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1 Mixture Regression Estimation Based on the Mixture of Normal and t Distributions

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



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