1 Mixture Regression Estimation Based on the Mixture of Normal and t Distributions
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122
A.K. Laha and A.C. Pravida Raja
is a circular random variable with p.d.f. f (θ ) and 0 ≤ γ < 0.5 is fixed. Let α, β be
two points on the unit circle satisfying
α
1. β f (θ )dθ = 1 − 2γ
μ
2. d1 (α, β) ≤ d1 (μ, ν) for all μ, ν satisfying ν f (θ )dθ = 1 − 2γ where d1 (φ, ξ ) is
the length of the arc starting from ξ and ending at φ traversed in the anti-clockwise
direction.
The γ -circular trimmed mean (γ -CTM) is then defined as
μγ = arg[
1
1 − 2γ
α
β
exp(iθ )f (θ )dθ ]
where γ is the trimming proportion. Laha and Mahesh (2011) proved that the γ CTM (μγ ) is SB-robust at the family of distributions = {vM(μ, κ), κ >0} when
the measure of dispersion is S(F) = Eγ ,F (d(Θ, μ)) where F ∈ and 0 ≤ γ < 0.5
Ko and Guttorp (1988) proved that K(F) = A−1 (ρF ) where
ρF =
EF2 (cosΘ) + EF2 (sinΘ) = A(κ)
is not SB-robust
√ at the family = {vM(μ, κ), κ >0} when the measure of dispersion
is S(F) = 1 − A(κ). Laha and Mahesh (2012) discussed robust estimation of κ
for vM distribution. They showed that K(F) is not SB-robust w.r.t. the dispersion
measure S(F) = EF (d(Θ, μ)). They proposed a new trimmed estimator for κ which
is defined as follows: Let f (θ ; μ, κ) be the p.d.f. of vM(μ, κ) distribution and α(κ)
and β(κ) be symmetrically placed around μ such that
α(κ)
β(κ)
f (θ ; μ, κ)dθ = 1 − 2γ
where γ is the trimming proportion such that γ ∈ [0, 0.5). Define
g ∗ (κ) = Eγ ,F (d(Θ, μ)) =
α(κ)
β(κ)
d(θ, μ)f (θ ; μ, κ)dθ
Then the new trimmed estimator for κ is defined as
Tγ (F) = g ∗−1 [Eγ ,F (d(Θ, μ))]
Laha and Mahesh (2012) proved that if Θ ∼ vM(0, κ), d(θ ) = π − |π − θ | and
g ∗ (κ) = Eγ ,F (d(Θ)), then Tγ (F) = g ∗−1 [Eγ ,F (d(Θ))] is SB-robust at the family
of distributions ∗ = {vM(0, κ); 0 < m ≤ κ ≤ M} w.r.t. the dispersion measure
α(κ)
S(F) = Eγ ,F (d(Θ)) = (1 − 2γ )−1 β(κ) d(θ )dF. Similar results on SB-robustness
of mean and concentration parameter of Wrapped Normal distribution can be seen
in Laha et al. (2013).
SB-Robustness of Estimators
123
4 SB-Robustness of FAP and ASN0 of Control Chart
In the construction of control charts, one of the most important consideration is that
of the False Alarm Probability (FAP). This is the probability that an observation
falls outside the control limits of a control chart when the process distribution has
not changed. The FAP is controlled at some chosen level α which is conventionally
taken to be 0.0027. Suppose that a control chart has been constructed with N(μ0 , σ0 )
as the underlying distribution of the quality characteristic and rational subgroup size
1. Then, the Lower Control Limit (LCL) and Upper Control Limit (UCL) of the X¯
chart are μ0 − 3σ0 and μ0 + 3σ0 , respectively. The FAP can be represented as a
functional given below.
T (F) = EF (I (LCL,UCL) (X)) = PF (X ≤ LCL or X ≥ UCL) where
I
(LCL,UCL) (X)
=
1 if X ≤ LCL or X ≥ UCL
0
otherwise.
(1)
Here, (LCL, UCL) denotes the complement of the interval (LCL,UCL). When F =
N(μ0 , σ0 ) we have
T (F) = 1 − [Φ(
LCL − μ0
UCL − μ0
) − Φ(
)] = α
σ0
σ0
(2)
Let = {N(μ, σ0 ) : k1 ≤ μ ≤ k2 }, μ0 ∈ [k1 , k2 ]. In the theorem below we discuss the SB-robustness of FAP at the family .
Theorem 1 1. T (F) = EF (I (LCL,UCL) (X)) is B-robust.
2.
√ T (F) is SB-robust at the family when the measure of dispersion is S(F) =
T (F)(1 − T (F)).
Proof Let G ε = (1 − ε)F + εδx . Then the functional corresponding to G ε can be
written as
T (G ε ) = E(1−ε)F+εδx (I (LCL,UCL) (X))
= (1 − ε)EF (I (LCL,UCL) (X)) + ε(I
= (1 − ε)T (F) + ε
= (1 − ε)T (F)
(LCL,UCL) (x))
if x ∈ (LCL, UCL)
if x ∈ (LCL, UCL)
As defined earlier
IF(x; T , F) = lim
ε→0
T (G ε ) − T (F)
ε
124
A.K. Laha and A.C. Pravida Raja
Substituting T (G ε ) in the above we get,
IF(x; T , F) =
(F)
limε→0 [(1−ε)T (F)]+ε−T
if x ∈ (LCL, UCL)
ε
[(1−ε)T (F)]−T (F)
limε→0
if x ∈ (LCL, UCL)
ε
A simple calculation yields
IF(x; T , F) =
1 − T (F) if x ∈ (LCL, UCL)
−T (F) if x ∈ (LCL, UCL)
(3)
Therefore, γ (T , F) = supx |IF(x; T , F)| < ∞ as |IF(x; T , F)| < 1, ∀x. Hence T (F)
is B-robust.
Now, the standardised influence function of T at F is
⎧
⎨ 1−T (F) if x ∈ (LCL, UCL)
T (F)
(4)
SIF(x; T , F, S) =
⎩ − T (F) if x ∈ (LCL, UCL)
1−T (F)
0
0
Now, when F = N(μ, σ0 ), let t1 = μ0 −μ+3σ
and t2 = μ0 −μ−3σ
. Then, we can
σ0
σ0
write T (F) = 1 − [Φ(t1 ) − Φ(t2 )]. Now it can be seen using simple calculus that
min T (F) > 0 and the minimum is attained at μ = μ0 . Also we note the following:
1. T (F) = h(μ) = 1 − [Φ(t1 ) − Φ(t2 )]
2. h is a decreasing function of μ in the interval (−∞, μ0 ] and is an increasing
function of μ in the interval [μ0 , ∞). Further limμ→∞ h(μ) = 1 and limμ→−∞
h(μ) = 1.
Thus we conclude that max T (F) < 1 and hence γ ∗ (T , , S) < ∞. Hence the theorem.
Remark 1 The FAP is not SB-robust at the family
w.r.t. the dispersion measure S(F).
∗
= {N(μ, σ0 ) : −∞ < μ < ∞}
Theorem 2 T (F) = EF (I (LCL,UCL) (X))
√ is not SB-robust at the family 1 when the
measure of dispersion is S(F) = T (F)(1 − T (F)) where 1 = {N(μ0 , σ ) :
σ > 0}.
Proof From Eq. 4 above, we have
⎧
⎨
SIF(x; T , F, S) =
⎩−
1−T (F)
T (F)
T (F)
1−T (F)
if x ∈ (LCL, UCL)
if x ∈ (LCL, UCL)
Note that when F = N(μ0 , σ ), we have T (F) = g(σ ) = 1 − [Φ(t) − Φ(−t)]
where t = 3σσ 0 . Also we observe that as σ −→ 0, g(σ ) −→ 0 and as σ −→ ∞,
g(σ ) −→ 1. Therefore, γ ∗ (T , 1 , S) = ∞. Hence the theorem.
SB-Robustness of Estimators
125
Remark 2 The FAP is SB-robust at the family ∗1 = {N(μ0 , σ ) : 0 < m1 ≤ σ ≤ m2 }
w.r.t. the dispersion measure S(F).
In Theorem 3 below we discuss the SB-robustness of the FAP at the family
2 = {N(μ, σ ) : k1 ≤ μ ≤ k2 , 0 < m1 ≤ σ ≤ m2 }, μ0 ∈ [k1 , k2 ].
Theorem 3 T (F) = EF (I (LCL,UCL)
(X)) is SB-robust at the family
√
sure of dispersion is S(F) = T (F)(1 − T (F)).
Proof From Eq. 4 above, we have
⎧
⎨
SIF(x; T , F, S) =
⎩−
1−T (F)
T (F)
T (F)
1−T (F)
2
when the mea-
if x ∈ (LCL, UCL)
if x ∈ (LCL, UCL)
Note that when F = N(μ, σ ), we can write T (F) = 1 − [Φ(t3 ) − Φ(t4 )] where
0
0
and t4 = μ0 −μ−3σ
. It can be seen using simple calculus that min 2
t3 = μ0 −μ+3σ
σ
σ
T (F) = 1 − [Φ(t5 ) − Φ(−t5 )] > 0 and max 2 T (F) = 1 − [Φ(t6 ) − Φ(−t6 )] < 1
0
0
and t6 = 3σ
. Thus γ ∗ (T , 2 , S) < ∞. Hence the theorem.
where t5 = 3σ
m1
m2
Another important consideration for control chart performance is ASN0 , which is
the average run length before an out-of-control signal is given by the control chart.
It is expected that the ASN0 should be large when the process is in-control and it
should be small if the process is out-of-control. The ASN0 can be represented as a
functional given below.
T1 (F) = [EF (I
(LCL,UCL) (X))]
−1
=
1
T (F)
In Theorem 4 below we discuss the SB-robustness of ASN0 at the family
{N(μ, σ0 ) : k1 ≤ μ ≤ k2 }, μ0 ∈ [k1 , k2 ].
=
1
Theorem 4 1. T1 (F) = [EF (I (LCL,UCL) (X))]−1 = T (F)
is B-robust.
2. T1 (F) is SB-robust at the family with respect to the dispersion measure S ∗ (F) =
1−T (F)
.
(T (F))2
Proof Let G ε = (1 − ε)F + εδx where F = N(μ, σ0 ). The functional corresponding
to G ε can be written as
T1 (G ε ) =
(1−ε)EF (I
1
(LCL,UCL) (X))+ε
(1−ε)EF (I
1
(LCL,UCL) (X))
if x ∈ (LCL, UCL)
if x ∈ (LCL, UCL)
The influence function of T1 at F is
IF(x; T1 , F) = lim
ε→0
T1 (G ε ) − T1 (F)
ε