The estimating equation (6) can be written as ∑ i = 1 n     Ψ ( x i , y i ; β ^ ; η ^ ) = 0 , with Ψ ( x , y ; β ; η ) = w ( x ; η ) ψ b ( y − F ( x T β ) − c ( x T β , b ) ) x . Under appropriate regularity conditions, the classical asymptotic of M-estimators hold, see [22] . Let β 0 be the model parameter and E 0 denote expectation under the model; define M 0 ( β ) = E 0 { Ψ ( x , y ; β ; η 0 ) } , with η 0 = p lim η ^ . If C 0 = D M 0 ( β ) | β = β 0 , where D denote the differential and A 0 = E 0 { Ψ ( x , y ; β 0 ; η 0 ) Ψ ( x , y ; β 0 ; η 0 ) T } , then

n ( β ^ − β 0 ) → N p ( 0, C 0 − 1 A 0 C 0 − 1 ) .

This result is valid for non-adaptive and adaptive weights alike, as long as η ^ converges to η 0 in probability. For the weighted maximum likelihood estimator given by ψ b ( u ) = u , A 0 and C 0 have simple expressions

C 0 = − E { w ( x ; η 0 ) F ′ ( x T β 0 ) x x T } , (8)

A 0 = E { w 2 ( x ; η 0 ) F ( x T β 0 ) ( 1 − F ( x T β 0 ) ) x x T } . (9)

Using the asymptotic normality of β ^ it is possible to construct confidence ellipsoids for β 0 . First, we estimate the matrices (9) and (6) with

C ^ = − 1 n ∑ i = 1 n     w ( x ; η ^ ) F ′ ( x i T β ^ ) x i x i T ,

A ^ = 1 n ∑ i = 1 n     w ( x i ; β ^ ) F ( x i T β ^ ) ( 1 − F ( x i T β ^ ) ) x i x i T .

Then the estimated asymptotic variance of n ( β ^ − β 0 ) is V ^ = C ^ − 1 A ^ C ^ − 1 . The asymptotic confidence ellipsoid of level ( 1 − α ) for β 0 is given ξ ( β 0 ) = { β ∈ R p : n ( β ^ − β ) T V ^ − 1 ( β ^ − β ) ≤ χ p ,1 − α 2 } . This can be generalized to linear transformations of β 0 .

This subsection focuses on the asymptotic features of the suggested estimator β ^ n described in the preceding section. We will show that the estimator is asymptotically consistent based on some general assumptions about the moments of predictor variables. Suppose that β 0 , μ 0 and Σ 0 are the actual values of β , μ and Σ respectively, and the independent sample ( x 1 , y 1 ) , ⋯ , ( x n , y n ) follows the logistic model P ( y i = 1 ) = μ i , i = 1 , ⋯ , n . We define functions

ℙ n ψ β = Ψ n ( β ) = 1 n ∑ i = 1 n     ψ β , η ^ ( x i , y i )

and

Ψ ( β ) = P ψ β , 0 = P ϕ β ,

where ψ β ,0 is the calculated by Equation (7) with η replaced with η 0 = ( 0, μ 0 , Σ 0 ) and p indicates the joint probability distribution of the (x,y)’s. Theorem 4.2 establishes that β ^ n is consistent, which makes use of the conclusions of Lemma 1, 2, and 3 mentioned below. To prove the Lemmas and the Theorems, the following assumptions must be made:

B1: μ ^ → p μ 0 and Σ ^ → p Σ 0 .

B2: E ( x x ) T is nonsingular.

B3: E G 0 ( ‖ x ‖ 4 ) < ∞ .

B4: The m ( x ) is a continuous weight function and has a first derivative that is bounded, m ( 0 ) = 1 and m 1 ( 0 ) = 0 .

B1 is met for the vast majority of well-known initial robust estimators, such as the Minimum Covariance Determinant used in the simulation experiments. In the following lemmas and theorem, the asymptotic characteristics are assumed to be as n → ∞ . The proof for Lemma 1 is given in [21] .

Lemma 1. In case B1 is holds, then α n = 0 p ( 1 ) .

Lemma 2. In case B2 is holds, then ‖ Ψ ( β n ) ‖ → 0 implies ‖ β n − β 0 ‖ → 0 for any sequence { β n } ∈ Θ .

Proof of Lemma 2. Consider the following

‖ Ψ ( β n ) ‖ = ‖ P ψ β n , 0 ‖ = ‖ P ϕ θ n ‖ = ‖ P ( y − ( x T β n ) ) x ‖ = ‖ P ( y − ( x T β 0 ) + ( x T β 0 ) − ( x T β n ) ) x ‖ = ‖ P ( ( x T β 0 ) ) x + P ( ( x T β 0 ) − ( x T β n ) ) x ‖

If ‖ Ψ ( β n ) ‖ → 0 , then Ψ ( β n ) → 0 . Since P ( y − ( x T β n ) ) x = 0 , we may see from the equality above that P ( ( x T β 0 ) − ( x T β n ) ) x → 0 . Note

P ( ( x T β 0 ) − ( x T β n ) ) x = P ( ( 1 ) ( c ) ( x T β n − x T β 0 ) x ) ≤ 1 4 P ( x T ( β n − β 0 ) x ) ,

where c ∈ ( x T β 0 , x T β n ) and h ′ ( μ ) is the initial derivation for logistic link, h ′ ( μ ) = e x ( e x = 1 ) 2 ∈ ( 0 , 1 / 4 ) . We also have X = ( x 1 , x 2 , ⋯ , x p ) T , β 0 − β n = β d = ( β d 1 , β d 2 , ⋯ , β d p ) T .

P ( x T ( β 0 − β n ) x ) = P ( ∑ i = 1 p     β d i [ x 1 x i x 2 x i ⋮ x p x i ] ) = ∑ i = 1 p     β d i P ( [ x 1 x i x 2 x i ⋮ x p x i ] ) → 0.

Since P ( x x T ) = E ( x x T ) is nonsingular, we have β 0 − β 1 → 0 , is shown by the fact that ‖ β n − β 0 ‖ → 0 .

Lemma 3. Suppose that B3 and B4 are true. Then the class { ψ η , β : β ∈ Θ , ‖ η − η 0 ‖ < δ } is P-Glivenko-Cantelli for some δ > 0 , where η 0 = ( 0, μ 0 , Σ 0 ) .

Proof of Lemma 3. To demonstrate that a class Ϝ of vector-valued functions ψ : ( x , y ) → ℝ p to be Glivenko-Cantelli, we must display each of the coordinate classes φ i : ( x , y ) → ℝ with ψ = ( ψ 1 , ⋯ , ψ p ) T ranging over Ϝ ( i = 1,2, ⋯ , p ) is Glivenko-Cantelli.

The class Ϝ = { ψ γ : γ = ( β , η ) = ( β , α , μ , Σ ) , α ∈ [ 0,1 ] , μ ∈ ℝ q , Σ ∈ S + q , β ∈ Θ , ‖ η − η 0 ‖ < δ } is a set of measurable functions that are indexed by a bounded subset in Γ ⊂ Θ × ℝ q × ℝ × ℝ q × q , and S + q represents a collection of positive semidefinite matrices described in ℝ q × q . This is because Σ is basically a variance-covariance matrix of continuous predictor variables, hence it is positive semidefinite and symmetric. For the norm, we use

‖ η − η n ‖ = ( α 2 + ‖ μ − μ 0 ‖ 2 + ‖ Σ − Σ 0 ‖ + ‖ β − β 0 ‖ 2 ) 1 2 , with ‖   .   ‖ indicates the Euclidean norm of vectors μ , β and for matrix Σ , ‖   .   ‖ represents the induced norm in general. For two values γ i = ( β i , η i ) , ( i = 1 , 2 ) of γ , we have

| ψ γ 1 i ( x , y ) − ψ γ 2 i ( x , y ) | = | w η 1 ( x ) ϕ β 1 i ( x , y ) − w η 2 ( x ) ϕ β 2 i ( x , y ) | = | w η 1 ( x ) ϕ β 1 i ( x , y ) − w η 1 ( x ) ϕ β 1 i ( x , y ) + w η 1 ( x , y ) ϕ β 2 i ( x , y ) − w η 2 ( x ) ϕ β 2 i ( x , y ) | ≤ | w η 1 ( x ) ϕ β 1 i ( x , y ) − w η 1 ( x ) ϕ β 2 i ( x , y ) | + | w η 1 ( x ) ϕ β 2 i ( x , y ) − w η 2 ( x ) ϕ β 2 i ( x , y ) | . (10)

But

| w η 1 ( x ) ϕ β 1 i ( x , y ) − w η 1 ( x ) ϕ β 2 i ( x , y ) | = w η 1 ( x ) | ( y − ( x T β 1 ) ) x i − ( y − ( x T β 2 ) ) x i | ≤ | ( x T β 1 ) − ( x T β 2 ) | | x i | ≤ K 0 | x i | |   x T β 2 − x T β 1 | ≤ K 0 | x i | ‖ c ‖ ‖ β 2 − β 1 ‖ , (11)

where K 0 represents the upper limit of the first derivation of the link function h ′ ( μ ) .

Making use of the Mean Value Theorem, for each x 1 and x 2 , there exists c ∈ ( x 1 , x 2 ) such that

‖ μ ( x 1 ) − μ ( x 2 ) ‖ = h ′ ( c ) | x 1 − x 2 | < K 0 | x 1 − x 2 | .

In addition, we have

| w η 1 ( x ) ϕ β 2 i ( x , y ) − w η 2 ( x ) ϕ β 2 ( x , y ) | = | w η 1 ( x ) − w η 2 ( x ) | | y − ( x T β 2 ) | | x i | = | w α 1 , μ 1 , Σ 1 ( x ) − w α 1 , μ 2 , Σ 2 ( x ) + w α 1 , μ 2 , Σ 2 ( x ) − w α 2 , μ 2 , Σ 2 ( x ) | | y − ( x T β 2 ) | | x i | ≤ ( | w α 1 , μ 1 , Σ 1 ( x ) − w α 1 , μ 2 , Σ 2 ( x ) | + | w α 1 , μ 2 , Σ 2 ( x ) − w α 2 , μ 2 , Σ 2 ( x ) | ) | x i | , (12)

then, we get

| w α 1 , μ 2 , Σ 2 ( x ) − w α 2 , μ 2 , Σ 2 ( x ) | = | m ( α 1 ( Z − μ 2 ) T Σ 2 − 1 ( Z − μ 2 ) ) − m ( α 2 ( Z − μ 2 ) T Σ 2 − 1 ( Z − μ 2 ) ) | ≤ K 1 | α 2 − α 1 | ( Z − μ 2 ) T Σ 2 − 1 ( Z − μ 2 ) , (13)

and

w α 1 , μ 1 , Σ 1 ( x ) − w α 1 , μ 2 , Σ 2 ( x ) = | m ( α 1 ( Z − μ 1 ) T Σ 1 − 1 ( Z − μ 1 ) ) − m ( α 1 ( Z − μ 2 ) T Σ 2 − 1 ( Z − μ 2 ) ) | ≤ K 1 α 1 | ( Z − μ 1 ) T Σ 1 − 1 ( Z − μ 1 ) − ( Z − μ 2 ) T Σ 2 − 1 ( Z − μ 2 ) | ≤ K 1 | ( Z − μ 1 ) T Σ 1 − 1 ( Z − μ 1 ) − ( Z − μ 1 ) T Σ 2 − 1 ( Z − μ 1 ) |         + K 1 | ( Z − μ 1 ) T Σ 2 − 1 ( Z − μ 1 ) − ( Z − μ 2 ) T Σ 2 − 1 ( Z − μ 2 ) | . (14)

Given that Σ is a positive semidefinite matrix, as well Σ − 1 . Indicate what the eigenvalues of Σ 2 − 1 are as λ 1 ≥ λ 2 ≥ ⋯ ≥ λ p ≥ 0 . Consequently, Σ 2 − 1 has a set of orthogonal eigenvectors, say p 1 , ⋯ , p q , s.t. Σ 2 − 1 p i = λ i p i . There exists an orthogonal matrix in matrix format Q s.t.

Q − 1 Σ 2 − 1 Q = Q T Σ 2 − 1 Q = Λ .

Then we get

| ( Z − μ 1 ) T Σ 2 − 1 ( Z − μ 1 ) − ( Z − μ 2 ) T Σ 2 − 1 ( − μ 2 ) | = | ∑ i = 1 q     λ i ( p i T ( − μ 1 ) ) 2 − ∑ i = 1 q     λ i ( p i T ( Z − μ 2 ) ) 2 | = | ∑ i = 1 q     λ i p i T ( 2 Z − μ 1 − μ 2 ) p i T ( μ 2 − μ 1 ) | ≤ ∑ i = 1 q     λ i ‖ p i ‖ 2 ‖ 2 Z − μ 1 − μ 2 ‖ ‖ μ 2 − μ 1 ‖

≤ λ 1 ‖ μ 2 − μ 1 ‖ ‖ 2 Z − μ 1 − μ 2 ‖ ∑ i = 1 q ‖ p i ‖ 2 = ρ ( Σ 2 − 1 ) ‖ μ 2 − μ 1 ‖ ‖ 2 Z − μ 1 − μ 2 ‖ ∑ i = 1 q ‖ p i ‖ 2 ≤ ‖ Σ 2 − 1 ‖ ‖ μ 2 − μ 1 ‖ ‖ 2 Z − μ 1 − μ 2 ‖ , (15)

where ρ ( Σ 2 − 1 ) is the spectral radius of Σ 2 − 1 and ρ ( A ) ≤ ‖ A ‖ applicable to any induced norm. Similarly Σ 2 − 1 − Σ 1 − 1 is also symmetric, thus indicate the eigenvalues of Σ 2 − 1 − Σ 1 − 1 as p 1 * , ⋯ , p q * s.t. ( Σ 2 − 1 − Σ 1 − 1 ) p i * = λ i * p i * . There exists an orthogonal matrix in matrix format Q * s.t.

( Q * ) − 1 ( Σ 2 − 1 − Σ 1 − 1 ) Q * = ( Q * ) T ( Σ 2 − 1 − Σ 1 − 1 ) Q * = Λ * .

Next, we have

| ( Z − μ 1 ) T Σ 2 − 1 ( Z − μ 1 ) − ( Z − μ 1 ) T Σ 1 − 1 ( Z − μ 1 ) | = | ( Z − μ 1 ) T ( Σ 2 − 1 − Σ 1 − 1 ) ( Z − μ 1 ) | = | ( Z − μ 1 ) T Q * Λ * ( Q * ) T ( Z − μ 1 ) | = | [ ( Q * ) T ( Z − μ 1 ) ] T Λ * [ ( Q * ) T ( Z − μ 1 ) ] |

= | ∑ i = 1 q     λ i ( ( p i * ) T ( Z − μ 1 ) ) 2 | ≤ max i = 1 , ⋯ , q | λ i * | ∑ i = 1 q ( ( p i * ) T ( Z − μ 1 ) ) 2 ≤ ρ ( Σ 2 − 1 − Σ 1 − 1 ) ‖ Z − μ 1 ‖ 2 ∑ i = 1 q ‖ ( p i * ) 2 ‖ ≤ ‖ Z − μ 1 ‖ 2 ‖ Σ 2 − 1 − Σ 1 − 1 ‖ . (16)

[23] presented a simple and popular M-estimator that minimizes a “bounded” version of the sum of residuals. The estimating equation is

θ ^ M = ∑ i = 1 n     ψ ( r i ( θ ^ ) σ ) x i = 0 (17)

Then, using (12), (13), (14), (15) and (17) we get

‖ w η 1 ( x ) ϕ β 2 i ( x , y ) − w η 2 ( x ) ϕ β 2 ( x , y ) ‖ < K 1 | α 2 − α 1 | ( Z − μ 2 ) T Σ 2 − 1 ( Z − μ 2 ) | x i |       + K 1 ‖ Σ 2 − 1 ‖ ‖ μ 2 − μ 1 ‖ ‖ 2 Z − μ 1 − μ 2 ‖ ‖ x i ‖       + K 1 ‖ Z − μ 1 ‖ 2 ‖ Σ 2 − 1 − Σ 1 − 1 ‖ | x i | . (18)

Since | α 2 − α 1 | , ‖ μ 2 − μ 1 ‖ , ‖ β 2 − β 1 ‖ < ‖ γ 2 − γ 1 ‖ , from (11), (10) and (16) now, we can provide a bound for ψ :

| ψ γ 1 i ( x , y ) − ψ γ 2 ( x , y ) | < K 0 ‖ β 2 − β 1 ‖ | x i | ‖ x ‖ + K 1 | α 2 − α 1 | ( Z − μ 2 ) T Σ 2 − 1 ( Z − μ 2 ) | x i |         + K 1 ‖ Σ 2 − 1 ‖ ‖ μ 2 − μ + 1 ‖ ‖ 2 Z − μ 1 − μ 2 ‖ | x i | + K 1 ‖ Z − μ 1 ‖ 2 ‖ Σ 2 − 1 − Σ 1 − 1 ‖ | x i | ≤ ( K 0 | x i | ‖ x ‖ + K 1 ( Z − μ 2 ) T Σ 2 − 1 ( Z − μ 2 ) | x i |         + K 1 ‖ Σ 2 − 1 ‖ ‖ 2 Z − μ 1 − μ 2 ‖ | x i | + K 1 ‖ Z − μ 1 ‖ 2 | x i | ) ‖ γ 2 − γ 1 ‖ = L i ( x ) ‖ γ 2 − γ 1 ‖ ,

for every γ 1 , γ 2 . We have constructed a Lipschitz criterion for each ψ i , i = 1 , ⋯ , p , so for ψ we have | ψ γ 1 ( x , y ) − ψ γ 2 ( x , y ) | < L ( x ) ‖ γ 2 − γ 1 ‖ , every γ 1 , γ 2 ,

where

L ( x ) = K 0 x ‖ x ‖ + K 1 ( Z − μ 2 ) T Σ 2 − 1 ( Z − μ 2 ) x       + K 1 ‖ Σ 2 − 1 ‖ ‖ 2 Z − μ 1 − μ 2 ‖ | x i | + K 1 ‖ Z − μ 1 ‖ 2 x .

Now investigate the bracketing entropy in relative to the L r ( P ) -norm

‖ ψ i ‖ P r = ( P | ψ | r ) 1 r .

Use brackets of the type [ ψ γ − ε L , ψ γ + ε L ] for γ spanning over a sufficient chosen subset of Γ and these brackets have L r ( P ) − s i z e   2 ε ‖ L ‖ p r . If γ ranges over a grid of mesh width ε over Γ then the brackets [ ψ γ − ε L , ψ γ + ε L ] range over Ϝ . Based on the Lipschitz condition we get

ψ γ 1 − ε L ≤ ψ γ 2 ≤ ψ γ 1 + ε L ,     if   ‖ γ 2 − γ 1 ‖ ≤ ε ,

Hence, many brackets are required as in radial balls ε 2 to cover Γ , alternatively we require less than ( d i a m Γ / ε ) 2 p + 2 cubes with size ε to cover parameter space Γ . If P | L i | r < ∞ , then a constant J exists, depends only on Γ and P, in such a way that the bracketing numbers fulfill

N [ ] ( ε , Ϝ , L R ( P ) ) ≤ J ( d i a m   Γ ε ) p ,     every   0 < ε < d i a m   Γ .

Given that all ψ ∈ Ϝ are continuous functions, they can be measured. If B3 is fulfilled, then P | L | < ∞ , and as a result, class Ϝ is P-Glivenko-Cantelli from the Theorem (19.4) (Glivenko-Cantelli) in [24] .¨

If B1, B2, B3 and B4 are hold, then estimators β ^ n are used as the solving to the equation Ψ ( β ^ n ) = 0 converges to β 0 in probability.

Proof of Theorem 1. denote

Ψ ( β ) = P ψ β , 0 = P ϕ β Ψ ( β , η ) = P ψ β , η = P ϕ β W η

Ψ ( β ) = Ψ n ( β , η ) = 1 n ∑ i = 1 n     ψ β , η Ψ n ( β , η ^ ) = 1 n ∑ i = 1 n     ψ β , η ^ .

Note that Ψ ( β ^ n ) = ( Ψ ( β ^ n ) − Ψ n ( β ^ n ) ) + Ψ n ( β ^ n ) . Then we can establish that

sup β ∈ Θ ‖ Ψ n ( β ) − Ψ ( β ) ‖ = 0 p ( 1 ) , then ‖ Ψ ( β ^ n ) ‖ = 0 p ( 1 ) since Ψ n ( β ^ n ) = 0 p ( 1 ) . Thus, based upon Lemma 2, we have ‖ β ^ n − β 0 ‖ = 0 p ( 1 ) . To show sup β ∈ Θ ‖ Ψ n ( β ) − Ψ ( β ) ‖ = 0 p ( 1 ) , we consider it as follows:

sup β ∈ Θ ‖ Ψ n ( β , η ^ ) − Ψ ( β ) ‖ = sup β ∈ Θ ‖ Ψ n ( β , η ^ ) + Ψ ( β , η ) + Ψ n ( β , η ) − Ψ ( β , η ) + Ψ ( β , η ) − Ψ ( β ,0 ) ‖ ≤ J 1 + J 2 + J 3 ,

where

J 1 = sup β ∈ Θ ‖ Ψ n ( β , η ^ ) − Ψ n ( β , η ) ‖

J 2 = sup β ∈ Θ ‖ Ψ n ( β , η ) − Ψ ( β , η ) ‖

J 3 = sup β ∈ Θ ‖ Ψ ( β , η ) − Ψ ( β , 0 ) ‖ .

Lemma 3 informs us that Ϝ is P-Glivenko-Cantelli class, so J 2 → 0 . For J 1 ,

J 1 = sup β ∈ Θ ‖ 1 n ∑ i = 1 n     ϕ β ( x i , y i ) ( w η ^ ( x i ) − w η ( x i ) ) ‖ ≤ 1 n ∑ i = 1 n sup β ∈ Θ ‖ ϕ β ( x i , y i ) ( w η ^ ( x i ) − w η ( x i ) ) ‖ = 1 n ∑ i = 1 n | w η ^ ( x i ) − w η ( x i ) | sup β ∈ Θ ‖ ϕ β ( x i , y i ) ‖ .

Using (16), we get

| w η ^ ( x i ) − w η ( x i ) | < K 1 | α ^ − α | ( Z − μ ) T Σ − 1 ( Z − μ )   + K 1 ‖ Σ − 1 ‖ ‖ μ ^ − μ ‖ ‖ 2 Z − μ ^ − μ ‖   + K 1 ‖ Z − μ ^ ‖ 2 ‖ Σ ^ − 1 − Σ − 1 ‖ . (19)

Since η ^ − η = ( α ^ − α , μ ^ − μ , Σ ^ − Σ ) = 0 p ( 1 ) , we have α ^ − α = 0 p ( 1 ) , μ ^ − μ = 0 p ( 1 ) and Σ ^ − Σ = 0 p ( 1 ) . Also, Σ ^ − 1 − Σ = Σ ^ − 1 ( Σ − Σ ^ ) Σ − 1 , as a result Σ ^ − 1 − Σ − 1 = 0 p ( 1 ) . Therefore, (19) follows that | w η ^ ( X i ) | = 0 p ( 1 ) . Because B2 and B3 are fulfilled, we also have sup β ∈ Θ ‖ ϕ β ( x , y ) ‖ 2 < ∞ , then we get J 1 → p 0 . For J 3

J 3 = sup β ∈ Θ ‖ P ϕ β ( x , y ) ( w η ( x ) − w η 0 ( x ) ) ‖ ≤ sup β ∈ Θ P ‖ ϕ β ( x , y ) ( w η ( x ) − w η 0 ( x ) ) ‖ ≤ sup β ∈ Θ P 1 2 ‖ ϕ β ( x , y ) ‖ 2 P 1 2 | w η ( x ) − w η 0 ( x ) | 2 = P 1 2 | w η ( x ) − w η 0 ( x ) | 2 sup β ∈ Θ P 1 2 ‖ ϕ β ( x , y ) ‖ 2 ,

and the Cauchy-Schwarz inequality was employed in the final inequality. We already know

w η ( x ) − w η 0 ( x ) = 1 − m ( η ( Z − μ ) T Σ − 1 ( Z − μ ) ) → 0.

Then η ^ = 0 p ( 1 ) m ( 0 ) = 1 and | w η ^ ( x ) − w η 0 ( x ) | 2 is bounded by 1, using the dominated convergence theorem we get

l i m n → ∞ P 1 2 | w η ( x ) − w η 0 ( x ) | 2 → 0.

We also have

sup β ∈ Θ P ‖ ϕ β ( x , y ) ‖ 2 = sup β ∈ Θ P ‖ ( y − x T β ) x ‖ 2 ≤ P ‖ x ‖ 2 .

and from assumption B3, we obtain sup β ∈ Θ P ‖ ϕ β ( x , y ) ‖ 2 < ∞ . It then follows that J 3 → 0 .¨

A Monte Carlo simulation analysis was performed for this subsection to assess the efficiency and robustness of the suggested estimator β ^ n . For the initial robust estimators of the scatter and location Σ ^ and μ ^ utilized the minimum covariance determinant (MCD). We calculated the following estimators for comparison: MLE, conditionally unbiased bounded influence (CUBI) of [2] , and the Mallows-type estimator (Mallows) of [3] . In the simulation, the following weight function m ( x ) was applied:

m ( x ) = a ( 1 − ( x − 1 ) 6 ) + b ,

where a = 0.8 and b = 0.2 .

Three models are involved in the simulation study: a clean logistic regression model, a contaminated model with a 10% contamination rate, and a contaminated model with a 20% contamination rate. First, clean model, the standard normal distribution was used to generate two predictor variables, x 1 ~ N ( 0,1 ) and x 2 ~ N ( 0,1 ) . Three sample sizes were used: ( n = 100 , 300 , 500 ) and p = 2 . We generate the response variable according to the Bernoulli distribution with a parameter equal to π i = exp ( β 0 + β 1 x 1 + β 2 x 2 ) / ( 1 + exp ( β 0 + β 1 x 1 + β 2 x 2 ) ) . The values of the true parameters β are taken ( 0,1.6,1.2 ) for three models. Second, the percentage of contamination in data equal 10%, and their predicted variables are generated from a normal distribution with ( μ = 0 ) and ( σ = 1 ). Third, the percentage of contamination in data equals 20% in a similar manner to the above.

The performance of these estimators is evaluated using Bias and mean squared error for various models. Nevertheless, the estimator with the smallest Bias and MSE is the best. Each scenario simulation included over 1000 repetitions. Consequently, for each parameter, the following are the calculations for bias and mean squared error:

Bias = | 1 1000 ∑ i = 1 1000 β i − β | ,

and

MSE = 1 1000 ∑ i = 1 n | β ^ i − β | 2 .

The numerical results, displayed in this paper, are based on simulation studies, and two real data applications. This numerical result is expected to evaluate, the performance of a proposed model. Table 1 shows the bias and mean squared errors of the four estimation techniques for the clean model. The findings show that the bias and MSE of the MLE, Mallows, and CUBIF estimators are relatively similar, while the WMLT estimator performs worse than the other estimators. When the sample size increases, the bias and mean squared errors are observed to decrease. As shown in Table 2, under 10% of the data were contaminated, so the new robust approach WMLT has the greatest overall performance among all comparable estimators for varied sample sizes. Table 3 demonstrates that when 20% of the data are contaminated, our proposed technique (WMLT) outperforms other estimators in terms of bias and mean squared errors. Due to the sensitivity of anomalies, conventional maximum likelihood estimates perform inadequately in the contaminated model. In conclusion, the proposed method outperforms all other methods compared with contaminated data. Furthermore, the new estimator performs reasonably well with clear data.

This study uses data from [25] which includes information from 33 people who died from acute myeloid leukemia. Each patient was measured for three variables: AG, WBC, and time. The response variable represents the survival time in weeks of the patient; it was converted into a binary variable with (Y = 1) indicating patients whose survival time exceeded 52 weeks and (Y = 0) indicating those who did not. WBC represents the white blood cell concentration of the patient. Whereas AG (present = 1, absent = 0) measured the presence or absence of a morphologic characteristic of white blood cells. The observation number 17 appears to be atypical. Using AG and WBC as predictor variables and binary survival time y as the response variable, a logistic regression model was constructed. The estimators analyzed here are weighted maximum likelihood technique (WMLT), MLE, MLE17 (MLE17 is the maximum likelihood estimator for clean data when observation 17 is excluded.), Mallows (estimator of the Mallows type), and CUBIF (conditionally unbiased bounded-influence function estimator).

Table 4 demonstrates that the MLE is extremely sensitive to influential observations. Furthermore, eliminating observation 17 lowered the impact of WBC to near nil. For the leukemia data, the new estimator (WMLT) demonstrated the greatest performance among all other estimators. However, Mallow’s estimators are reasonably close to the MLE17.

The Erythrocyte Sedimentation Rate (ESR) data. In this data, the primary objective was to determine if the levels of two plasma proteins (Fibrinogen and γ.Globulin) were responsible for the increase in ESR in healthy individuals. The research was conducted by the Institute of Medical Research in Kuala Lumpur, Malaysia, on 32 patients, and the original data was collected by [26] . The response of zero indicates a healthy person, whereas the response of one indicates an unwell person. Here, the continuous variables (FIB and γ.GLO) are compared to the binary response (ESR). In the original ESR data, [27] identified two outliers (cases 13 and 29) in X-space. Cases 14 and 15 are influential observations. Therefore, removing instances 14 and 15 would result in cases with no overlap.

Case 13 was modified so that ( y , x 1 , x 2 ) = ( 3 , 6 , 37 ) in order to execute uncontaminated data analysis. From the uncontaminated data, the ESR data were contaminated where the occurrences (Y = 1) and non-occurrences (Y = 0) were replaced with each other for cases 14 and 15, and this might only leave one out of the three overlapping cases for the ESR data.

Under contaminated ESR data, β₀ and se(β₀) of all estimators are impacted by outliers as compared to the other parameters (see Table 5). The results shown in Table 5 also indicate that the MLE is primarily influenced by outliers. Following the modification of the tainted data, only one overlapping observation, case 13, remains. This is the reason why the coefficients and standard errors of the WMLT that downweights this observation are so large. Even though the WMLT has the smallest χ² value, the WMLT estimator should also be taken into consideration. The results shown in Table 5 indicate that the WMLT is a suitable estimator for the ESR data because its estimates are relatively closer to the MLE for uncontaminated data.