Comparing the efficiency of estimates in concrete errors-in-variables models under unknown nuisance parameters
We consider a regression of y on x given by a pair of mean and variance functions with a parameter vector θ to be estimated that also appears in the distribution of the regressor variable x. The estimation of θ is based on an extended quasi score (QS) function. Of special interest is the case where...
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| Цитувати: | Comparing the efficiency of estimates in concrete errors-in-variables models under unknown nuisance parameters / A. Kukush, A. Malenko, H. Schneeweiss // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 69–81. — Бібліогр.: 11 назв.— англ. |
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irk-123456789-45142009-11-25T12:00:28Z Comparing the efficiency of estimates in concrete errors-in-variables models under unknown nuisance parameters Kukush, A. Malenko, A. Schneeweiss, H. We consider a regression of y on x given by a pair of mean and variance functions with a parameter vector θ to be estimated that also appears in the distribution of the regressor variable x. The estimation of θ is based on an extended quasi score (QS) function. Of special interest is the case where the distribution of x depends only on a subvector α of θ, which may be considered a nuisance parameter. A major application of this model is the classical measurement error model, where the corrected score (CS) estimator is an alternative to the QS estimator. Under unknown nuisance parameters we derive conditions under which the QS estimator is strictly more аfficient than the CS estimator. We focus on the loglinear Poisson, the Gamma, and the logit model. 2007 Article Comparing the efficiency of estimates in concrete errors-in-variables models under unknown nuisance parameters / A. Kukush, A. Malenko, H. Schneeweiss // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 69–81. — Бібліогр.: 11 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4514 en Інститут математики НАН України |
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We consider a regression of y on x given by a pair of mean and variance functions with a parameter vector θ to be estimated that also appears in the distribution of the regressor variable x. The estimation of θ is based on an extended quasi score (QS) function. Of special interest is the case where the distribution of x depends only on a subvector α of θ, which may be considered a nuisance parameter. A major application of this model is the classical measurement error model, where the corrected score (CS) estimator is an alternative to the QS estimator. Under unknown nuisance parameters
we derive conditions under which the QS estimator is strictly more аfficient than the CS estimator. We focus on the loglinear Poisson, the Gamma, and the logit model. |
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Article |
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Kukush, A. Malenko, A. Schneeweiss, H. |
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Kukush, A. Malenko, A. Schneeweiss, H. Comparing the efficiency of estimates in concrete errors-in-variables models under unknown nuisance parameters |
| author_facet |
Kukush, A. Malenko, A. Schneeweiss, H. |
| author_sort |
Kukush, A. |
| title |
Comparing the efficiency of estimates in concrete errors-in-variables models under unknown nuisance parameters |
| title_short |
Comparing the efficiency of estimates in concrete errors-in-variables models under unknown nuisance parameters |
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Comparing the efficiency of estimates in concrete errors-in-variables models under unknown nuisance parameters |
| title_fullStr |
Comparing the efficiency of estimates in concrete errors-in-variables models under unknown nuisance parameters |
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Comparing the efficiency of estimates in concrete errors-in-variables models under unknown nuisance parameters |
| title_sort |
comparing the efficiency of estimates in concrete errors-in-variables models under unknown nuisance parameters |
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Інститут математики НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/4514 |
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Comparing the efficiency of estimates in concrete errors-in-variables models under unknown nuisance parameters / A. Kukush, A. Malenko, H. Schneeweiss // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 69–81. — Бібліогр.: 11 назв.— англ. |
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| first_indexed |
2025-07-02T07:44:28Z |
| last_indexed |
2025-07-02T07:44:28Z |
| _version_ |
1836520328598126592 |
| fulltext |
Theory of Stochastic Processes
Vol.13 (29), no.4, 2007, pp.69–81
ALEXANDER KUKUSH, ANDRII MALENKO, AND HANS
SCHNEEWEISS
COMPARING THE EFFICIENCY OF ESTIMATES
IN CONCRETE ERRORS-IN-VARIABLES
MODELS UNDER UNKNOWN NUISANCE
PARAMETERS
We consider a regression of y on x given by a pair of mean and
variance functions with a parameter vector θ to be estimated that
also appears in the distribution of the regressor variable x. The es-
timation of θ is based on an extended quasi score (QS) function. Of
special interest is the case where the distribution of x depends only
on a subvector α of θ, which may be considered a nuisance parame-
ter. A major application of this model is the classical measurement
error model, where the corrected score (CS) estimator is an alter-
native to the QS estimator. Under unknown nuisance parameters
we derive conditions under which the QS estimator is strictly more
efficient than the CS estimator. We focus on the loglinear Poisson,
the Gamma, and the logit model.
1. Introduction
Suppose that the relation between a response variable y and a covariate
(or regressor) x is given by a pair of conditional mean and variance functions:
E (y|x) =: m(x, θ), V (y|x) =: v(x, θ). (1)
Here θ is an unknown d-dimensional parameter vector to be estimated. The
parameter θ belongs to an open parameter set Θ. The variable x has a
density ρ(x, θ) with respect to a σ-finite measure ν on a Borel σ-field on
the real line. We assume that v(x, θ) > 0, for all x and θ, and that all the
functions are sufficiently smooth. Such a model is called a mean-variance
Alexander Kukush is supported by the Swedish Institute grant SI-01424/2007.
Invited lecture.
2000 Mathematics Subject Classifications: 62J05, 62J12, 62F12, 62F10, 62H12, 62J10.
Key words and phrases: Mean-variance model, measurement error model, quasi score
estimator, corrected score estimator, nuisance parameter, optimality property.
69
70 A.KUKUSH, A.MALENKO, AND H.SCHNEEWEISS
model, cf. Carroll et al. (1995). We want to estimate θ on the basis of an
i.i.d. sample (xi, yi), i = 1, . . . , n.
General statements and results on the polynomial EIVM can be found in
Shklyar et al. (2007) for known nuisance parameter α and in Kukush et al.
(2006) for unknown α. Here we consider other special cases of the models,
which can be treated as mean-variance model (1), i.e. the loglinear Poisson,
the Gamma, and the logit model. We focus on the case of unknown mean
and variance of the latent variable. The case of known nuisance parameters
is considered in Kukush and Schneeweiss (2006).
We assume regularity conditions, which make it possible to differentiate
integrals with respect to parameters and which guarantee that the consid-
ered estimators, generated by unbiased scores, are consistent and asymp-
totically normal with asymptotic covariance matrices that are given by the
sandwich formula, see Carroll et al. (1995). These regularity conditions are
discussed in Kukush and Schneeweiss (2005) for a nonlinear measurement
error model. See also the discussion concerning the sandwich formula in
Schervish (1995), p. 428.
We use the symbols E to denote the expectation of random values, vec-
tors, and matrices and V to denote the variance or the covariance matrix.
We often omit the arguments of functions, e.g., instead of ρ(x, θ) we write ρ
for simplicity. All vectors are considered to be column vectors. We use sub-
scripts to indicate partial derivatives, e.g., ρθ = ∂ρ
∂θ
. For any scalar function
its derivative with respect to a vector is a column vector and for a vector it
is a matrix. We compare real matrices in Lowener order, i.e., for symmetric
matrices A and B of equal size, A < B and A ≤ B means that B − A is
positive definite and B − A is positive semidefinite, respectively.
The paper is organized as follows. Section 2 contains general results
on mean-variance models and measurement error model. In Section 3 spe-
cial cases of Poisson, Gamma, and logit EIVM are treated, and Section 4
concludes.
2. General results
The estimation of θ in the mean-variance model (1) cannot be performed
by the maximum likelihood (ML) approach because the conditional distri-
bution of y given x is by assumption not known. Instead an estimator of
θ is based on an unbiased estimating (or score) function, which we suppose
to be given. A typical example of such an estimating function is a member
of a general class of estimating functions. Let L be the class of all unbiased
linear-in-y score functions (for short: linear score (LS) functions):
SL(x, y; θ) := yg(x, θ)− h(x, θ), (2)
where unbiasedness means that ∀ θ ∈ Θ : E SL(x, y; θ) = 0. Here g and h
are vector-valued functions of dimension d, the same dimension as θ. The
EFFICIENCY COMPARISON OF ESTIMATORS 71
expectation is meant to be carried out under the same θ as the θ of the
argument.
The estimator of θ based on SL is called linear score (LS) estimator θ̂L
and is given as the solution to the equation
n∑
i=1
SL(xi, yi; θ̂L) = 0.
Under general conditions θ̂L exists and is consistent and asymptotically
normal. The asymptotic covariance matrix (ACM) ΣL of θ̂L is given by the
sandwich formula, cf. Heyde (1997),
ΣL = A−1
L BLA−�
L , AL = − E SLθ, BL = E SLS�
L . (3)
AL is supposed to be nonsingular (this is the identifiability condition).
A quasi score functions is defined as follows, see Kukush et al. (2006):
SQ(x, y; θ) :=
(y − m)mθ
v
+ lθ, l := log ρ(x, θ). (4)
The QS estimator θ̂Q of θ is defined as the solution to the equation
n∑
i=1
SQ(xi, yi, θ̂Q) = 0. (5)
The quasi-score function (4) belongs to L, therefore the estimator θ̂Q is
consistent and asymptotically normal under regularity conditions.
Theorem 2.1 (Optimality of QS) Let SL be a score function from the
class L and SQ be the quasi-score function (4). Then
ΣQ ≤ ΣL.
Moreover, ΣL = ΣQ for all θ if, and only if, θ̂L = θ̂Q a.s.
Theorem 2.2 (Strict Optimality of QS)Under the conditions of Theo-
rem 2.1
rank (ΣL − ΣQ) = rank
[(
mgi − hi
vgi
)
,
(
lθi
mθi
)
, i = 1, . . . , d
]
− d, (6)
where rank [·] is the maximum number of linearly independent random vec-
tors inside the square brackets. In particular,
ΣQ < ΣL
if, and only if, the random vectors in (6) are linearly independent.
72 A.KUKUSH, A.MALENKO, AND H.SCHNEEWEISS
If
span
{(
mgi − hi
vgi
)
, i = 1, d
}
∩ span
{(
lθi
mθi
)
, i = 1, d
}
=
{(
0
0
)}
,
then
rank (ΣL − ΣQ) = rank
[(
hi
gi
)
, i = 1, . . . , d
]
.
As an immediate consequence, we have the following corollary:
Corollary 2.1 A sufficient condition for ΣQ < ΣL is that the random
variables
{(mg − h)i, i = 1, . . . , d, lθj
, j ∈ Bθ} (7)
are linearly independent, where {lθj
, j ∈ Bθ} is a basis of span {lθj
, j =
1, . . . , d}.
A reader can find the proofs of Theorems 2.1 and 2.2 in Kukush et al.
(2006).
2.1 Measurement error model
A measurement error model is a model where the response variable y
depends on a latent (unobservable) variable ξ with distribution p(ξ, α). Here
θ is split into two subvectors,
θ = (β�, α�)�, β ∈ R
k, α ∈ R
d−k. (8)
In such a case we call β the unknown parameter of interest and α – the
unknown nuisance parameter.
The variable ξ can be observed only indirectly via a surrogate variable
x, which is related to ξ through a measurement equation of the form
x = ξ + δ, (9)
where the measurement error δ is independent of ξ and y and E δ = 0.
Additionally, we assume δ ∼ N(0, σ2
δ ) with known σ2
δ .
The dependence of y on ξ is either given by a conditional distribution
of y given ξ or simply by a conditional mean function supplemented by a
conditional variance function:
E (y|ξ) = m∗(ξ, β), V (y|ξ) = v∗(ξ, β). (10)
Note that m∗ and v∗ do not depend on α. From (10) we can derive the
conditional mean and variance functions of y given x:
m(x, β, α) := E (y|x) = E [m∗(ξ, β)|x] (11)
v(x, β, α) := V (y|x) = E [v∗(ξ, β)|x] + V [m∗(ξ, β)|x]. (12)
EFFICIENCY COMPARISON OF ESTIMATORS 73
To compute these, we need to know the conditional distribution of ξ
given x, which we can derive from the unconditional distribution of ξ,
p(ξ, α), and the measurement equation (9).
The quasi-score function (4) takes the form
SQ =
(
(y − m)v−1mβ
(y − m)v−1mα + lα
)
. (13)
An important special case for p(ξ, α) is the normal distribution ξ ∼
N(μξ, σ
2
ξ ), σ2
ξ > 0. In this case, x ∼ N(μ, σ2), μ = μξ, σ2 = σ2
ξ + σ2
δ ,
α = (μ, σ)�, and ξ|x ∼ N(μ(x), τ 2) with
μ(x) = Kx + (1 − K)μ, (14)
τ 2 = Kσ2
δ , (15)
where K = σ2
ξ/σ
2 is the reliability ratio, 0 < K < 1.
The subvector lα in the score function SQ takes the special form
lα = (lμ, lσ)� =
(
x − μ
σ2
,
(x − μ)2
σ3
− 1
σ
)�
. (16)
Among the linear score functions, the so-called corrected score (CS)
function is of particular interest. It is given by special functions g and h.
Suppose we can find functions g = g(x, β) and h = h(x, β) such that
E [g|ξ] = v∗−1m∗
β (17)
E [h|ξ] = m∗v∗−1m∗
β. (18)
Then, because of E (yg−h) = E E [(yg−h)|y, ξ] = E (y−m∗)v∗−1m∗
β = 0,
SC :=
(
yg − h
lα
)
is a linear score function within the class L. It is called the corrected score
function of the measurement error model. In a number of important cases
such functions g and h can be found in closed form. But there are also cases
where g and h do not exist, see Stefanski (1989).
2.2 Pre-estimation
In the measurement error model with θ� = (β�, α�), we could also
define a modified QS estimator, which is based on a score function that
instead of (13) consists of the two subvectors (y−m)v−1mβ and lα, implying
an estimator of α which uses the second subvector only. This means that α
would be pre-estimated using only the data xi, not the data yi. We can then
74 A.KUKUSH, A.MALENKO, AND H.SCHNEEWEISS
substitute the resulting estimator α̂ in the first subvector, (y−m)v−1mβ , and
use this to estimate β. We might call this estimator of β a QS estimator with
pre-estimated nuisance parameters or simply pre-estimated QS estimator.
Such a two-step estimation procedure is, of course, simpler to apply than
the one we propose, but according to Theorem 2.1 it is at most as efficient
and often less efficient than the latter one.
There are, however, cases where pre-estimation of the nuisance parame-
ter is in accordance with our QS approach and does not reduce the efficiency
of QS. Suppose that
mα = Amβ (19)
with some nonrandom matrix A (which may depend on θ).
Corollary 2.2 Suppose in a model with nuisance parameters as described
by (8, 9) condition (19) holds, then a sufficient condition for Σ
(β)
Q < Σ
(β)
L is
that the two systems of random variables
{mβi
, i = 1, . . . , k} and {(mg − h)i, i = 1, . . . , k, lαj
, j = 1, . . . , d − k}
are both linearly independent.
For later use, we formulate an extension of Corollary 2.2, which deals
with the case where only part of mα is linearly related to mβ . It can be
proved in the same way as Corollary 2.2.
Corollary 2.3 Suppose in a model with nuisance parameters the nuisance
parameter vector α is subdivided into two subvectors α′ ∈ R
r and α′′ ∈
R
(d−k−r) such that mα′′ = Amβ with some nonrandom matrix A. Suppose
further that there exists a nonrandom nonsingular square matrix B such
that l̃α′′ := Blα′′ is a function of x and α′′ only. Let θ′ = (β�, α′�)�. Then
a sufficient condition for Σ
(θ′)
Q < Σ
(θ′)
L is that the two system of random
variables
{mβi
, i = 1, . . . , k, mαj
, j = 1, . . . , r}
and
{(mg − h)i, i = 1, . . . , k, lαj
, j = 1, . . . , d − k}
are both linearly independent.
The proof of Corollaries 2.2 and 2.3 can be found in Kukush et al. (2006).
3. Special cases
Consider the mean-variance measurement error model of Section 2.1
and assume that the error free mean function m∗ is a function of a linear
predictor in ξ:
m∗(ξ, β) = m̃(β0 + β1ξ), β = (β0, β1)
�. (20)
EFFICIENCY COMPARISON OF ESTIMATORS 75
The mean function m = m(x, β, α) can then be computed as follows:
m = E (m∗|x) = E [m̃{β0 + β1(Kx + (1 − K)μ + τγ)}|x] , (21)
where γ ∼ N(0, 1) and γ is independent of x.
This is a Generalized Linear Model (GLM).
We have mμ = β1(1−K)mβ0 and thus by Corollary 2.3 the QS estimator
of μ is just empirical mean, μ̂ = 1
n
∑n
i=1 xi.
Now suppose that in GLM
m̃′′ = c0m̃
′ (22)
with some constant c0. Then by Corollary 2.3 we obtain the QS estimator
of σ2 is just empirical variance. This property holds for the Poisson and the
Gamma models, but it does not hold for Logit one. We give indirect proof of
the fact that in the Logit model QS estimator of σ2 is not empirical variance
and σ has to be estimated together with other unknown parameters.
3.1 Poisson model
In the loglinear Poisson measurement error model, y|ξ ∼ Po(λ) with
λ = exp(β0 + β1ξ), and x = ξ + δ. Here m∗ = v∗ = λ.
For QS, we have, cf. Shklyar and Schneeweiss (2005),
m(x, θ) = exp
{
β0 + β1μ(x) +
β2
1τ
2
2
}
, v(x, θ) = m2(x, θ)(eβ2
1τ2−1)+m(x, θ)
with μ(x) and τ 2 from (14) and (15), respectively. The β-component of the
CS function is, cf. Shklyar and Schneeweiss (2005),
S
(β)
C = yg−h, g = (1, x)�, h = exp
{
β0 + β1x − 1
2
β2
1σ
2
δ
}
(1, x−σ2
δβ1)
�.
We know that μ and σ2 can be pre-estimated and therefore ΣC − ΣQ is of
the form
ΣL − ΣQ =
(
Σ
(β)
L − Σ
(β)
Q 0
0 0
)
. (23)
We can apply Corollary 2.2. For β1 �= 0, the variables {(mg − h)0, (mg −
h)1, lμ, lσ} are linearly independent, since the functions{
1, x, x2, eβ1Kx, eβ1x, xeβ1Kx, xeβ1x
}
are linearly independent. For the same reason, mβ0 and mβ1 are linearly
independent under β1 �= 0:
mβ0 = econst · eβ1Kx, mβ1 = const · eβ1Kx + const · xeβ1Kx.
Thus by Corollary 2.2, Σ
(β)
Q < Σ
(β)
C under β1 �= 0.
76 A.KUKUSH, A.MALENKO, AND H.SCHNEEWEISS
3.2 Gamma model
In the loglinear Gamma measurement error model, y|ξ follows a Gamma
distribution G(ω, π) with ω = exp(β0 + β1ξ), π > 0, and x = ξ + δ:
f(y|η) =
1
Γ(π)
(π
ω
)π
yπ−1 exp
(
−yπ
ω
)
, y > 0.
Here m∗ = ω and v∗ = π−1ω2, where π−1 corresponds to the dispersion
parameter ϕ, which, according to Kukush et al. (2006), we can assume to
be known. For QS, we have
m(x, θ) = exp
{
β0 + β1μ(x) + β2
1τ
2/2
}
,
v(x, θ) =
(
1 +
1
π
)
exp{2β0+2β1μ(x)+2β2
1τ
2}−exp{2β0+2β1μ(x)+β2
1τ
2}.
The β-component of the CS function is
S
(β)
C = yg−h, g = exp
{
−β0 − β1x − 1
2
β2
1σ
2
δ
}
(1, x+β1σ
2
δ )
�, h = (1, x)�,
cf. Kukush et al. (2005). As in Section 3.1, we can apply Corollary 2.2. For
β1 �= 0, the variables {(mg−h)0, (mg−h)1, lμ, lσ} are linearly independent,
since the functions
{
1, x, x2, eβ1(1−K)x, xeβ1(1−K)x
}
are linearly independent. In addition, as in Section 3.1, mβ0 and mβ1 are
linearly independent under β1 �= 0. Thus by Corollary 2.2, Σ
(β)
Q < Σ
(β)
C
under β1 �= 0.
3.3 Logit model
In the logit measurement error model, y is a binary variable following
a binomial distribution, the mean of which is a logistic function of a linear
predictor in ξ:
y ∼ B(1, π), π = H(η) = (1 + e−η)−1, η = β0 + β1ξ, x = ξ + δ.
For this model, m∗ = π, v∗ = π(1 − π).
For QS, we need the mean and variance functions of y given x, which
are given by
m = E
[{1 + exp(−β0 − β1(Kx + (1 − K)μ + τγ)}−1|x]
, v = m(1 − m),
(24)
where γ ∼ N(0, 1), and γ is independent of x.
EFFICIENCY COMPARISON OF ESTIMATORS 77
We can then construct the quasi score function (13) for θ = (β0, β1, μ, σ)�
with lα from (16). As y is binary, the QS estimator of θ is just the ML es-
timator. Note that, according to properties of GLM, the QS estimator of μ
is the empirical mean x. We cannot say the same for the QS estimator of
σ2, see below.
To find the CS estimator, we start from the maximum likelihood score
function for β in the error free model, which is given by
S
(β)
M =
(
y − 1
1 + e−η
)
(1, ξ)�.
Due to complex zeros in the denominator one cannot solve the deconvolution
problem
E(S
(β)
C |y, ξ) = S
(β)
M .
Therefore we construct a modified corrected score (C*S) function for β, as
a function S
(β)
C∗ = S
(β)
C∗ (y, x, β) such that
E (S
(β)
C∗ |y, ξ) = S
(β)
M (1 + e−η) = {y(1 + e−η) − 1}(1, ξ)�.
S
(β)
C∗ is of the form S
(β)
C∗ = ygc − hc, where gc and hc are functions of x and
β such that
E (gc|ξ) = (1 + e−β0−β1ξ)(1, ξ)�, E (hc|ξ) = (1, ξ)�.
The solutions to these deconvolution problems are
gc = (1 + ea−β1x, x + (x + β1σ
2
δ )e
a−β1x)�, hc = (1, x)�, (25)
where a = −β0 − β2
1σ
2
δ/2. Function S
(β)
C∗ has to be supplemented by the
subvector lα, which yields the conventional estimators of the nuisance pa-
rameters μ and σ2: μ̂C∗ = x and σ̂2
C∗ = s2
x.
In addition to the QS and CS estimators, we also consider the conditional
score (DS) estimator, cf. Carroll et al. (1995). Let z = x + yσ2
δβ1, η∗ =
β0 + β1z. Then
E (y|z) = m∗ := H(η∗ − β2
1σ
2
δ/2)
V (y|z) = v∗ := H(1 − H).
The conditional score function for β is then given by, cf. Carroll et al.
(1995), S
(β)
D = (y − m∗)(1, z)t. It is obviously unbiased. By using the fact
that y is binary, the conditional score function can be written as a linear
function of y: S
(β)
D = ygd − hd, where
gd = {1 − H(β0 + β1x + β2
1σ
2
δ/2)}(1, x + β1σ
2
δ )
� +
+ H(β0 + β1x − β2
1σ
2
δ/2)(1, x)�,
hd = −H(β0 + β1x − β2
1σ
2
δ/2)(1, x)�.
78 A.KUKUSH, A.MALENKO, AND H.SCHNEEWEISS
If S
(β)
D is supplemented by the subvector (lμ, lσ)
�, then DS is a member of
the class L of linear score functions. The conditional score estimators of μ
and σ2 are μ̂D = x and σ̂2
D = s2
x.
Now, according to Theorem 2.1,
ΣQ ≤ ΣC∗ and ΣQ ≤ ΣD. (26)
But we can also compare Σ
(β,σ)
C∗ and Σ
(β,σ)
D to Σ
(β,σ)
Q , where these matrices
are the ACMs of the corresponding estimators of (β0, β1, σ)�.
Since μ̂C∗ = μ̂D = μ̂Q, we have for the μ-components Σ
(μ)
C∗ = Σ
(μ)
D = Σ
(μ)
Q ,
and thus by (26),
rank
(
Σ
(β,σ)
C∗ − Σ
(β,σ)
Q
)
= rank (ΣC∗ − ΣQ) ,
rank
(
Σ
(β,σ)
D − Σ
(β,σ)
Q
)
= rank (ΣD − ΣQ) .
Theorem 3.1 In the logit model, Σ
(β,σ)
Q ≤ Σ
(β,σ)
C∗ and Σ
(β,σ)
Q ≤ Σ
(β,σ)
D . When
β1 �= 0, the inequalities become strict inequalities.
In particular, under β1 �= 0, Σ
(σ)
Q < Σ
(σ)
C∗ and Σ
(σ)
Q < Σ
(σ)
D . This means
that in the logit model σ̂2
Q is an asymptotically more efficient estimator of
σ2 than σ̂2
C∗ = σ̂2
D = s2
x.
3.4 Proof of Theorem 3.1
The first statement is a direct consequence of Theorem 2.1. So we need
only prove the strict inequalities under β1 �= 0.
First we prove the linear independence of
[lμ, lσ, (mgc − hc)0, (mgc − hc)1],
then the linear independence of
[lμ, lσ, (mgd − hd)0, (mgd − hd)1],
and finally the linear independence of
[mβ0 , mβ1, mσ],
where
lμ ∝ x − μ, lσ ∝ (x − μ)2 − σ2.
By Corollary 2.3 with α′ = σ and α′′ = μ, these facts will yield that
Σ
(β,σ)
Q < Σ
(β,σ)
C∗ and Σ
(β,σ)
Q < Σ
(β,σ)
D .
Consider the case β1 > 0 (the case β1 < 0 can be treated similarly).
EFFICIENCY COMPARISON OF ESTIMATORS 79
1) From (24), we have, as x → −∞,
m(x) = E [H(β0+β1ξ)|x] ∼ exp{β0+β1(Kx+(1−K)μ)} E eβ1τγ = Ceβ1Kx.
Together with (25) it follows that
(mgc − hc)(x) ∼ const · eβ1(K−1)x(1, x)� as x → −∞.
Thus the functions lμ, lσ, (mgc − hc)0, (mgc − hc)1 have different asymptotic
behavior, as x → −∞, and are therefore linearly independent.
2) As to the asymptotic behavior of (mgd − hd), we have, as x → −∞,
(gd)0 → 1, (gd)1 ∼ x,
(hd)0 ∼ const · eβ1x, (hd)1 ∼ const · xeβ1x,
and thus
(mgd − hd)0 ∼ const · eβ1Kx, (mgd − hd)1 ∼ const · xeβ1Kx.
Again the functions lμ, lσ, (mgd−hd)0, (mgd−hd)1 have different asymptotic
behavior as x → −∞ and are therefore linearly independent.
3) We have, by (24),
mβ0 = E [H ′|x] = E [H ′{β0 + β1(Kx + (1 − K)μ + τγ)}|x],
mβ1 = (Kx + (1 − K)μ) E [H ′|x] + τ 2 E [H ′′|x],
mσ = β1Kσ(x − μ) E [H ′|x] + β1ττσ E [H ′′|x],
where H(i) = H(i)(β0 + β1ξ). This system of equations can also be
written in matrix form:⎛
⎝ mβ0
mβ1
mσ
⎞
⎠ =
⎛
⎝ 1 0 0
(1 − K)μ K τ 2
−β1Kσμ β1Kσ β1ττσ
⎞
⎠
⎛
⎝ E [H ′|x]
x E [H ′|x]
E [H ′′|x]
⎞
⎠ (27)
Because of τ 2 = Kσ2
δ , see (15), and Kσ �= 0, the transformation matrix on
the right hand side of (27) is nonsingular if β1 �= 0. By the properties of
the logistic function, we have
H ′ = H − H2, H ′′ = H ′ − 2(H2 − H3).
Therefore the vector on the right hand side of (27) is a nonsingular linear
transformation of the vector of functions (f1(x), f2(x), f3(x))�, where
f1(x) = E [H −H2|x], f2(x) = x E [H −H2|x], f3(x) = E [H2 −H3|x].
To prove the linear independence of [mβ0 , mβ1, mσ] it thus suffices to show
that [f1, f2, f3] are linearly independent. But this is guaranteed by the
fact that these functions have different asymptotic behavior, as x → −∞.
Indeed, E [Hr|x] ∼ const · erβ1Kx and thus
f1(x) ∼ const · eβ1Kx, f2(x) ∼ const · xeβ1Kx, f3(x) ∼ const · e2β1Kx.�
80 A.KUKUSH, A.MALENKO, AND H.SCHNEEWEISS
4. Conclusions
We studied the Poisson, the Gamma, and the logit errors-in-variance
models with unknown nuisance parameters. For the Poisson and the Gamma
models, we showed that the Quasi-Score estimator for β is strictly more ef-
ficient than the Corrected Score one for β. For the logit model, we proved
that the compound Quase-Score estimator for β and σ is strictly more ef-
ficient than both Corrected Score and Conditional Score estimators for β
and σ. In particular in the logit model the Quasi-Score estimator for σ is
different from the empirical variance of x. For the Gamma and the Pois-
son models the Quasi-Score estimators for σ coincides with the empirical
variance of x. All the three models are GLMs, therefore the Quasi-Score
estimator of μ is just the empirical mean of x.
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EFFICIENCY COMPARISON OF ESTIMATORS 81
Department of Mathematical Analysis, Kyiv National Taras
Shevchenko University, Kyiv, Ukraine.
E-mail address: alexander kukush@univ.kiev.ua
Department of Probability Theory and Mathematical Statistics,
Kyiv National Taras Shevchenko University, Kyiv, Ukraine
E-mail address: exipilis@yandex.ru
University of Muenchen, Germany
E-mail address: Hans.Schneeweiss@stat.uni-muenchen.de
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