWANZIG
∣∣∣g(2) (x) − g(2) (x + δ)
∣∣∣ ≤ Lδ. (5)
The aim is to estimate g(x) for a given x ∈ (0, 1).
Further we introduce kernel functions K(.) with∫ ∞
−∞
K(u)du = 1, K(u) = K(−u),
∫ ∞
−∞
K(u)2du = μK < ∞ (6)
and with compact supported Fourier transform
ΦK (t) =
∫
exp(itu)K(u)du, ΦK (t) = 0 for t < a, t > b. (7)
Furthermore we assume that the second derivative of the kernel exists and
that for some constants μ′
K , μ′′
K , cK , c′K∫ ∞
−∞
K ′(u)2du = μ′
K < ∞;
∫ ∞
−∞
K ′′(u)2du = μ′′
K < ∞, (8)
∣∣∣u4K(u)
∣∣∣ ≤ cK ,
∣∣∣u4K ′(u)
∣∣∣ ≤ c′K . (9)
Examples for kernels fulfilling these conditions (6) - (9) are
K2(u) = 48 cos(u)
πu4 (1 − 15
u2 ) − 144 sin(u)
πu5 (2 − 5
u2 ) and K3(u) = 3
8π
(
sin(u
4 )
(u
4 )
)4
.
Note the sinc kernel K1(u) = 1
π
sin(u)
u
does not fulfill the condition (9).
We set the bandwidth hn = C(log n)−
1
2 , C sufficiently large. That is the
optimal bandwidth in the model (1) - (3), see [2].
In ordinary local linear regression common local weights are
wi (x) =
K
(
ξi−x
hn
)
∑n
i=1 K
(
ξi−x
hn
) . (10)
In errors-in-variables models the design points ξi are unknown, that is why
we have chosen
w∗
i (x) =
K∗
hn
(
xi−x
hn
)
∑n
i=1 K∗
hn
(
xi−x
hn
) , (11)
where K∗
hn
is the deconvolution kernel of K defined by
K∗
hn
(u) =
1
2π
∫
exp(−itu +
1
2
σ2 t2
h2
n
) ΦK (t) dt. (12)
Note, that the dominators in (10) and (11) are positive with increasing
probability, because they are consistent density estimators, compare also
(22).
LOCAL LINEAR ESTIMATION 319
For weighted means and weighted quadratic forms with weights (10) we
use an analog notation as in [3]: xw, ξw, mξξ, myx, ..., mgg. If the weights (11)
are used, then we write x∗
w, ξ
∗
w, m∗
ξξ, m∗
yx, ..., m
∗
gg. Thus xw =
∑n
i=1 wi(x)xi,
x∗
w =
∑n
i=1 w∗
i (x)xi and mxy =
∑n
i=1 wi(x)(xi − xw)(yi − yw) and m∗
xy =∑n
i=1 w∗
i (x)(xi − x∗
w)(yi − y∗
w) and so on.
Denote the local linear approximation of g(ξ) by t(ξ) = β0 + β1(ξ − x).
Then the local naive estimator is defined as
ĝnaive(x) = y∗
w − m∗
xy
m∗
xx
(x∗
w − x). (13)
Note that ĝnaive(x) = t̂naive(0) = β̂0,naive, where
(
β̂0,naive, β̂1,naive
)T
= arg min
β0,β1
n∑
i=1
w∗
i (x) (yi − t(xi))
2 .
The local total least squares estimator is defined for m∗
xy �= 0 as
ĝtls(x) = y∗
w − β̂1,tls (x∗
w − x) (14)
with
β̂1,tls =
m∗
yy − m∗
xx +
√
(m∗
yy − m∗
xx)
2 + 4
(
m∗
xy
)2
2m∗
xy
. (15)
Note that ĝtls(x) = t̂tls(0) = β̂0,tls, where
(
β̂0,tls, β̂1,tls
)T
= arg min
β0,β1
n∑
i=1
w∗
i (x) min
ξ
[
(yi − t(ξ))2 + (xi − ξ)2
]
.
3. Main result
In this section the main theorems are proved. The proofs are based on
auxiliary results shown in Section 4.
The following theorem states the consistence of the local naive estima-
tor. Further it is shown that the naive estimator is really an estimator of
the tangent.
Theorem. In the model (1) - (5) and for kernels with (6) - (9) it holds
1. β̂1,naive = g′(x) + OP
(
(log n)−
1
2
)
2. ĝnaive(x) = g(x) + OP
(
(log n)−
1
2
)
320 SILVELYN ZWANZIG
Proof.
1. Using (37) and (38) we get
β̂1,naive =
m∗
xy
m∗
xx
=
−σ2g′(x)
−σ2
+ Op (hn) = g′(x) + Op (hn) .
2. Remember (13). Applying (35), (36) and that β̂1,naive is stochastically
bounded, we obtain the statement.
The following theorem delivers the results for the local total least squares
estimator. The estimator is consistent but does not yield a consistent of the
tangent.
Theorem. In the model (1) - (3) and for kernels with (6) - (9) it holds
1. for g′(x) �= 0
β̂1,tls = − 1
g′(x)
(
1 +
√
1 + g′(x)2
)
+ OP
(
(log n)−
1
2
)
.
2. ĝtls(x) = g(x) + OP
(
(log n)−
1
2
)
Proof.
1. Introduce
F (x) =
⎧⎪⎨⎪⎩
F1(x) for m∗
xy > 0
undefined for m∗
xy = 0
F2(x) for m∗
xy < 0
, (16)
where F1(x) = 1
2
x + 1
2
√
x2 + 4 and F2(x) = 1
2
x − 1
2
√
x2 + 4. Both
functions are increasing. F1 is convex and F2 is concave. The first
derivatives are bounded by 1 for all x. It holds
β̂1,tls = F
(
m∗
yy − m∗
xx
m∗
xy
)
.
Consider zn =
m∗
yy−m∗
xx
m∗
xy
. From (37), (38) and (39) it follows for g′(x) �=
0 that zn = −z + Op (hn) , where z = − 2
g′(x)
. Denote
βlim = − 1
g′(x)
(
1 +
√
1 + g′(x)2
)
.
It holds for g′(x) < 0 that F1(z) = βlim and for g′(x) > 0 that
F2(z) = βlim. Suppose g′(x) > 0, then from (37) follows that
limn→∞ P
(
m∗
xy > 0
)
= 0. We have for T > 0
P
(∣∣∣β̂1,tls − βlim
∣∣∣ > T hn
)
= P
(∣∣∣β̂1,tls − βlim
∣∣∣ > T hn/m∗
xy < 0
)
+ o(1).
LOCAL LINEAR ESTIMATION 321
Further
P (|F2(zn) − βlim| > T hn) = P (|F2(zn) − F2(z)| > T hn) = o(1)
for T → ∞. Assume g′(x) < 0, then limn→∞ P
(
m∗
xy < 0
)
= 0. For
T > 0
P
(∣∣∣β̂1,tls − βlim
∣∣∣ > T hn
)
= P (|F1(zn) − F1(z)| > T hn) + o(1).
Hence
lim
T→∞
lim sup
n→∞
P
(∣∣∣β̂1,tls − βlim
∣∣∣ > T hn
)
= 0.
2. Remember (14). Applying (35), (36) and that β̂1,tls is stochastically
bounded, we obtain the statement.
3. Auxiliary results
In [2] it is shown that ExK
∗
h(
x−a
h
) = K( ξ−a
h
). Furthermore we have the
following integral equations.
Lemma. Under x = ξ + δ, δ ∼ N(0, σ2) and for kernels with (6) - (9) it
holds for all a that the expectation with respect to δ is
1.
Ex/ξK
∗
h(
x − a
h
) (x − ξ) =
σ2
h
K
′
(
ξ − a
h
) (17)
2.
Ex/ξK
∗
h(
x − a
h
) (x − ξ)2 =
σ4
h2
K
′′
(
ξ − a
h
) + σ2K(
ξ − a
h
). (18)
Proof.
1. Note that
K ′(u) =
1
2π
∫
(−it) exp(−itu) ΦK (t) dt. (19)
Using exp(−it(x−a
h
)) = exp(−it( ξ−a
h
) exp(−it(x−ξ
h
) and (12) we get
Ex/ξK
∗
h(
x − a
h
)(x − ξ) =
1
2π
∫
exp(−it(
ξ − a
h
))
ΦK (t)
Φδ
(
t
h
)I2 (t) dt
where Φδ (t) = exp(−1
2
σ2t2) and I2(t) =
∫
u exp(−i t
h
u) ϕδ (u) dx with
ϕδ (u) = 1√
2π
exp(−u2
2
). Using the quadratic decomposition we get
exp(−i
t
h
u) ϕδ (u) = Φδ
(
t
h
)
ϕ
(−itσ2
h
,σ2)
(u) , (20)
322 SILVELYN ZWANZIG
where
ϕ(
−itσ2
h
,σ2
) (u) =
1√
2π
σ exp(− 1
2σ2
(
u + it
σ2
h
)2
).
Thus
I2(t) = Φδ
(
t
h
) ∫
u ϕ(
−itσ2
h
,σ2
)(u)du = −it
σ2
h
Φδ
(
t
h
)
.
Summarizing and applying (19) we obtain
Ex/ξK
∗
h(
x − a
h
)(x − ξ) =
σ2
h
1
2π
∫
(−it) exp(−it(
ξ − a
h
)) ΦK (t) dt
=
σ2
h
K ′
(
ξ − a
h
)
.
2. Note that
K ′′(u) = − 1
2π
∫
t2 exp(−itu) ΦK (t) dt. (21)
In the same way as above we get
Ex/ξK
∗
h(
x − a
h
)(x − ξ)2 =
1
2π
∫
exp(−it(
ξ − a
h
))
ΦK (t)
Φδ
(
t
h
)I3 (t) dt,
with I3(t) =
∫
u2 exp(−i t
h
u)ϕδ (u) du. Applying (20) we obtain
I3(t) = Φδ
(
t
h
) ∫
u2ϕ
(−itσ2
h
,σ2)
(u) du =
(
σ2 − σ4
(
t
h
)2
)
Φδ
(
t
h
)
.
Summarizing and using (21) we get
Ex/ξK
∗
h(
x − a
h
)(x − ξ)2
=
1
2π
∫
exp(−it(
ξ − a
h
)) ΦK (t)
(
σ2 − σ4
(
t
h
)2
)
dt
=
σ4
h2
K ′′
(
ξ − a
h
)
+ σ2K
(
ξ − a
h
)
.
Lemma. In the model (1) - (5) and for kernels with (6) - (9) and hn =
C (log n)−
1
2 , C sufficiently large, it holds that:
1.
1
nhn
n∑
i=1
K∗
hn
(
xi − x
hn
)
= 1 + OP
(
(log n)−
1
2
)
(22)
LOCAL LINEAR ESTIMATION 323
2.
1
nhn
n∑
i=1
K∗
hn
(
xi − x
hn
)
xi = x + OP
(
(log n)−
3
2
)
(23)
3.
1
nhn
n∑
i=1
K∗
hn
(
xi − x
hn
)
x2
i −
1
nhn
n∑
i=1
K
(
ξi − x
hn
)
ξ2
i (24)
= −σ2 + OP
(
(log n)−
1
2
)
4.
1
nhn
n∑
i=1
K∗
hn
(
xi − x
hn
)
yi =
1
nhn
n∑
i=1
K
(
ξi − x
hn
)
g(ξi)+OP
(
(log n)−
1
2
)
(25)
5.
1
nhn
n∑
i=1
K∗
hn
(
xi − x
hn
)
y2
i (26)
=
1
nhn
n∑
i=1
K
(
ξi − x
hn
) (
g(ξi)
2 + σ2
)
+ OP
(
(log n)−
1
2
)
6.
1
nhn
n∑
i=1
K∗
hn
(
xi − x
hn
)
xiyi − 1
nhn
n∑
i=1
K
(
ξi − x
hn
)
ξig(ξi)(27)
= −σ2g′(x) + OP
(
(log n)−
3
2
)
Proof.
The proofs are based on a sum of i.i.d. random variables of the form
S = 1
nh
∑n
i=1 Zi. For Vn =
∑n
i=1 V ar (Zi) we have
S = ES + Op
(
1
(nh)2
Vn
)
. (28)
Here only the main steps are given, but all details are shown in [5].
1. First we consider S1 = 1
nh
∑n
i=1 Z1,i with Z1,i = K∗
h
(
xi−x
h
)
. We have
for the expectation with respect to δi
Exi/ξi
Z1,i = Exi/ξi
K∗
h
(
xi − x
h
)
= K
(
ξi − x
h
)
.
324 SILVELYN ZWANZIG
Further E
(
K∗
h
(
xi−x
h
))2 ≤ hVK (h) with VK (h) =
∫
K∗
h(u)2du. From
the Parseval equality and (12) it follows that
VK (h) =
∫ ∣∣∣ΦK∗
h
(t)
∣∣∣2 dt =
∫
exp(σ2 t2
h2
) ΦK (t)2 dt.
For kernels K with compactly supported Fourier transform, (7), we
get
VK (h) ≤ max
t∈[a,b]
(exp(σ2 t2
h2
) ΦK (t)2) ≤ const exp(c0h
−2). (29)
Thus for h = hn,
V ar(S1) = 1
nhn
VK (hn) ≤ const exp(c0h
−2
n − ln n − ln(hn)) < const√
n
.
The expectation of S1 with respect to all δi is the ordinary density
estimator of pξ : ES1 = p̂ξ (x) with
p̂ξ (x) =
1
nhn
n∑
i=1
K
(
ξi − x
hn
)
.
Using the model assumption (3) we get (for more details see the
report [5])
p̂ξ (x) = 1 + OP
(
(log n)−
1
2
)
. (30)
2. Here S2 = 1
nh
∑n
i=1 Z2,i with Z2,i = K∗
h
(
xi−x
h
)
xi. From (17) it follows
that
Exi/ξi
Z2,i =
σ2
h
K ′
(
ξi − x
h
)
+ K
(
ξi − x
h
)
ξi. (31)
Further E (Z2,i)
2 ≤ E
(
K∗
h
(
xi−x
h
)
xi
)2 ≤ hc2(ξi)VK (h) . Thus using
a similar argumentation as in (29) we get that the V ar (S2) < const√
n
.
The conditional expectation of S2 can be interpreted as a sum of an
ordinary estimator of p′ξ and of the expectation of an ordinary kernel
estimator
f̂(x) =
1
nh
∑
K
(
ξi − x
h
)
xi (32)
of f(x) = x. Using the model assumption (3) we get (for more details
see the report [5]) for h = hn
ES2 = x + OP
(
h3
n +
1
nhn
)
= x + OP
(
(log n)−
3
2
)
. (33)
LOCAL LINEAR ESTIMATION 325
3. Now we have a sum of independent r.v. S3 = 1
nh
∑n
i=1 Z3,i with Z3,i =
K∗
h
(
xi−x
h
)
x2
i . Applying (17), (18) and (31), we obtain Exi/ξi
Z3,i =
K(
ξi − x
h
)
(
σ2 + ξ2
i
)
+ 2ξi
σ2
h
K ′
(
ξi − x
h
)
+
σ4
h2
K
′′
(
ξi − x
h
).
Further we apply similar argumentation as above and use
p̂′ξ (x) = 0 + OP
(
hn +
1
nhn
)
and
1
nhn
n∑
i=1
σ2
hn
K ′
(
ξi − x
hn
)
ξi = −σ2 + OP
(
h3
n +
1
nh3
n
)
.
We estimate V ar (S3) < const√
n
in a similar way as above.
4. Can be shown similarly, see [5].
We use Exi/ξi
Z4,i = K
(
ξi−x
h
)
g (ξi) .
5. Can be shown similarly, see [5].
We use Exi/ξi
Z5,i = K
(
ξi−x
h
) (
g (ξi)
2 + σ2
)
.
6. Can be shown similarly, see [5] with Z6,i = K∗
h
(
xi−x
h
)
yixi. We use
Exi/ξi
Z6,i = E
(
K∗
h
(
xi − x
h
)
xi
)
g (ξi)
=
(
K
(
ξi − x
h
)
ξi +
σ2
h
K ′
(
ξi − x
h
))
g (ξi) .
and that
1
nhn
n∑
i=1
σ2
hn
K ′
(
ξi − x
hn
)
g (ξi) = g′(x) + OP
(
h3
n +
1
nh3
n
)
.
Now we summarize the results for means and quadratic forms with
weights w∗
i .
Lemma. In the model (1) - (5) and for kernels with (6) - (9) and
hn = C (log n)−
1
2 , C sufficiently large, it holds that:
mξξ = OP
(
(log n)−1
)
(34)
xw∗ = x + OP
(
(log n)−
1
2
)
(35)
yw∗ = g(x) + OP
(
(log n)−
1
2
)
(36)
m∗
xx = −σ2 + OP
(
(log n)−
1
2
)
(37)
m∗
xy = −σ2g′(x) + OP
(
(log n)−
1
2
)
(38)
m∗
yy = σ2 + OP
(
(log n)−
1
2
)
(39)
326 SILVELYN ZWANZIG
Proof.
1. We decompose mξξ =
∑n
i=1 wi (x) (ξi − x)2 −
(
ξw − x
)2
thus mξξ ≤∑n
i=1 wi (x) (ξi − x)2 .
Consider 1
nh
∑n
i=1 K
(
ξi−x
h
)
(ξi − x)2 as S = 1
nh
∑n
i=1 Zi . We get from
(9) that
EZi ≤ h2
∫ 1−x
h
0−x
h
|K (u)| u2 du ≤ h2cK
∣∣∣∣∣
∫ 1−x
h
0−x
h
1
|u2|du
∣∣∣∣∣ = O(h3).
The variance term is estimated by E
(
K
(
ξ−x
h
)
(ξ − x)2
)2
= O(h).
Remember (30) we get
∑n
i=1 wi (x) (ξi − x)2 = Op (hn) .
2. Applying (22) and (23) we get
xw∗ =
1
1 + Op (hn)
(
x + Op
(
h3
n
))
= x + Op (hn) .
3. Analogously we estimate yw∗ by using (22) (25), (30)
yw∗ =
1
nhn
∑n
i=1 K
(
ξi−x
hn
)
g (ξi) + Op (hn)
1 + Op (hn)
= gwp̂ξ (x) + Op (hn) = gw + Op (hn) .
Furthermore it holds gw = g(x)+Op
(
h2
n + 1
nhn
)
, compare for instance
Theorem 4 in [5]. Thus yw∗ = g(x) + Op (hn) .
4. Using the decomposition m∗
xx =
∑n
i=1 w∗
i x
2
i − (xw∗)2 . Because of (22),
(24) it holds that
n∑
i=1
w∗
i x
2
i =
1
1 + Op (hn)
1
nhn
n∑
i=1
K∗
hn
(
xi − x
hn
)
x2
i
= −σ2 +
1
nhn
n∑
i=1
K
(
ξi − x
hn
)
ξ2
i + Op (hn) .
Applying (30) we obtain
∑n
i=1 w∗
i x
2
i =
∑n
i=1 wiξ
2
i − σ2 + Op (hn) .
Thus m∗
xx = mξξ−σ2 +Op (hn) . Then the statement (37) follows from
(34).
5. Using the decomposition m∗
xy =
∑n
i=1 w∗
i xiyi − yw∗xw∗ and (22), (27),
and (30) we get
∑n
i=1 w∗
i xiyi =
1
1 + Op (hn)
(
1
nhn
n∑
i=1
K
(
ξi − x
hn
)
ξig(ξi) − σ2g′(x) + Op (hn)
)
=
n∑
i=1
wiξig(ξi) − σ2g′(x) + Op (hn) .
LOCAL LINEAR ESTIMATION 327
Under (5) we can show (compare Theorem 4 in [5]) that
mξg = g′(x)mξξ + Op
(
h2
n +
1
nhn
)
.
Then (38) follows from (34), (35), and (36).
6. We consider now m∗
yy =
∑n
i=1 w∗
i y
2
i − (yw∗)
2 . From (22) and (26) we
get
n∑
i=1
w∗
i y
2
i =
1
nhn
n∑
i=1
K
(
ξi − x
hn
)
(g(ξi)
2 + σ2) + 1 + Op (hn) .
Thus by (30)
n∑
i=1
w∗
i y
2
i =
n∑
i=1
wig(ξi)
2 + σ2 + Op (hn) .
Under (5) we can show (compare Theorem 4 in [5]) that
mgg = g′(x)2mξξ + Op
(
h
5
2
n +
1
nhn
)
.
Then (39) follows from (34), (35) and (36).
Bibliography
1. J. Fan and I. Gjibels (1996), Local Polynomial Modeling and Its Application,
London: Chapman and Hall.
2. J. Fan and Y. Truong (1993), Nonparametric regression with errors in vari-
ables. Ann. Statist. 21. 1900-1925.
3. W.A. Fuller (1987), Measurement Errors Models. Wiley, New York.
4. J. Staudenmayer and D. Ruppert (2004), Local polynomial regression and
simulation extrapolation. J.R.Soc.B. 66, Part 1, 17 -30.
5. S. Zwanzig (2006), On local linear estimation in nonparametric errors-in-
variables models. U.U.D.M. Report 2006-12. Uppsala University.
Department of Mathematics, Uppsala University, SE-75106 Uppsala,
Box 480 Sweden.
E-mail address: zwanzig@math.uu.se
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