Assessing robustness of inference in symmetrical nonlinear regression models

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Assessing robustness of inference in symmetrical nonlinear regression models

Luis H. Vanegas
Universidad Nacional de Colombia

Luz M. Rondon
Universidad Nacional de Colombia

Francisco José A. Cysneiros
Universidad Federal de Pernambuco

Reporte Interno de Investigación No. 16
Departamento de Estadística
Facultad de Ciencias
Universidad Nacional de Colombia
Bogotá, COLOMBIA

Assessing robustness of inference in
symmetrical nonlinear regression models
Luis Hernando Vanegas
a,∗
Luz Marina Rondon
a
Francisco Jos´e A.Cysneiros
b
a
Universidad Nacional de Colombia,Facultad de Ciencias,Departamento de
Estad´ıstica - Carrera 30 No.45-03,Bogot´a - Colombia
b
Departamento de Estat´ıstica,CCEN-UFPE - Cidade Universit´aria - Recife,PE -
Brazil 50740-540
ABSTRACT
This paper describes how diagnostic procedures were derived for symmetrical non-
linear regression models,continuing the work carried out by Cysneiros and Vanegas
(2008) and Vanegas and Cysneiros (2010),who showed that the parameters es-
timates in nonlinear models are more robust with heavy-tailed than with normal
errors.In this paper,we focus in assessing if the robustness of this kind of models
is also observed in the inference process (i.e.partial F-test).Symmetrical nonlinear
regression models includes all symmetric continuous distributions for errors covering
both light- and heavy-tailed distributions such as Student-t,logistic-I and -II,power
exponential,generalized Student-t,generalized logistic and contaminated normal.
Firstly,a statistical test is shown to evaluating the assumption that the error terms
all have equal variance.The results of a simulation study which describes the be-
haviour of the test for heteroscedasticity proposed in the presence of outliers is
then given.To assess the robustness of inference process,we present the results of a
simulation study which described the behavior of partial F-test in the presence of
outliers.Also,some diagnostic procedures are derived to identify influential obser-
vations on the partial F-test.A dataset described in Venables and Ripley (2002)
is also analysed.Diagnostic analysis indicates that a power exponential nonlinear
model seems to fit the data better than other symmetrical nonlinear models.
Key words:Symmetric distribution,heavy-tailed error,testing heteroscedasticity,
partial F-test,robust model.
1 INTRODUCTION
It is well known that normal linear and nonlinear regression models can be highly influenced
by extreme observations in response variable (see Cook and Weisberg (1982),Barnett and
Lewis (1994)).As an alternative to this type of analysis,can be considered models in which
the distribution of the error presents heavier tails than normal,which can accommodate
these observations and reduce and control their influence on the parameters estimates and
on statistical inference.Examples of distributions with heavier tails than normal can include
Student-t,logistic-II and power exponential (with positive index parameter).The symme-
trical regression models framework can be used for applying models where the error follow
distributions of this kind,which cover both light- and heavy-tailed distributions for error.
Considerable contributions have been made to symmetrical regression models in the recent
years.For instance,Cordeiro,Ferrari,Uribe-Opazo and Vasconcellos (2000) corrected maxi-
mumlikelihood estimates in symmetrical nonlinear regression models,while Galea,Paula and
Uribe-Opazo (2003) developed local influence measurements in symmetrical linear regres-
sion models.Cysneiros and Paula (2004) proposed restricted tests in linear models having
t-multivariate distribution.Cordeiro (2004) corrected likelihood ratio tests in symmetrical
nonlinear regression models;Galea,Paula and Cysneiros (2005) developed a standardized
residual and proposed some local influence measurements in symmetrical nonlinear regres-
sion models and Cysneiros and Paula (2005) proposed restricted tests in symmetrical linear
regression models.Cysneiros,Paula and Galea (2007) developed estimation procedures and
diagnostics measurements in heteroscedastic symmetrical linear regression models.Cysneiros
and Vanegas (2008) recently proposed standardized residuals for symmetrical nonlinear re-
gression models and carried out an analytical,empirical study to describe these residuals’
behavior.Vanegas and Cysneiros (2010) developed diagnostic procedures based on case-
deletion and mean-shift outlier models in symmetrical nonlinear regression models,showing
that,in the presence of outliers in response variable,the parameters estimates in nonlinear
models are more robust with heavy-tailed than with normal errors.In this paper,we focus
in assessing if the robustness of this kind of models is also observed in the inference process.

Corresponding author.
Email address:lhvanegasp@unal.edu.co (Luis Hernando Vanegas).
2
This paper deals with diagnostic procedures for symmetrical nonlinear regression models,
thereby continuing the work carried out in Cysneiros and Vanegas (2008) and Vanegas and
Cysneiros (2010).A statistical test for heteroscedasticity is proposed.Also,some diagnostic
procedures are described for identifying influential observations on partial F-test.The paper
is organized as follows.Section 2 introduces symmetrical nonlinear regression class.Section
3 describes a statistical test for evaluating the presence of heteroscedasticity in error terms.
Besides,presents the results of a simulation study illustrating the behavior of the proposed
heteroscedasticity test in the outliers presence.To assess the robustness of inference process,
section 4 presents the results of a simulation study describing partial F-test behavior in the
presence of extreme observations in response variable.In addition,describes some diagnostic
procedures based on the convergence of iterative parameter estimation and,in the case-
deletion model for identifying influential observations on the partial F-test.Section 5 analyzes
a dataset described in Venables and Ripley (2002),in which diagnostic procedures are
applied.This dataset consists of 52 obese patients on a weight reduction programme who
tended to lose adipose tissue at a diminishing rate,where weight is measured in kilograms,
and the time since the start of the programme,in days.Section 6 deals with some concluding
remarks.
2 SYMMETRICAL NONLINEAR REGRESSION MODELS
Suppose Y
1
,...,Y
n
as n independent random variables where density function is given by
f
Y
i
(y) =
1

φ
g{(y −
i
)
2
/φ},y ∈ IR,
with 
i
∈ IR and φ > 0 location and dispersion parameters,respectively.The function
g:IR −→[0,∞) is such that
R

0
g(u)du < ∞and is typically known as the density generator.
For instance,for Student-t distribution with ν degrees of freedomwe have g(u) ∝ (ν+u)

ν+1
2
and for power exponential distribution with index parameter −1 < k ≤ 1 we have g(u) ∝
exp(−
1
2
u
1/(1+k)
).We denote Y
i
∼ S(
i
,φ,g).The symmetrical nonlinear regression model is
defined as
Y
i
= (β;x
i
) +ǫ
i
,i = 1,...,n,
where 
i
(β) = (β;x
i
) is an injective and twice differentiable function with respect to
β = (β
1
,...,β
p
)
T
.We also suppose that the derivative matrix D
β
= ∂µ/∂β has rank p
3
(p < n) for all β ∈ Ω
β
⊂ IR
p
,with Ω
β
a compact set with interior points,ǫ
i
∼ S(0,φ,g) and
x
i
is the vector of explanatory variables.The characteristic function of Y
i
can be expressed as
ς
y
(y) = E(e
ity
) = e
itµ
ϕ(t
2
φ),t ∈ IR for some function ϕ(),with ϕ(u) ∈ IR for u > 0.Where,
E(Y
i
) = 
i
and V ar(Y
i
) = ξφ,where ξ > 0 is a constant given by ξ = −2ϕ

(0) with ϕ

(0) =
{∂ϕ(u)/∂u}
u=0
(see,for instance,Fang,Kotz and Ng (1990)).For example,for power
exponential distribution with index parameter k we have ξ = 2
1+k
Γ[3(1+k)/2]/Γ[(1+k)/2],
with Γ() the Gamma function.All extra parameters will be considered to be known or fixed
in this paper.
The log-likelihood function for the parameters vector θ = (β
T
,φ)
T
is given by
L(θ;y) =
n
X
i=1
l(y
i
;
i
;φ) = −
n
2
log φ +
n
X
i=1
log[g(z
2
i
)],
where z
i
= φ
−1/2
{y
i
−
i
(β)}.The score functions for β and φ have,respectively,the forms
U
β
(θ) = φ
−1
D
T
β
D(v)(y−µ) and U
φ
(θ) = (2φ)
−1

−1
Q
V
(β,φ)−n} with µ = (
1
,...,
n
)
T
,
y = (y
1
,...,y
n
)
T
the observed responses,D(v) = diag{v
1
,...,v
n
} with v
i
= v(z
i
) = −2
g

(z
2
i
)
g(z
2
i
)
,
g

(z
2
i
) = {
∂g(u)
∂u
}
u=z
2
i
and Q
V
(β,φ) = (y−µ)
T
D(v)(y−µ).The Fisher information matrix for
θ can be expressed as K
θθ
= diag{K
ββ
,K
φφ
},where K
ββ
=
4d
g
φ
D
T
β
D
β
and K
φφ
=
n

2
(4f
g
−1)
with 4d
g
= E(v
2
(z)z
2
),4f
g
= E(v
2
(z)z
4
) and z ∼ S(0,1,g).Thus,the maximum likelihood
estimators of β and φ are asymptotic independents.For power exponential distribution with
index parameter k we founded 4d
g
= 2
1−k
Γ[(3 − k)/2](1 + k)
−2
/Γ[(k + 1)/2] and 4f
g
=
(k + 3)/(k + 1).Some expressions for v(),d
g
and f
g
for symmetrical distributions can be
found in Cysneiros and Paula (2005).The maximum likelihood estimates of θ,
ˆ
θ = (
ˆ
β
T
ˆ
φ)
T
,
can be obtained by solving U(
ˆ
θ) =

U
T
β
(
ˆ
θ),U
φ
(
ˆ
θ)

T
= 0.Some iterative procedures can be
used such as Newton-Raphson,BFGS and Fisher scoring method.The iterative process for
ˆ
θ takes the form
β
(m+1)
=

D
T(m)
β
D
(m)
β

−1
D
T(m)
β

D
(m)
β
β
(m)
+D(ρ
(m)
)
h
y −µ(β
(m)
)
i

=

D
T(m)
β
D
(m)
β

−1
D
T(m)
β
˜
Z
(m)
,(1)
φ
(m+1)
=
1
n
Q
V

(m+1)

(m)
),m= 0,1,2,...,
with D(ρ) = diag{ρ(z
1
),...,ρ(z
n
)} and ρ(z
i
) = v(z
i
)/4d
g
.If the function g(u) is monoton-
4
ically decreasing for u > 0,then v() > 0 and ρ(z) > 0 for all z ∈ IR.Also,v(z) = v(−z)
and ρ(z) = ρ(−z) for all z ∈ IR.When the error of model has heavy-tailed distribution,
the values of the weighting ρ(z
i
) and v(z
i
) in β and φ,respectively,has small values for
|z
i
| large.Thus,models with heavy tailed distribution can reduce the influence of extreme
observations while the weights are equal for all observations in normal nonlinear regression
model,consequently,estimates in these models are more sensitive to extreme observations.
It is easy to show that in Student-t,logistic-II,and power exponential (k > 0) distributions
the values of v(z) and ρ(z) decrease for |z| large.For example,for the power exponential
distribution with index parameter k we have v(z) = (1 + k)
−1
|z|
−k/(k+1)
.
ˆ
β is a consistent
estimator of β and

n(
ˆ
β −β)
d
−→N
p
(0,J
−1
ββ
) in suitable regularity conditions (see Cox and
Hinkley (1974)),where J
ββ
= lim
n→∞
1
n
K
ββ
.Then,
ˆ
K
−1
ββ
=
ˆ
φ
4d
g
(D
T
ˆ
β
D
ˆ
β
)
−1
is a consistent
estimator of the asymptotic variance-covariance matrix of
ˆ
β.Also
ˆ
φ is a consistent estimator
of the φ and

n(
ˆ
φ −φ)
d
−→N(0,J
−1
φφ
),where J
φφ
= lim
n→∞
1
n
K
φφ
.Then,
ˆ
K
−1
φφ
=
4
ˆ
φ
2
n(4f
g
−1)
is a
consistent estimator of the asymptotic variance of
ˆ
φ.The regularity conditions do not hold
for some symmetric distributions such as Kotz,generalized Kotz and double exponential
(Cordeiro,Ferrari,Uribe-Opazo and Vasconcellos,2000).
Vanegas and Cysneiros (2010) have shown that the heavy-tailed models can be more robust
against extreme observations than the normal model.They also developed diagnostic proce-
dures based on case-deletion and mean-shift outlier models.For instance,they founded that
one-step approximation
ˆ
θ
I
(i)
of
ˆ
θ
(i)
could be expressed as
ˆ
θ
I
(i)
=






ˆ
β
I
(i)
ˆ
φ
I
(i)






=






ˆ
β −
ˆ
φ
1/2
ρ(ˆz
i
)ˆz
i
(D
T
ˆ
β
D
ˆ
β
)
−1
ˆ
d
i
/(1 −
ˆ
h
ii
)
ˆ
φ −2
ˆ
φ(v(ˆz
i
)ˆz
2
i
−1)/(n −1)(4f
g
−1)






,
where
ˆ
h
ii
=
ˆ
d
T
i
(D
T
ˆ
β
D
ˆ
β
)
−1
ˆ
d
i
was the (i,i)-th element of
ˆ
H = D
ˆ
β
(D
T
ˆ
β
D
ˆ
β
)
−1
D
T
ˆ
β
,
ˆ
d
T
i
was the
i-th row of D
ˆ
β
and D
ˆ
β
= {D
β
}
β=
ˆ
β
.To identify influential observations we can use the
following measurements for
ˆ
β
j
(j = 1,...,p) and
ˆ
φ,respectively,
t
β
j
,i
=
ˆ
β
I
j(i)

ˆ
β
j
q
ˆ
V ar(
ˆ
β
j
)
=
q
(4d
g
)
ˆ
h
ii
ρ(ˆz
i
)ˆz
i
1 −
ˆ
h
ii
ˆ
ψ
j,i
(2)
5
and
t
φ,i
=
ˆ
φ
I
(i)

ˆ
φ
q
ˆ
V ar(
ˆ
φ)
=
s
n
4f
g
−1
v(ˆz
i
)ˆz
2
i
−1
n −1
,(3)
where
ˆ
β
I
j(i)
= a
T
j
ˆ
β
I
(i)

j,i
=
a
T
j
(D
T
β
D
β
)
−1
d
i

a
T
j
(D
T
β
D
β
)
−1
a
j

h
ii
is the linear correlation coefficient between
ˆ
β
j
and d
T
i
ˆ
β,a
j
= (a
1
,...,a
p
)
T
with a
j
= 1 and a
s
= 0 for all s 6= j.We have that
GD
β
j
,i
= t
2
β
j
,i
and GD
φ,i
= t
2
φ,i
,with GD
β
j
,i
and GD
φ,i
the univariate versions of Generalized
Cook Distance for
ˆ
β
j
and
ˆ
φ,respectively (see Vanegas and Cysneiros (2010)).Large values
of GD
β
j
,i
or GD
φ,i
indicate that the i-th observation has a disproportionate influence on
ˆ
β
j
or
ˆ
φ,respectively.
3 TESTING FOR HETEROSCEDASTICITY
Many authors have discussed the detection and testing of variance heterogeneity,for instance,
Cook and Weisberg (1983) for normal linear regression models,Lin and Wei (2003) for
normal nonlinear regression models and Cysneiros,Paula and Galea (2007) for symmetrical
linear regression models.The standard assumption for the model that the error terms all
have equal variance is now evaluated.For this,it was assumed that,in the same way as 
i
,
the dispersion parameter depended on a set of explanatory variables through the following
structure
φ
i
= φτ(g
i
,λ),i = 1,...,n,
where τ
i
(λ) = τ(g
i
,λ) was an injective and twice differentiable function with respect to
λ = (λ
1
,...,λ
q
)
T
.It was supposed that the derivative matrix D
λ
= ∂τ/∂λ had rank q with
τ = (τ
1
,...,τ
n
)
T
and g
i
the vector of explanatory variables.Then,the score function for
θ = (β
T

T
,φ)
T
in this model could be written as
U(θ) =











U
β
(θ)
U
λ
(θ)
U
φ
(θ)











6
where U
β
(θ) = D
T
β
D
(f)
(y−µ),U
λ
(θ) = φD
T
λ
mand U
φ
(θ) = τ
T
m,with D
(f)
= diag{f
1
,...,f
n
},
f
i
= v(z
i
)/φ
i
,m = (m
1
,...,m
n
)
T
and m
i
= (v(z
i
)z
2
i
−1)/2φ
i
.Likewise,the Fisher matrix
information for θ was K
θθ
(θ) = diag{K
ββ
,K

} where
K

=






K
λλ
K
λφ
K
φλ
K
φφ






,
K
ββ
= D
T
β
W
1
D
β
,K
λλ
= φ
2
D
T
λ
W
2
D
λ
,K
λφ
= φD
T
λ
W
2
τ,K
φφ
= τ
T
W
2
τ,with W
1
=
4d
g
diag{1/φ
1
,...,1/φ
n
} and W
2
=
4f
g
−1
4
diag{1/φ
2
1
,...,1/φ
2
n
}.It was assumed that exist
λ

in the parametric space of λ such that τ
i


) = 1 for i = 1,...,n.Then,for testing
variance heterogeneity the following hypothesis was evaluated
H
0
:λ = λ

(4)
H
1
:λ 6= λ

The Score test was used as this only requires estimating model parameters under the null
hypothesis which provides efficiency from the computational point of view.The Score test
for evaluating (4),denoted ξ
λ
,could be expressed as
ξ
λ
=
n
U
T
λ
(θ) V ar(
ˆ
λ)U
λ
(θ)
o
θ=
ˆ
θ

(5)
where
ˆ
θ

= (
ˆ
β
T

◦ T
,
ˆ
φ)
T
was the maximum likelihood estimate of θ under H
0
(i.e.constant
variance for error terms) and V ar(
ˆ
λ) was the appropiate submatrix of K
−1
θθ
(θ).Asymptot-
ically and under H
0

λ
followed Chi-square distribution with q degrees of freedom.Large
values of ξ
λ
suggested evidence of heteroscedasticity.
Theorem 1 The Score test for evaluating hypothesis (4) could be expressed as
ξ
λ
=
1
(4f
g
−1)
n
ˆm

T
ˆm

o
θ=
ˆ
θ

(6)
where ˆm

= H

λ
m

,H

λ
= D

λ

D

T
λ
D

λ

D

T
λ
,D

λ
=

I −
11
T
n

D
λ
,m

= (m

1
,...,m

n
)
T
and
m

i
= v(z
i
)z
2
i
.
7
This result extended the heteroscedasticity test developed by Cook and Weisberg (1983)
and Lin and Wei (2003) for normal regression cases to the symmetric nonlinear regression
models.When the error of the model had heavy-tailed distribution,weights v(ˆz
i
) in m

i
had
small values for |ˆz
i
| large,while that in normal regression v(ˆz
i
) = 1 for all observations.Thus,
models where the error has heavy-tailed distribution could reduce the influence of extreme
observations on ξ
λ
.Cook and Weisberg (1983) suggested two options for the functional form
of τ
i
(λ)
i)
τ
i
= exp(g
T
i
λ),i = 1,...,n.(7)
In this case λ

= 0 and D
λ
= D
(τ)
G,with D
(τ)
= diag{τ
1
,...,τ
n
} and G= (g
1
,...,g
n
)
T
.
Then,ξ
λ
was given by the expression (6),where H

λ
= G

(G

T
G

)
−1
G

T
and G

=

I −
11
T
n

G.
ii)
τ
i
=
q
Y
j=1
g
λ
j
ij
,g
ij
> 0,i = 1,...,n;j = 1,...,q.(8)
Here,λ

= 0 and D
λ
= D
(τ)
G
l
,with G
l
the matrix of natural logarithms of G.Then,ξ
λ
was given by the expression (6),where H

λ
= G

l
(G

T
l
G

l
)
−1
G

T
l
and G

l
=

I −
11
T
n

G
l
.
3.1 Simulation study 1
Some Monte Carlo simulations have been developed for studying the performance of the
partial F-test in nonlinear models in the presence of outliers in the response variable.We
considered the Michaelis-Menten model,which can be expressed as
Y
i
=
β
1
x
i
β
2
+x
i

i
,i = 1,...,n,
where β
1
> 0,β
2
> 0,x
i
> 0 and ǫ
i
∼ S(0,φ,g) were independent and identically distributed
variables.For errors,we generated 10,000 samples of N(0,φ) with sample sizes n = 40.
The explanatory variable x was generated following uniform distribution in the interval
(0,100) and their values were fixed throughout the simulations.The values of parameters
β
1
= 1,β
2
= 6.41 and φ = 0.1.To guarantee the presence of one outlier we constructed
Y

i
= β

1
x
i
/(β
2
+ x
i
) + ǫ
i
,where i = 6 and β

1
= 1.1,1.2,1.3,1.4,1.5,1.6,1.7 and 1.8.The
heteroscedasticity test ξ
λ
for the two functional forms of τ
i
considered in this section were
8
calculated in each one of the replications.δ
α
=#{ξ
λ
> ξ
α
}/100 was then computed,being the
percentage of replications where the heteroscedasticity test suggested that the assumption of
variance constant of the error terms was violated to the 100α% level of significance,with ξ
α
such that#{ξ
λ
> ξ
α


1
= β
1
}/10,000 = α.Tables 1 and 2 present the values of δ
α
for α =
0.1,0.05 and 0.01 levels where τ
i
had the functional forms given by (7) and (8),respectively,
with q = 1 and g
i
= x
i
,i = 1,...,n.
It can be observed that the percentage of replications in all the scenarios studied where
ξ
λ
suggest that the assumption of variance constant of the error terms had been violated,
increased when the difference between β

1
and β
1
also increased.However,for the normal
distribution,the increase occured quicker,which indicated that this distribution has more
sensitived than the other distributions considered for extreme observations in the response
variable.Additionally,the values of δ
α
were higher in the normal model than in the other
models in all scenarios.The influence of outliers on ξ
λ
increased in the Student-t models
when ν,the degrees of freedom,also increased.The same occured for power exponential
models when k,the index parameter,decreased.This was because the Student-t distribution
tended to normal when ν tended to infinity,while the power exponential distribution tended
to normal when k tended to zero.Then,we concluded that,in the presence of outliers in
response variable,the heteroscedasticity test in nonlinear models is more robust with heavy-
tailed than with normal errors.
4 INFLUENTIAL OBSERVATIONS ON PARTIAL F-TEST
4.1 Simulation study 2
The simulation study that began in Section 3 is now continued.In each replication,partial
F-test,F
1
(1) = (
ˆ
β
1
−1)
2
/
ˆ
V ar(
ˆ
β
1
),were obtained for evaluating the hypothesis H
0

1
=
1 vs.H
a

1
6= 1 in normal,Student-t with ν = 4,6,8 and 10 degrees of freedom,power
exponential k =0.6,0.7,0.8 and 0.9 index parameter and Logistic-II models.δ
α
=#{F
1
(1) >
F
α
}/100 was then computed,being the percentage of replications where partial F-test for
β
1
suggested that H
0
must be rejected to the 100α% level of significance,with F
α
such that
#{F
1
(1) > F
α
|H
0
was true}/10,000 = α.Table 3 presents the values of δ
α
for α = 0.1,0.05
and 0.01 levels.
9
Table 1
Performance of the heteroscedasticity test with functional form (7) in the presence of extreme
observations in the response variable
Distribution
β

1
−β
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
δ
0.1
Normal
10.48
11.30
13.36
16.88
22.12
29.90
39.34
50.34
Student-t (ν = 4)
10.33
10.51
11.06
12.20
13.55
14.72
16.26
17.65
Student-t (ν = 6)
10.09
10.68
11.52
12.93
15.02
17.37
19.82
22.53
Student-t (ν = 8)
10.10
10.78
11.75
13.62
16.38
19.37
23.07
26.64
Student-t (ν = 10)
10.20
10.98
12.13
14.10
17.55
20.76
25.59
30.34
Power Exp.(k = 0.6)
10.52
11.50
13.45
16.55
21.61
28.79
37.99
48.01
Power Exp.(k = 0.7)
10.39
11.54
13.38
16.26
21.38
28.38
37.34
47.20
Power Exp.(k = 0.8)
10.24
11.43
13.19
15.97
21.25
27.91
36.63
46.55
Power Exp.(k = 0.9)
10.27
11.45
13.24
16.14
21.15
27.61
36.40
45.90
Logistic-II
10.19
10.85
11.81
13.73
16.92
20.72
25.65
31.15
δ
0.05
Normal
5.28
5.83
7.04
9.66
14.15
20.86
29.47
39.13
Student-t (ν = 4)
4.92
4.96
5.39
6.12
6.89
7.90
8.97
9.94
Student-t (ν = 6)
4.96
5.20
5.67
6.82
8.34
10.06
11.80
13.69
Student-t (ν = 8)
5.08
5.36
6.04
7.35
9.40
11.56
14.37
17.15
Student-t (ν = 10)
5.10
5.37
6.22
7.65
10.05
13.08
16.51
19.83
Power Exp.(k = 0.6)
5.12
5.57
6.78
9.23
13.17
19.05
26.78
36.54
Power Exp.(k = 0.7)
5.11
5.43
6.66
9.18
12.91
18.59
26.15
35.80
Power Exp.(k = 0.8)
5.09
5.40
6.54
9.09
12.90
18.47
25.82
35.21
Power Exp.(k = 0.9)
5.04
5.35
6.50
9.07
12.88
18.33
25.52
34.72
Logistic-II
5.05
5.26
6.06
7.48
9.72
12.92
16.75
21.02
δ
0.01
Normal
1.10
1.18
1.82
2.94
4.76
8.30
13.24
20.72
Student-t (ν = 4)
1.01
1.02
1.14
1.35
1.59
1.80
2.10
2.37
Student-t (ν = 6)
1.01
1.03
1.31
1.60
1.94
2.41
3.05
3.52
Student-t (ν = 8)
1.05
1.21
1.51
1.85
2.37
3.16
4.09
5.04
Student-t (ν = 10)
1.13
1.24
1.59
1.98
2.67
3.65
5.04
6.83
Power Exp.(k = 0.6)
1.12
1.30
1.71
2.65
4.48
7.53
12.05
18.81
Power Exp.(k = 0.7)
1.14
1.30
1.76
2.70
4.52
7.61
11.91
18.54
Power Exp.(k = 0.8)
1.09
1.26
1.70
2.60
4.29
7.18
11.25
17.50
Power Exp.(k = 0.9)
1.12
1.23
1.71
2.51
4.01
6.84
10.56
16.51
Logistic-II
1.05
1.16
1.52
1.86
2.54
3.74
5.40
7.48
It can be observed that the percentage of replications where F
1
(1),in all the models and for
all levels of significance studied,suggesting that H
0
must be rejected,δ
α
,increased when the
difference between β

1
and β
1
also increased.However,the increase for the normal distribution
occured quicker than for the other distributions considered.In all scenarios the values of δ
α
10
Table 2
Performance of the heteroscedasticity test with functional form (8) in the presence of extreme
observations in the response variable
Distribution
β

1
−β
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
δ
0.1
Normal
10.49
12.29
15.99
22.53
31.86
42.75
55.95
69.09
Student-t (ν = 4)
10.38
10.73
12.03
14.09
16.85
20.00
23.02
26.03
Student-t (ν = 6)
10.39
11.29
13.09
16.16
20.40
25.25
30.96
36.40
Student-t (ν = 8)
10.41
11.64
13.67
17.44
22.86
29.27
37.12
44.49
Student-t (ν = 10)
10.42
11.76
14.19
18.48
24.98
32.49
41.79
51.52
Power Exp.(k = 0.6)
10.41
12.25
15.55
21.70
30.86
41.35
53.59
66.69
Power Exp.(k = 0.7)
10.36
12.16
15.51
21.53
30.42
41.02
53.02
66.24
Power Exp.(k = 0.8)
10.33
12.13
15.44
21.45
30.05
40.52
52.67
65.43
Power Exp.(k = 0.9)
10.31
12.06
15.37
21.28
29.74
40.02
52.32
64.58
Logistic-II
10.40
11.60
13.60
17.73
23.90
31.77
40.89
51.28
δ
0.05
Normal
5.49
7.05
10.09
15.83
24.53
35.29
47.56
60.97
Student-t (ν = 4)
5.23
5.87
6.88
8.69
10.41
12.59
14.91
17.02
Student-t (ν = 6)
5.37
6.28
7.91
10.28
13.48
17.45
21.71
25.98
Student-t (ν = 8)
5.42
6.52
8.61
11.55
15.67
21.26
27.14
33.66
Student-t (ν = 10)
5.46
6.66
8.90
12.31
17.35
23.90
31.24
40.05
Power Exp.(k = 0.6)
5.48
6.99
10.03
15.21
22.77
33.39
45.19
58.28
Power Exp.(k = 0.7)
5.45
6.92
10.02
14.98
22.50
32.96
44.46
57.46
Power Exp.(k = 0.8)
5.36
6.87
10.01
14.87
22.27
32.50
43.90
56.73
Power Exp.(k = 0.9)
5.35
6.82
9.99
14.83
22.14
32.11
43.68
55.79
Logistic-II
5.45
6.46
8.67
11.82
16.55
23.30
31.36
40.89
δ
0.01
Normal
1.36
2.30
3.97
7.16
12.83
21.13
31.18
43.36
Student-t (ν = 4)
1.12
1.47
2.13
3.08
4.12
5.11
6.11
7.12
Student-t (ν = 6)
1.23
1.90
2.79
4.15
5.83
7.76
9.77
12.28
Student-t (ν = 8)
1.25
1.91
2.94
4.69
7.01
9.48
12.79
16.72
Student-t (ν = 10)
1.27
1.98
3.24
5.00
7.85
11.31
15.62
21.25
Power Exp.(k = 0.6)
1.25
2.04
3.68
6.50
11.32
18.63
28.80
40.21
Power Exp.(k = 0.7)
1.24
2.02
3.57
6.47
11.07
18.26
28.02
39.57
Power Exp.(k = 0.8)
1.23
2.00
3.51
6.31
10.67
17.58
27.14
38.33
Power Exp.(k = 0.9)
1.16
2.00
3.48
6.29
10.44
17.29
26.56
37.58
Logistic-II
1.16
1.84
2.97
4.72
7.50
11.00
15.96
22.72
were higher in the normal model.For example,with α = 0.1 it was observed that the value
of δ
α
reached 28.9% in the normal model,while for the other models it was always less than
18%.Similar trends were observed for other levels of α.In the Student-t models the influence
of outliers on partial F-test increased when ν,the degrees of freedom,increased too.The
11
same occured for the power exponential models when k,the index parameter,decreased.
This pattern indicated that partial F-test in models with heavy-tailed distributions was less
sensitive in the presence of outliers in the response variable than in the normal model.It could
be observed that,unlike the other models considered,the influence of outliers in the Student-
t and power exponential model on partial F-test was not strictly increased when (β

1
−β
1
)
increased.Then,we concluded that,in the presence of outliers in response variable,the
partial F-test in nonlinear models is more robust with heavy-tailed than with normal errors,
which is new because in the past,some works had concluded the robustness of heavy-tailed
models,but only on parameter estimates.
4.2 Diagnostic procedures
The Wald’s test can be used for evaluating the hypothesis H
0

j
= γ vs.H
a

j
6= γ.The
statistic of this test is the following
F
j
(γ) =

ˆ
β
j
−γ

2
ˆ
V ar

ˆ
β
j

= t
2
j
(γ),(9)
where t
j
(γ) = (
ˆ
β
j
−γ)/
q
ˆ
V ar(
ˆ
β
j
).Asymptotically and under H
0
,F
j
(γ) and t
j
(γ) follow Chi-
square(1) and standard normal distributions,respectively.The case-deletion model (CDM),
proposed by Cook and Weisberg (1982),can be used to study the influence of i-th observa-
tion on F
j
(γ) in normal linear regression model framework.The following measurement of
influence can then be calculated
Δ
j(i)
(γ) =
F
j(i)
(γ) −F
j
(γ)
F
j
(γ)
,F
j
(γ) 6= 0,(10)
where F
j(i)
(γ) was the partial F-test calculated with
ˆ
θ
(i)
,the estimate of θ obtained when
the i-th observation has been excluded from the dataset.Large values of |Δ
j(i)
(γ)| indicated
that the i-th observation had a disproportionate influence on F
j
(γ).However,the calcu-
lation of Δ
j(i)
(γ),i = 1,...,n,could be computationally expensive,especially when n is
large.An approximation of these values was thus obtained by substituting
ˆ
θ
(i)
by one-step
approximation,
ˆ
θ
I
(i)
,in the expression (10).
12
Table 3
Performance of the partial F-test in the presence of extreme observations in the response variable
Distribution
β

1
−β
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
δ
0.1
Normal
11.25
12.79
14.18
16.77
19.27
22.26
25.49
28.92
Student-t (ν = 4)
10.19
11.10
13.42
14.47
15.02
15.32
14.91
14.44
Student-t (ν = 6)
10.27
11.12
13.44
14.92
15.79
16.33
16.29
15.89
Student-t (ν = 8)
10.76
11.98
13.56
14.93
16.07
16.71
16.98
16.98
Student-t (ν = 10)
10.93
12.60
13.60
15.14
16.51
17.45
17.80
17.93
Power Exp.(k = 0.6)
11.13
12.53
13.21
14.11
14.68
15.01
15.06
14.83
Power Exp.(k = 0.7)
10.93
12.46
13.14
14.07
14.50
14.71
14.56
14.30
Power Exp.(k = 0.8)
10.87
12.04
13.04
13.86
14.23
14.09
13.84
13.47
Power Exp.(k = 0.9)
10.88
11.32
13.03
13.80
14.16
14.07
13.82
13.46
Logistic-II
10.94
11.94
13.38
14.71
15.70
16.35
16.48
16.52
δ
0.05
Normal
6.03
6.90
8.35
9.85
11.77
14.39
16.78
19.44
Student-t (ν = 4)
5.62
6.73
7.73
8.37
8.89
8.90
8.66
8.25
Student-t (ν = 6)
5.70
6.74
7.76
8.63
9.26
9.78
9.76
9.28
Student-t (ν = 8)
5.73
6.77
7.84
8.64
9.47
10.05
10.27
10.24
Student-t (ν = 10)
5.74
6.83
8.04
8.96
9.85
10.71
11.02
11.13
Power Exp.(k = 0.6)
5.99
6.73
7.39
8.02
8.30
8.53
8.57
8.54
Power Exp.(k = 0.7)
5.88
6.68
7.34
8.01
8.25
8.32
8.33
8.03
Power Exp.(k = 0.8)
5.83
6.66
7.33
7.99
8.13
8.18
8.12
7.84
Power Exp.(k = 0.9)
5.76
6.61
7.28
7.70
7.83
7.94
7.70
7.38
Logistic-II
5.76
6.80
7.88
8.56
9.39
9.83
10.10
10.11
δ
0.01
Normal
1.55
1.93
2.33
2.72
3.41
4.14
5.00
6.00
Student-t (ν = 4)
1.34
1.74
2.10
2.30
2.52
2.56
2.39
2.22
Student-t (ν = 6)
1.34
1.83
2.15
2.42
2.79
2.87
2.81
2.68
Student-t (ν = 8)
1.39
1.87
2.26
2.57
2.94
3.08
3.16
3.15
Student-t (ν = 10)
1.43
1.89
2.30
2.63
3.05
3.29
3.36
3.36
Power Exp.(k = 0.6)
1.48
1.79
2.15
2.34
2.46
2.52
2.54
2.47
Power Exp.(k = 0.7)
1.46
1.86
2.13
2.33
2.40
2.50
2.51
2.43
Power Exp.(k = 0.8)
1.41
1.85
2.11
2.32
2.39
2.41
2.39
2.31
Power Exp.(k = 0.9)
1.40
1.84
2.03
2.31
2.37
2.32
2.29
2.20
Logistic-II
1.45
1.91
2.26
2.54
2.93
3.05
3.05
3.02
Theorem 2 The influence measurement Δ
j(i)
(γ) based on the one-step approximation of
ˆ
θ
(i)
,denoted
ˆ
θ
I
(i)
,can be expressed as
Δ
I
j(i)
(γ) ≈

1 −t
β
j
,i
/t
j
(γ)

2

1 −2
q
4f
g
−1
n
t
φ,i


1 +
ˆ
ψ
2
j,i
ˆ
h
ii
1−
ˆ
h
ii

−1,t
j
(γ) 6= 0 (11)
13
According to (11) it was concluded that the i-th observation was not influential on F
j
(γ)
when the three following conditions were satisfied:i) this was not influential on
ˆ
β
j
(i.e.
t
2
β
j
,i
≈ 0),ii) was not influential on
ˆ
φ (i.e.t
2
φ,i
≈ 0),and iii) was not located on a remote
region of the subspace defined by columns of D
ˆ
β
(i.e.
ˆ
h
ii
≈ 0).The result obtained in (11)
was the extension of the expression developed by Cook and Weisberg (1982) for the normal
linear regression models to symmetrical nonlinear regression models.
The convergence of the iterative process for estimating θ may also be used for studying the
influence of i-th observation on F
j
(γ).Then,from (1) we have
ˆ
β = (D
T
ˆ
β
D
ˆ
β
)
−1
D
T
ˆ
β
˜
Z
= (D
T
ˆ
β
D
ˆ
β
)
−1
D
T
ˆ
β
n
D
ˆ
β
ˆ
β +D(ˆρ)
h
y −µ(
ˆ
β)
io
,(12)
where β = (β
∗T

j
)
T
and D
β
= (D
β
∗,d
(j)
) can be defined with β

the vector of location
parameters when the j-th has been excluded,D
β
∗ = ∂µ/∂β

and d
(j)
= ∂µ/∂β
j
.From (12)
and using that (I −
ˆ
H

) is a symmetric and idempotent matrix we can express
ˆ
β as






ˆ
β

ˆ
β
j






=






D
T
ˆ
β

D
ˆ
β

D
T
ˆ
β

ˆ
d
(j)
ˆ
d
T
(j)
D
ˆ
β

ˆ
d
T
(j)
ˆ
d
(j)






−1






D
T
ˆ
β

ˆ
d
T
(j)






˜
Z
=







(D
T
ˆ
β

D
ˆ
β

)
−1
D
T
ˆ
β

h
I −(
ˆ
R
T
(j)
ˆ
R
(j)
)
−1
ˆ
d
(j)
ˆ
R
T
(j)
i
˜
Z
(
ˆ
R
T
(j)
ˆ
R
(j)
)
−1
ˆ
R
T
(j)
(I −
ˆ
H

)
˜
Z







,(13)
where H

= D
β
∗(D
T
β
∗D
β
∗)
−1
D
T
β
∗ and
ˆ
R
(j)
= (I −
ˆ
H

)
ˆ
d
(j)
.Also,from the Section 2,the
variance of
ˆ
β
j
is the respective element of K
−1
ββ
,thus V ar(
ˆ
β
j
) = φ(R
T
(j)
R
(j)
)
−1
/4d
g
.From
(13) the expression (9) can then be rewritten as
F
j
(γ) =
h 
ˆ
R
T
(j)
ˆ
R
(j)

−1
ˆ
R
T
(j)

I −
ˆ
H


˜
Z−γ
i
2
ˆ
φ

ˆ
R
T
(j)
ˆ
R
(j)

−1
/4d
g
=
h 
ˆ
R
T
(j)
ˆ
R
(j)

−1
ˆ
R
T
(j)

I −
ˆ
H


˜
Z−γ
ˆ
d
(j)
i
2
ˆ
φ

ˆ
R
T
(j)
ˆ
R
(j)

−1
/4d
g
14
Given that
˜
Z = D
ˆ
β

ˆ
β

+
ˆ
β
j
ˆ
d
(j)
+D(ˆρ)
h
y −µ(
ˆ
β)
i
we can write the matrix (I−
ˆ
H

)(
˜
Z−γ
ˆ
d
(j)
)
as
ˆ
R
(z)
= (I −
ˆ
H

)
n
D(ˆρ)
h
y −µ(
ˆ
β)
i
+(
ˆ
β
j
−γ)
ˆ
d
(j)
o
.Defining
ˆ
R

(z)
=
q
4d
g
/
ˆ
φ
ˆ
R
(z)
we have
F
j
(γ) as the following
F
j
(γ) =
h 
ˆ
R
T
(j)
ˆ
R
(j)

−1
ˆ
R
T
(j)
ˆ
R

(z)
i
2

ˆ
R
T
(j)
ˆ
R
(j)

−1
,(14)
i.e.,F
j
(γ) can be considered to be the statistical test for evaluating the significance of
the normal linear regression through the origin of
ˆ
R

(z)
on
ˆ
R
(j)
when error variance is 1.
Therefore,a graph of
ˆ
R

(z)
against
ˆ
R
(j)
can reveal which observations were contributing to
the relationship assessed by F
j
(γ) and which were being diverted from the same.In the
normal linear case with intercept,(14) means that the partial F-test can be interpreted as
F
j
(0) = n
(1 −R
∗2
)
(1 −R
2
)
ρ
2
X
j
Y X
∗,
where ρ
X
j
Y X
∗ was the partial correlation coeficient between the response and the j-th ex-
planatory variable while R
2
and R
∗2
were the coefficients of determination for the model
with all the covariates and without the j-th,respectively.
5 EXAMPLE
We considered a dataset analyzed in Venables and Ripley (2002) which showed that obese
patients on a weight reduction programme tended to lose adipose tissue at a diminishing rate.
The two variables were x,for time (in days) since the start of the programme and y,being
the patients’ weight in kilograms measured in standard conditions.The dataset pertained
to 48 years-old male patients 193 cm height with a large body frame.The model used for
analyzing this dataset was given by
Y
i
= β
1
+
β
2
2
x
i

3

i
,i = 1,...,52,
where β
1
was ultimate lean weight,or asymptote,β
2
was total amount to be lost and β
3
was time taken to lose half the amount remaining to be lost.The data are illustrated in
Figure 1.We consider two distributions in addition to the normal for errors in the model:
Student-t with 4 degrees of freedom and power exponential where k = 0.9.Table 4 shows
the parameters estimates and standard errors for the models fitted.It can be observed that
15
the ultimate lean weight estimate,
ˆ
β
1
,was smaller in normal model,which was consistent
with the other parameters estimates since also in this model
ˆ
β
2
and
ˆ
β
3
were the higher and
smaller,respectively.For location and scale parameters the standard errors were smaller in
the heavy-tailed error models.
Fig.1.Scatter plot for the weight loss data
0 50 100 150 200 250
110120130140150160170180
Time (in days)
Weight (in kilograms)
Table 4
Parameter estimates (standard errors) for symmetrical nonlinear models fitted on weight loss data
Distribution
β
1
β
2
β
3
φ
Normal
81.373
102.684
141.910
0.754
(2.202)
(2.021)
(5.139)
(0.148)
Student-t(4)
82.529
101.612
139.413
0.481
(2.013)
(1.844)
(4.704)
(0.124)
Power exponential(0.9)
83.675
100.674
136.717
0.131
(1.620)
(1.479)
(3.781)
(0.035)
Three observations (22,39 and 44) were identified in residuals plots (Figures 2.a - 5.a) that
could be considered outliers in all the models except in the power exponential.These plots
also show an increase in the magnitude of the residuals when the weight of the patients
decreased,which suggested that the error’s variance was not constant.This behavior was
observed with less intensity in the model with power exponential error distribution.Table 5
shows the percentage changes in parameter estimates when the outliers were eliminated from
the dataset.It can be seen that the changes were smaller with the heavy-tailed error models
than in the normal,especially with power exponential distribution.In plots for GD
β,i
(Figures
16
2.b - 5.b) and GD
φ,i
,(the latter is not shown here) it can also be observed that the influence
of the observations was smaller with the heavy-tailed error models.It could be interesting to
evaluate whether the time taken to lose half the amount remaining to be lost was different to
145 days,i.e.,evaluating H
0

3
= 145 vs.H
1

3
6= 145.Table 6 shows F
3
(145) with and
without outliers.Here,again,the power exponential model appearsed more robust regarding
outliers than in the other models considered.The graphs of |Δ
I
3(i)
(145)| (Figures 2.c - 5.c)
indicate that F
3
(145) was more robust in heavy-tailed error models,especially in the power
exponential,where the outliers did not have a disproportionate influence.Also,the influence
on F
3
(145) can be observed in plots of
ˆ
R
(j)
versus
ˆ
R

(z)
(Figures 2.d - 5.d).The dotted line
in these plots represents the regression of
ˆ
R

(z)
on
ˆ
R
(j)
.We observed that the observations
did not show the significance of F
3
(145) so clearly in the normal and the Student-t models
as they did in the power exponential model.This plot also showed the same observations as
influentials as in plot |Δ
I
3(i)
(145)|.
Table 5
Percentage changes in parameter estimates when observations 22,39 and 44 were eliminated from
the dataset.
Distribution
β
1
β
2
β
3
φ
Normal
1.655
−1.234
−2.238
−31.029
Student-t(4)
0.956
−0.722
−1.299
−23.738
Power exponential(0.9)
−0.073
0.061
0.121
−26.505
Table 6
F
3
(145) values and p-values with and without observations 22,39 and 44
Distribution
with all observations
without outliers
F
3
(145)
p-value
F
3
(145)
p-value
Normal
0.3613
0.5477
2.2733
0.1315
Student-t(4)
1.4105
0.2350
3.3001
0.0692
Power exponential(0.9)
4.7985
0.0284
6.8244
0.0090
Theorem 2 was used to assess the assumption for the model that the error terms all had
equal variance with the functional forms described in (7) and (8).Table 7 shows ξ
λ
with
and without outliers when q = 1 and g
i
= x
i
,i = 1,...,n.It was observed that there was
17
Fig.2.Residual plots t

D
(ˆz
i
) (a),index plots GD
β,i
(b),index plots |Δ
I
3(i)
(145)| (c) and
ˆ
R
(j)
vs
ˆ
R

(z)
plot (d) for fitted model with normal errors on weight loss data
110 120 130 140 150 160 170 180
-3-2-10123
22
39
44
0 10 20 30 40 50
0.000.100.200.30
22
39
44
46
49
50
51
0 10 20 30 40 50
0.00.51.01.5
22
31 39
44
46
49
50
51
-0.04 -0.02 0.00 0.02
-2-1012
22
31
39
44
46
49
50
51
t

D
(ˆz
i
)
GD
β,i
ˆ

I
3(i)
(145)|
ˆ
R

(z)
ˆ
R
(j)Index
Index
(a) (b)
(c) (d)
Table 7
Heteroscedasticity test ξ
λ
values and p-values with and without observations 22,39 and 44
Functional
Distribution
with all observations
without outliers
form of τ
i
ξ
λ
p-value
ξ
λ
p-value
exp(g
T
i
λ)
Normal
6.598
0.011
4.703
0.030
Student-t(4)
7.253
0.007
4.723
0.029
Power exponential(0.9)
1.936
0.164
1.217
0.269
q
Q
j=1
g
λ
i
ij
Normal
4.995
0.025
3.192
0.074
Student-t(4)
5.400
0.020
3.160
0.075
Power exponential(0.9)
1.460
0.227
0.799
0.371
statistical evidence in the normal and Student-t models against the equal variance assumption
for the error terms.The heteroscedasticity test was highly influenced by the outliers in those
models.By contrast,the heteroscedasticity test was robust to outliers and was not significant
18
Fig.3.Residual plots t

D
(ˆz
i
) (a),index plots GD
β,i
(b),index plots |Δ
I
3(i)
(145)| (c) and
ˆ
R
(j)
vs
ˆ
R

(z)
plot (d) for fitted model with Student-t(4) errors on weight loss data
110 120 130 140 150 160 170 180
-3-2-10123
22
39
44
0 10 20 30 40 50
0.000.100.200.30
4
5
39
44
46
49
50
51
0 10 20 30 40 50
0.00.51.01.5
4
5
22
46
49
50
51
-0.04 -0.02 0.00 0.02
-2-1012
22
39
44
46
49
50
51
t

D
(ˆz
i
)
GD
β,i
ˆ

I
3(i)
(145)|
ˆ
R

(z)
ˆ
R
(j)
Index
Index
(a)
(b)
(c)
(d)
for power exponential model.Figure 5 show the values for |Δ
I
3(i)
(150)| and |Δ
3(i)
(150)| for
all models considered.The dotted line can be used as reference since it has 0 intercept and
slope 1.These measurements showed high agreement.In conclusion,the power exponential
nonlinear model seemed to fit the data better than the other symmetrical nonlinear models.
6 CONCLUDING REMARKS
This paper described how diagnostic procedures were derived for symmetrical nonlinear
regression models.We developed a statistical test to assess heteroscedasticity in the error
terms and some diagnostic procedures for identifying influential observations on partial F-
test.Simulations experiments on the Michaelis-Menten model showed that,in the presence
of outliers in the response variable,the inference (i.e.partial F-test and heteroscedasticity
test) in nonlinear models are more robust with heavy-tailed than with normal errors.
19
Fig.4.Residual plots t

D
(ˆz
i
) (a),index plots GD
β,i
(b),index plots |Δ
I
3(i)
(145)| (c) and
ˆ
R
(j)
vs
ˆ
R

(z)
plot (d) for fitted model with power exponential(0.9) errors on weight loss data.
110 120 130 140 150 160 170 180
-3-2-10123
22
39
44
0 10 20 30 40 50
0.000.100.200.30
37
44
50
0 10 20 30 40 50
0.00.51.01.5
2
39
44
49
51
-0.04 -0.02 0.00 0.02
-2-1012
22
39
44
49
50
t

D
(ˆz
i
)
GD
β,i
ˆ

I
3(i)
(145)|
ˆ
R

(z)
ˆ
R
(j)Index
Index
(a) (b)
(c) (d)
Fig.5.Influence measurements |Δ
3(i)
(145)| and |Δ
I
3(i)
(145)| for the fitted models on weight loss
data
0.0 0.5 1.0 1.5
0.00.51.01.5
0.0 0.5 1.0 1.5
0.00.51.01.5
0.0 0.5 1.0 1.5
0.00.51.01.5
Normal Student-t(4) Power Exp(0.9)

I
3(i)
(145)|

I
3(i)
(145)|

I
3(i)
(145)|

3(i)
(145)||Δ
3(i)
(145)||Δ
3(i)
(145)|
20
APPENDIX
A Proof of Theorem 1
The following was obtained after algebraic manipulations
{V ar(ˆγ)}
θ=
ˆ
θ

=
4
4f
g
−1
"
D
T
λ

I −
11
T
n
!
D
λ
#
−1
λ=λ

and
{U
λ
(θ)}
θ=
ˆ
θ

=
1
2
[D
T
λ
(m

−1)]
θ=
ˆ
θ

Then,substituting in (5) the following was obtained
ξ
λ
=
1
4f
g
−1



(m

−1)
T
D
λ
"
D
T
λ

I −
11
T
n
!
D
λ
#
−1
D
T
λ
(m

−1)



θ=
ˆ
θ

Since

I −
11
T
n

is a symmetric and idempotent matrix,the following could be written
ξ
λ
=
1
4f
g
−1

(m

−1)
T
D
λ

D

T
λ
D

λ

−1
D
T
λ
(m

−1)

θ=
ˆ
θ

From the convergence of iterative estimation of scale parameter 1
T
m

= n,which implied
that (m

−1) could be written as

I −
11
T
n

m

.Replacing in the expression above,ξ
λ
could
be express as
ξ
λ
=
1
4f
g
−1

m

T
D

λ

D

T
λ
D

λ

−1
D

T
λ
m


θ=
ˆ
θ

Since H

λ
= D

λ

D

T
λ
D

λ

−1
D

T
λ
was a symmetric and idempotent matrix,then
ξ
λ
=
1
4f
g
−1
n
ˆm

T
ˆm

o
θ=
ˆ
θ

with
ˆ
m

= H

λ
m


B Proof of Theorem 2
F
j(i)
(γ) can be written as
F
j(i)
(γ) =

ˆ
β
j(i)
−γ

2
ˆ
φ
(i)
4d
g
a
T
j

D
(i)T
ˆ
β
(i)
D
(i)
ˆ
β
(i)

−1
a
j


ˆ
β
j(i)
−γ

2
ˆ
φ
(i)
4d
g
a
T
j

D
T
ˆ
β
D
ˆ
β

ˆ
d
i
ˆ
d
T
i

−1
a
j
,(B.1)
21
where
ˆ
β
j(i)
= a
T
j
ˆ
β
(i)
and D
(i)
β
is the derivative matrix after excluding the i-th observation
from the dataset.Substituting
ˆ
θ
(i)
by
ˆ
θ
I
(i)
in (B.1) F
I
j(i)
(γ) was obtained,given by
F
I
j(i)
(γ) ≈

a
T
j
ˆ
β
I
(i)
−γ

2
ˆ
φ
I
(i)
4d
g
a
T
j




D
T
ˆ
β
D
ˆ
β

−1
+

D
T
ˆ
β
D
ˆ
β

−1
ˆ
d
i
ˆ
d
T
i

D
T
ˆ
β
D
ˆ
β

−1
1−
ˆ
d
T
i

D
T
ˆ
β
D
ˆ
β

−1
ˆ
d
i


a
j
=


a
T
j
ˆ
β −
ˆ
φ
1/2
ρ(ˆz
i
)ˆz
i
a
T
j

D
T
ˆ
β
D
ˆ
β

−1
ˆ
d
i
(
1−
ˆ
h
ii)
−γ



2
ˆ
φ
4d
g

1 −
2(v(ˆz
i
)ˆz
2
i
−1)
(n−1)(4f
g
−1)





a
T
j

D
T
ˆ
β
D
ˆ
β

−1
a
j
+

a
j

D
T
ˆ
β
D
ˆ
β

−1
ˆ
d
i

2
1−
ˆ
h
ii




=




ˆ
β
j
−γ


ˆ
φ
1/2
ρ(ˆz
i
)ˆz
i
a
T
j

D
T
ˆ
β
D
ˆ
β

−1
ˆ
d
i
(
1−
ˆ
h
ii)



2
ˆ
φ
4d
g
a
T
j

D
T
ˆ
β
D
ˆ
β

−1
a
j

1 −
2(v(ˆz
i
)ˆz
2
i
−1)
(n−1)(4f
g
−1)





1 +

a
j

D
T
ˆ
β
D
ˆ
β

−1
ˆ
d
i

2
a
T
j

D
T
ˆ
β
D
ˆ
β

−1
a
j
(1−
ˆ
h
ii
)




=




(
ˆ
β
j
−γ)
r
ˆ
φ
4d
g
a
T
j

D
T
ˆ
β
D
ˆ
β

−1
a
j

(4d
g
)
1/2
ρ(ˆz
i
)ˆz
i
a
T
j

D
T
ˆ
β
D
ˆ
β

−1
ˆ
d
i
(
1−
ˆ
h
ii)
r
a
T
j

D
T
ˆ
β
D
ˆ
β

−1
a
j




2

1 −
2(v(ˆz
i
)ˆz
2
i
−1)
(n−1)(4f
g
−1)





1 +

a
T
j

D
T
ˆ
β
D
ˆ
β

−1
ˆ
d
i

2
(
1−
ˆ
h
ii)
r
a
T
j

D
T
ˆ
β
D
ˆ
β

−1
a
j




From (2) and (3) we have
F
I
j(i)
(γ) =
h
t
j
(γ) −t
β
j
,i
i
2

1 −2
q
4f
g
−1
n
t
φ,i

h
1 +
ˆ
ψ
2
j,i
ˆ
h
ii
1−
ˆ
h
ii
i
=
t
2
j
(γ)
h
1 −t
β
j
,i
/t
j
(γ)
i
2

1 −2
q
4f
g
−1
n
t
φ,i

h
1 +
ˆ
ψ
2
j,i
ˆ
h
ii
1−
ˆ
h
ii
i
,t
j
(γ) 6= 0.
22
Then,as Δ
I
j(i)
(γ) can be expressed as F
I
j(i)
(γ)/F
j
(γ) −1 we have
Δ
I
j(i)
(γ) ≈

1 −t
β
j
,i
/t
j
(γ)

2

1 −2
q
4f
g
−1
n
t
φ,i


1 +
ˆ
ψ
2
j,i
ˆ
h
ii
1−
ˆ
h
ii

−1,t
j
(γ) 6= 0.✷
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