Strong and weak consistency of least squares estimators in simple linear EV regression models

Journal of Statistical Planning and Inference 205 (2020) 64–73

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Strong and weak consistency of least squares estimators in simple linear EV regression models Pingyan Chen a , Luliang Wen b , Soo Hak Sung c ,

∗

a

Department of Mathematics, Jinan University, Guangzhou, 510630, PR China Department of Statistics, Jinan University, Guangzhou, 510630, PR China c Department of Applied Mathematics, Pai Chai University, Daejeon, 35345, South Korea b

article

info

Article history: Received 14 December 2018 Received in revised form 12 June 2019 Accepted 13 June 2019 Available online 19 June 2019

a b s t r a c t For a simple linear errors-in-variables regression model, we obtain a necessary and sufficient condition for the convergence rate in the strong consistency of the least squares estimator for each of the unknown parameters. We also obtain a necessary and sufficient condition for the convergence rate in the weak consistency. © 2019 Elsevier B.V. All rights reserved.

MSC: 62F12 60F15 Keywords: Strong consistency Weak consistency Simple linear EV regression model

1. Introduction To correct the effects of the sampling errors, Deaton (1985) proposed the errors-in-variables (EV) regression model. A simple linear EV regression model is

ηk = θ + β xk + εk , ξk = xk + δk , 1 ≤ k ≤ n,

(1.1)

where θ , β, x1 , . . . , xn are unknown parameters or constants, (εk , δk ), 1 ≤ k ≤ n, are random vectors and ξk , ηk , 1 ≤ k ≤ n, are observable variables. From (1.1), we have

ηk = θ + βξk + (εk − βδk ), 1 ≤ k ≤ n. As a usual regression model of ηk on ξk with the errors εk − βδk , we can obtain the least squares (LS) estimators of β and θ as

∑n

ξ − ξ¯ η − η¯ n ) , θˆn = η¯ n − βˆ n ξ¯n , (1.2) ∑ ξ − ξ¯ ∑n where ξ¯n = n−1 k=1 ξk . The notations of η¯ n , δ¯ n , and x¯ n are defined in the same way. Based on these notations, we have ∑n ∑n ∑n (δk − δ¯ n )εk + k=1 (xk − x¯ n )(εk − βδk ) − β k=1 (δk − δ¯ n )2 βˆ n − β = k=1 (1.3) ∑n ¯ 2 k=1 (ξk − ξn ) βˆ n =

n )( k k=1 ( k n 2 ( n) k=1 k

∗ Corresponding author. E-mail address: [email protected] (S.H. Sung). https://doi.org/10.1016/j.jspi.2019.06.004 0378-3758/© 2019 Elsevier B.V. All rights reserved.

P. Chen, L. Wen and S.H. Sung / Journal of Statistical Planning and Inference 205 (2020) 64–73

65

and

θˆn − θ = −¯xn (βˆ n − β ) − (βˆ n − β )δ¯n + ε¯ n − β δ¯n .

(1.4)

The EV model is somewhat more practical than the ordinary regression model, and hence has attracted more and more attention. Fuller (1987) summarized many early works for the EV models. The weak and strong consistency of the LS estimators for the unknown parameters have been studied by Hu et al. (2017), Liu and Chen (2005), Martsynyuk (2013), Miao et al. (2011), Shen (2017), Wang and Hu (2017), Wang et al. (2015, 2018), and Wu et al. (2017). This paper deals with the EV model with independent and identically distributed (i.i.d.) errors having finite variances. Liu and Chen (2005) obtained necessary and sufficient conditions for the weak and strong consistency of the least squares estimators for the unknown parameters β and θ as follows. Theorem 1.1. Under the model (1.1), assume that {(ε, δ ), (εn , δn ), n ≥ 1} are i.i.d. random vectors with E ε = E δ = 0 and 0 < E ε 2 , E δ 2 < ∞. Furthermore, assume that E(εδ ) − β E δ 2 ̸ = 0. Then a necessary and sufficient condition for βˆ n → β a.s. and βˆ n → β in probability is sn /n → ∞, and a necessary and sufficient condition for θˆn → θ in probability is nx¯ n /s∗n → 0, where and in the following, sn =

∑n

k=1 (xk

− x¯ n )2 , and s∗n = max{n, sn }, n ≥ 1.

Remark 1.1. The assumption E(εδ ) − β E δ 2 ̸ = 0 in Theorem 1.1 is omitted in Liu and Chen (2005). However, this is necessary in the proof of Theorem 1.1. Theorem 1.1 shows that the weak and strong consistency of βˆ n are equivalent. However, it is unknown whether the weak and strong consistency of θˆn are equivalent. Liu and Chen (2005) mentioned that the strong consistency of θˆn holds under some supplement conditions such as supn≥1 nx¯ 2n /s∗n < ∞. Hu et al. (2017) extended the strong consistency of βˆ n in Theorem 1.1 to the identically distributed ψ -mixing random vectors. Under the condition supn≥1 nx¯ 2n /s∗n < ∞, they also extended the strong consistency of θˆn to the identically distributed ψ -mixing random vectors. It is more interesting to obtain a convergence rate in the strong or weak consistency. Miao et al. (2011), among other things, obtained a convergence rate in the strong consistency of βˆ n as follows. Theorem 1.2. Under the model (1.1), assume that {(ε, δ ), (εn , δn ), n ≥ 1} are i.i.d. random vectors with E ε = E δ = 0 and √ 0 < E |ε|p , E |δ|p < ∞ for some p ≥ 2. Then n1−1/p / sn → 0 implies that

√

sn

n1/p

(βˆ n − β ) → 0 a.s.

√

√

√

If n1−1/p / sn → 0 for some p ≥ 2, then sn /n1/p = n1−2/p sn /n1−1/p → ∞, and so Theorem 1.2 gives a convergence rate in the strong consistency of βˆ n . Miao et al. (2011) also obtained a convergence rate in the weak consistency of βˆ n and obtained convergence rates in the weak and strong consistency of θˆn . However, their results, including Theorem 1.2, give only sufficient conditions. Martsynyuk (2013), among other things, obtained convergence rates in the weak and strong consistency assuming that the i.i.d. errors {ε, εn } and {δ, δn } are in the domain of attraction of the normal law. Hence it is possible that E ε 2 < ∞ or = ∞, and E δ 2 < ∞ or = ∞. Here we consider only the case with finite variances. Theorem 1.3 (Martsynyuk, 2013). Under the model (1.1), assume that {(ε, δ ), (εn , δn ), n ≥ 1} is a sequence of i.i.d. random vectors with E ε = E δ = 0 and 0 < E ε 2 , E δ 2 < ∞. Furthermore, assume that lim sup√ xn | < ∞ and sn /n → ∞. Let n→∞ |¯ {bn , n ≥ 1} be a sequence of positive constants satisfying bn → ∞ and lim supn→∞ bn / sn /n < ∞. Then bn (βˆ n − β ) → 0 a.s. Martsynyuk (2013) also obtained a convergence rate in the weak consistency of βˆ n and obtained convergence rates in the weak and strong consistency of θˆn . However, the results of Martsynyuk (2013), including Theorem 1.3, give only sufficient conditions. In the paper, we provide necessary and sufficient conditions for the convergence rates in the weak and strong consistency for the unknown parameters β and θ . 2. Main results To prove the main results, the strong law of large numbers for weighted sums of random variables is needed. One can refer to Thrum (1987).

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P. Chen, L. Wen and S.H. Sung / Journal of Statistical Planning and Inference 205 (2020) 64–73

Lemma 2.1. Let {X , Xn , n ≥ 1} be a sequence of i.i.d. random variables with EX = 0 and E |X |p < ∞ for some p ≥ 2. Let {ank , 1 ≤ k ≤ n, n ≥ 1} be an array of constants satisfying sup

n ∑

n≥1

|ank |2 < ∞.

k=1

Then n−1/p

n ∑

ank Xk → 0 a.s.

k=1

The following lemma provides the convergence rate of βˆ n → β a.s. Lemma 2.2. Under the model (1.1), assume that {(ε, δ ), (εn , δn ), n ≥ 1} is √ a sequence of i.i.d. random vectors with E ε = E δ = 0, 0 < E |ε|p , E |δ|p < ∞ for some p ≥ 2. Furthermore, assume that n1−1/p / sn → c ∈ [0, ∞]. Then

⎧ 0 a.s. ⎪ ⎪ ⎪ ⎪ c(E(εδ)−β E δ2 ) ⎪ a.s. ⎪ ⎪ ⎪ 1+c 2 E δ2 ⎪ ⎪ √ c(E( εδ ) − β E δ 2 ) a.s. ⎨ sn ˆ n − β ) → ∞ a.s. ( β ⎪ n1/p ⎪ ⎪ −∞ a.s. ⎪ ⎪ ⎪ E(εδ )−β E δ 2 ⎪ ⎪ a.s. ⎪ 2 ⎪ ⎩ dE δ

if c = 0, if 0 < c < ∞, p = 2, if 0 < c < ∞, p > 2,

√ √

if c = ∞, n1/p / sn → 0, E(εδ ) − β E δ 2 > 0, if c = ∞, n1/p / sn → 0, E(εδ ) − β E δ 2 < 0,

√ √

if c = ∞, n1/p / sn → d ∈ (0, ∞), if c = ∞, n1/p / sn → ∞.

0 a.s.

Proof. By the Kolmogorov strong law of large numbers, n 1∑ (δk − δ¯ n )εk → E(εδ ) a.s. n

(2.1)

n 1∑ (δk − δ¯ n )2 → E δ 2 a.s. n

(2.2)

k=1

and

k=1

√

Set ank = (xk − x¯ n )/ sn for 1 ≤ k ≤ n and n ≥ 1. Then

∑n

k=1

a2nk = 1 for all n ≥ 1, and hence by Lemma 2.1,

√ √ n n ∑ 1∑ sn sn 1 (xk − x¯ n )(εk − βδk ) = 1−1/p 1/p √ (xk − x¯ n )(εk − βδk ) = 1−1/p o(1) a.s. n n n sn n k=1

(2.3)

k=1

By the Kolmogorov strong law of large numbers and Lemma 2.1 again, we also have that n ∑

1

n

(ξk − ξ¯n )2

√ 1−1/p

sn

=

1 n1−1/p

k=1

( √

√ =

sn

n1−1/p sn

n1−1/p

(δk − δ¯ n ) + 2 2

k=1 n

n1/p 1 ∑

+√

sn n

√ =

sn +

sn

n ∑

n ∑

) (xk − x¯ n )δk

k=1

n1/p

δk2 − √ δ¯n2 + sn

k=1

n1/p

n1/p

sn

sn

2

n ∑

1

n1−2/p n

√ 1/p

sn

(xk − x¯ n )δk

k=1

+ √ (E δ 2 + o(1)) + √ o(1) + o(1) a.s.,

(2.4)

since 1 n1−2/p

n ∑

1 n1/p

√

sn

(xk − x¯ n )δk =

k=1

1 n1−2/p

o(1) = o(1)

by p ≥ 2. Hence, we have by (1.3) and (2.1)–(2.4) that

√ n

E(εδ ) − β E δ 2 +

sn

(βˆ n − β ) = √ 1/p

sn

n−(1−1/p)

+

n1/p (

√

sn

√

sn n−(1−1/p) o(1) + o(1)

)−1 E δ 2

a.s. √ + n1/p ( sn )−1 o(1) + o(1)

(2.5)

P. Chen, L. Wen and S.H. Sung / Journal of Statistical Planning and Inference 205 (2020) 64–73

Dividing the numerator and denominator by

√

(

sn

n

(βˆ n − β ) = 1/p

E(εδ ) − β E δ

) 2

√

67

sn n−(1−1/p) , we also have that

√

√

n1−1/p / sn + o(1) + o(1)n1−1/p / sn

√

1 −1 2 1−1/p ( s )−1 o(1) 1 + ns− n n E δ + nsn o(1) + n

a.s.

(2.6)

If 0 ≤ c < ∞, then the result follows from (2.6). If c = ∞, then the result follows from (2.5). □ Remark 2.1. Miao et al. (2011) proved Lemma 2.2 when c = 0. The following theorem gives a necessary and sufficient condition for the convergence rate in the strong consistency of βˆ n . Theorem 2.1. Under the model (1.1), assume that {(ε, δ ), (εn , δn ), n ≥ 1} is a sequence of i.i.d. random vectors with √ E ε = E δ = 0, 0 < E |ε|p , E |δ|p < ∞ for some p ≥ 2. Furthermore, assume that lim infn→∞ sn /n1/p > 0 and 2 E(εδ ) − β E δ ̸ = 0. Then

√

sn

n1/p

√

(βˆ n − β ) → 0 a.s. if and only if n1−1/p / sn → 0.

√

√

Proof. Sufficiency. If n1−1/p / sn → 0, then n−1/p sn (βˆ n − β ) → 0 a.s. by Lemma 2.2. √ Necessity. Suppose that n1−1/p / sn does not converge to 0. Taking a subsequence when necessary, we can assume √ that n1−1/p / sn → c ∈ (0, ∞]. If 0 < c < ∞, then we have by Lemma 2.2. that

√

sn

n

{

(βˆ n − β ) → 1/p

c(E(εδ )−β E δ 2 ) 1+c 2 E δ 2

if p = 2,

a.s.

c(E(εδ ) − β E δ ) a.s. 2

(2.7)

if p > 2.

√

√

√

If c = ∞, then lim supn→∞ n1/p / sn = d ∈ [0, ∞) because n1/p / sn ≤ n1−1/p / sn and the assumption √ 1/p √ lim infn→∞ sn /n1/p > 0. In this case, there exists a subsequence {nk } such that limk→∞ nk / snk = d. By Lemma 2.2,

√

sn k

1/p

nk

⎧ ⎪ ⎨∞ a.s. (βˆ nk − β ) → −∞ a.s. ⎪ ⎩ E(εδ)−β E δ2 a.s. dE δ 2

if d = 0, E(εδ ) − β E δ 2 > 0, if d = 0, E(εδ ) − β E δ 2 < 0,

(2.8)

if 0 < d < ∞. −1/p √

Eqs.(2.7) and (2.8), together with the assumption E(εδ ) − β E δ 2 ̸ = 0, imply that nk √ 0 a.s., and hence n−1/p sn (βˆ n − β ) does not converge to 0 a.s. □

snk (βˆ nk − β ) does not converge to

Remark 2.2. To estimate the convergence rate of βˆ n → β a.s., we have considered the convergence of n−1/p √ 0 a.s. Hence, it is natural to assume that lim infn→∞ n−1/p sn > 0.

√

sn (βˆ n −β ) →

Remark 2.3. Miao et al. (2011) proved the sufficiency of Theorem 2.1 (see Theorem 1.2). We now compare √ Theorem 2.1 with the corresponding one (see Theorem 1.3) of Martsynyuk (2013). From the condition lim sup b / sn /n < ∞ of n n→∞ √ Martsynyuk (2013), it follows that bn = O( sn /n). If p = 2, the convergence rate bn is the same as in Theorem 2.1. If √ p > 2, the convergence rate bn is less than that ( sn /n1/p ) of Theorem 2.1. Since the condition lim supn→∞ |¯xn | < ∞ in Theorem 1.3 is not used in Theorem 2.1, Theorem 2.1 generalizes and improves the result of Martsynyuk (2013). The following lemma provides the convergence rate of βˆ n → β in probability. Lemma 2.3. Under the model (1.1), assume that {(ε, δ ), (εn , δn ), n ≥ 1} is a sequence of i.i.d. random vectors with E ε = E δ = 0, √ 0 < E ε 2 , E δ 2 < ∞. Furthermore, assume that n1−1/p / sn → c ∈ [0, ∞] for some p ≥ 2. Then

⎧ 0 in probability ⎪ ⎪ ⎪ c(E(εδ)−β E δ2 ) ⎪ ⎪ in probability ⎪ ⎪ 1+c 2 E δ 2 ⎪ ⎪ ⎪ √ c(E(εδ ) − β E δ 2 ) in probability ⎨ sn (βˆ n − β ) → ∞ in probability 1 ⎪ n /p ⎪ ⎪ −∞ in probability ⎪ ⎪ 2 ⎪ ⎪ ⎪ E(εδ)−β2 E δ in probability ⎪ dE δ ⎪ ⎩ 0 in probability

if c = 0, if 0 < c < ∞, p = 2, if 0 < c < ∞, p > 2,

√ √ if c = ∞, n / sn → 0, E(εδ ) − β E δ 2 < 0, √ if c = ∞, n1/p / sn → d ∈ (0, ∞), √ if c = ∞, n1/p / sn → ∞, if c = ∞, n1/p / sn → 0, E(εδ ) − β E δ 2 > 0, 1/p

where Xn → ∞ in probability means that P(Xn < x) → 0 for any real x, and Xn → −∞ in probability means that P(Xn > x) → 0 for any real x. Proof. The proof is similar to that of Lemma 2.2. However, it does not use Lemma 2.1.

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P. Chen, L. Wen and S.H. Sung / Journal of Statistical Planning and Inference 205 (2020) 64–73

We first consider the case of 0 ≤ c < ∞. By the Kolmogorov strong law of large numbers, n ∑

1

n

√ 1/p

sn

k=1

n n1−1/p 1 ∑ (δk − δ¯ n )εk = √ · (δk − δ¯ n )εk → cE(εδ ) a.s., and hence, in probability. sn n k=1

Similarly, n ∑

1

n

√ 1/p

(δk − δ¯ n )2 → cE δ 2 in probability.

sn

k=1

Furthermore, n ∑

1

n

√ 1/p

(xk − x¯ n )(εk − βδk ) → 0 in probability,

sn

k=1

since

⏐ ⏐2 n ⏐ 1 ⏐ ∑ 1 ⏐ ⏐ E ⏐ 1/p √ (xk − x¯ n )(εk − βδk )⏐ = 2/p E(ε − βδ )2 → 0. ⏐n ⏐ sn n k=1

We also have that n 1 ∑ (ξk − ξ¯n )2 sn k=1

=

(

1

sn +

sn

n ∑

(δk − δ¯ n ) + 2 2

k=1

=1+

n sn

n

(δk − δ¯ n )2 +

k=1

n 2 ∑

sn

(xk − x¯ n )δk

k=1

1 + c 2 E δ 2 in probability

if p = 2,

1 in probability

if p > 2.

{ →

(xk − x¯ n )δk

k=1

n 1∑

·

)

n ∑

Hence, the result follows from (1.3). We next consider the case of c = ∞. By the Kolmogorov strong law of large numbers, n 1∑ (δk − δ¯ n )εk → E(εδ ) a.s., and hence, in probability n k=1

and n 1∑ (δk − δ¯ n )2 → E δ 2 a.s., and hence, in probability. n k=1

Furthermore, n 1∑ (xk − x¯ n )(εk − βδk ) → 0 in probability, n k=1

since

⏐ n ⏐2 ( √ )2 ⏐1 ∑ ⏐ sn 1 sn ⏐ ⏐ E⏐ (xk − x¯ n )(εk − βδk )⏐ = 2 E(ε − βδ )2 = 2/p E(ε − βδ )2 → 0. 1 − ⏐n ⏐ n n n 1/p k=1

We also have that n ∑

1

n

√ 1−1/p

=

sn

1 n1−1/p

(ξk − ξ¯n )2

k=1

( √

sn

sn +

n ∑

(δk − δ¯ n ) + 2

k=1

2

n ∑ k=1

) (xk − x¯ n )δk

P. Chen, L. Wen and S.H. Sung / Journal of Statistical Planning and Inference 205 (2020) 64–73

√ =

sn

n1−1/p

n1/p

+√

sn

·

n 1∑

n

(δk − δ¯ n )2 +

k=1

(xk − x¯ n )δk

√ 1−1/p

sn

k=1

√ √ if n1/p / sn → ∞.

if n1/p / sn → d ∈ [0, ∞),

dE δ 2 in probability

{ →

n ∑

2

n

69

∞ in probability

Hence, the result follows from (1.3). □ Remark 2.4. The conditions of Lemma 2.3 are weaker than those of Lemma 2.2. Lemma 2.3 requires only the finite second moments of ε and δ . The following theorem gives a necessary and sufficient condition for the convergence rate in the weak consistency of βˆ n . Theorem 2.2. Under the model (1.1), assume that {(ε, δ ), (εn , δn ), n√≥ 1} is a sequence of i.i.d. random vectors with E ε = E δ = 0, 0 < E ε 2 , E δ 2 < ∞. Furthermore, assume that lim infn→∞ sn /n1/p > 0 for some p ≥ 2 and E(εδ ) − β E δ 2 ̸ = 0. Then

√

sn

n1/p

√

(βˆ n − β ) → 0 in probability if and only if n1−1/p / sn → 0.

Proof. The proof is the same as that of Theorem 2.1 except that Lemma 2.3 is used instead of Lemma 2.2 and is omitted.

□

We next estimate the convergence rates in the weak and strong consistency of θˆn . To do this, the following lemma is needed. It is interesting in itself. Lemma 2.4. Under the model (1.1), assume that {(ε, δ ), (εn , δn ), n ≥ 1} is a sequence of i.i.d. random vectors with E ε = E δ = 0, 0 < E ε 2 , E δ 2 < ∞. Then min{1, E δ 2 } ≤ lim inf n→∞

n 1 ∑

s∗n

n 1 ∑ (ξk − ξ¯n )2 ≤ 1 + E δ 2 a.s. s∗n

(ξk − ξ¯n )2 ≤ lim sup n→∞

k=1

(2.9)

k=1

and

√ lim sup n→∞

s∗n n

|βˆ n − β| ≤

|E(εδ )| + |β|E δ 2 a.s., min{1, E δ 2 }

(2.10)

where s∗n = max{n, sn }. Proof. We first prove (2.9). It is clear that n n n s 2 ∑ 1 ∑ 1 ∑ ¯n )2 = n + ¯ ( ξ − ξ (x − x ) δ + (δk − δ¯ n )2 . k k n k s∗n s∗n s∗n s∗n k=1

k=1

√

(2.11)

k=1

Set ank = (xk − x¯ n )/ s∗n for 1 ≤ k ≤ n and n ≥ 1. Then sup n≥1

n ∑

a2nk = sup n≥1

k=1

sn s∗n

≤ 1.

Therefore by Lemma 2.1 with p = 2,

√ n n ∑ 1 ∑ n −1/2 ¯ (x − x ) δ = · n ank δk → 0 a.s. k n k s∗n s∗n k=1

(2.12)

k=1

By the Kolmogorov strong law of large numbers, n n 1∑ 1∑ 2 (δk − δ¯ n )2 = δk − δ¯n2 → E δ 2 a.s., n n k=1

k=1

which, together with the definition of s∗n , implies that

( min{1, E δ } ≤ lim inf 2

n→∞

sn s∗n

( ≤ lim sup n→∞

+

n 1 ∑

s∗n

) (δk − δ¯ n )

2

k=1

n 1 ∑ + (δk − δ¯ n )2 s∗n s∗n

sn

Hence, (2.9) holds by (2.11)–(2.13).

k=1

) ≤ 1 + E δ 2 a.s.

(2.13)

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P. Chen, L. Wen and S.H. Sung / Journal of Statistical Planning and Inference 205 (2020) 64–73

We now prove (2.10). By the Kolmogorov strong law of large numbers,

⏐ ⏐ ⏐ ⏐√ n n ⏐ 1 ∑ ⏐ ⏐ ⏐ n 1∑ ⏐ ⏐ ⏐ ¯n )εk ⏐⏐ ≤ |E(εδ )| a.s. lim sup ⏐ √ ∗ · (δk − δ¯ n )εk ⏐ = lim sup ⏐ ( δ − δ k ⏐ ⏐ nsn s∗n n n→∞ ⏐ n→∞ ⏐

(2.14)

√ n n n 1∑ 1 ∑ · (δk − δ¯ n )2 = lim sup (δk − δ¯ n )2 ≤ E δ 2 a.s. lim sup √ ∗ nsn s∗n n n→∞ n→∞

(2.15)

k=1

k=1

and

k=1

k=1

By Lemma 2.1 with p = 2, 1

n ∑

ns∗n

k=1

√

(xk − x¯ n )(εk − βδk ) → 0 a.s.

(2.16)

Therefore, (2.10) holds by (1.3), (2.9), and (2.14)–(2.16).

□

We need the following simple lemma. The proof is presented in Appendix. Lemma 2.5. Let {Xn , n ≥ 1} and {Yn , n ≥ 1} be sequences of random variables. Assume that 0 < a < lim infn→∞ |Xn | ≤ lim supn→∞ |Xn | < b < ∞ a.s. for some positive constants a and b. Then the following statements hold. (i) Xn Yn → 0 in probability if and only if Yn → 0 in probability. (ii) Xn Yn → 0 a.s. if and only if Yn → 0 a.s. The following lemma is needed to estimate the convergence rate in the strong consistency of θˆn . Lemma 2.6. Under the model (1.1), assume that {(ε, δ ), (εn , δn ), n ≥ 1} is a sequence of i.i.d. random vectors with E ε = E δ = 0, 0 < E |ε|p , E |δ|p < ∞ for some p ≥ 2. Furthermore, assume that supn≥1 sn x¯ 2n /s∗n < ∞ and E(εδ ) − β E δ 2 ̸ = 0. Then n−1/p

s∗n x¯ n (βˆ n − β ) → 0 a.s. if and only if n1−1/p x¯ n / s∗n → 0.

√

√

Proof. By the Kolmogorov strong law of large numbers,

√

s∗n

·

n1/p

n n x¯ n ∑ n1−1/p x¯ n 1 ∑ n1−1/p x¯ n ¯ ( δ − δ ) ε = · (δk − δ¯ n )εk = √ ∗ (E(εδ ) + o(1)) a.s. √ k n k ∗ ∗ sn sn n sn

(2.17)

n n x¯ n ∑ n1−1/p x¯ n 1 ∑ n1−1/p x¯ n 2 ¯ ( δ − δ ) = · (δk − δ¯ n )2 = √ ∗ (E δ 2 + o(1)) a.s. √ k n ∗ ∗ sn sn n sn

(2.18)

k=1

k=1

and

√

s∗n

·

n1/p

k=1

k=1

√

Set ank = x¯ n (xk − x¯ n )/ s∗n for 1 ≤ k ≤ n and n ≥ 1. By the assumption supn≥1 sn x¯ 2n /s∗n < ∞, sup

n ∑

n≥1

⏐ n ⏐ 2 ∑ ⏐ x¯ n (xk − x¯ n ) ⏐2 ⏐ √ ⏐ = sup sn x¯ n < ∞. ⏐ ⏐ ∗ ∗ s n≥1 n≥1 s

a2nk = sup

k=1

n

k=1

n

Hence we have by Lemma 2.1 that

√

s∗n

n1/p

·

n n x¯ n ∑ 1 ∑ ¯ (x − x )( ε − βδ ) = ank (εk − βδk ) = o(1) a.s. k n k k s∗n n1/p k=1

(2.19)

k=1

It follows by (1.3) and (2.17)–(2.19) that n−1/p

s∗n x¯ n (βˆ n − β ) =

√

n1−1/p x¯ n s∗n −1/2 (E(εδ ) − β E δ 2 + o(1)) + o(1) s∗n −1

∑n

ξ − ξ¯n )2

k=1 ( k

a.s.

By Lemmas 2.4 and 2.5, n−1/p

s∗n x¯ n (βˆ n − β ) → 0 a.s. if and only if n1−1/p x¯ n s∗n

√

−1/2

(E(εδ ) − β E δ 2 + o(1)) → 0.

(2.20)

Since E(εδ ) −β E δ 2 ̸ = 0, the right-hand side of (2.20) is equivalent to n1−1/p x¯ n s∗n −1/2 → 0. Hence the proof is complete.

□

The following theorem provides a necessary and sufficient condition for the convergence rate in the strong consistency of θˆn .

P. Chen, L. Wen and S.H. Sung / Journal of Statistical Planning and Inference 205 (2020) 64–73

71

Theorem 2.3. Under the model (1.1), assume that {(ε, δ ), (εn , δn ), n ≥ 1} is a sequence of i.i.d. random vectors with E ε = E δ = 0, 0 < E |ε|p , E |δ|p < ∞ for some p ≥ 2. Furthermore, assume that sn (log log n)/n1+2/p → 0, supn≥1 sn x¯ 2n /s∗n < ∞, and E(εδ ) − β E δ 2 ̸ = 0. Then

√

s∗n

n1/p

(θˆn − θ ) → 0 a.s. if and only if n1−1/p x¯ n / s∗n → 0.

√

Proof. By the classical Hartman–Wintner law of iterated logarithm, the assumption sn (log log n)/n2/p+1 → 0, and the definition of s∗n , we obtain that

√

s∗n

n1/p

√ |¯εn | =

s∗n log log n n2/p+1

√ ·

n log log n

|¯εn | → 0 a.s.

(2.21)

|δ¯n | → 0 a.s.

(2.22)

and

√

√

s∗n

n

|δ¯ | = 1/p n

s∗n log log n n2/p+1

√ ·

n log log n

We also have by Lemma 2.4 and the Hartman–Wintner law of iterated logarithm (or Lemma 2.1) that

√

√

s∗n

n

|(βˆ n − β )δ¯n | = 1/p

s∗n n

|βˆ n − β| · n1/2−1/p |δ¯n | → 0 a.s.

(2.23)

It follows by (1.4) and (2.21)–(2.23) that

√

√

s∗n

n

∗

sn 1 n /p

(θˆn − θ ) → 0 a.s. if and only if 1/p

x¯ n (βˆ n − β ) → 0 a.s.

(2.24)

√ By Lemma 2.6, the right-hand side of (2.24) is equivalent to n1−1/p x¯ n / s∗n → 0. Hence the proof is complete. □ The following lemma is needed to estimate the convergence rate in the weak consistency of θˆn . Lemma 2.7. Under the model (1.1), assume that {(ε, δ ), (εn , δn ), n ≥ 1} is a sequence of i.i.d. random vectors with E ε = E δ = 0, 0 < E ε 2 , E δ 2 < ∞. Furthermore, assume that sn x¯ 2n /(s∗n n2/p ) → 0 for some p ≥ 2 and E(εδ ) − β E δ 2 ̸ = 0. Then n−1/p

√

s∗n x¯ n (βˆ n − β ) → 0 in probability if and only if n1−1/p x¯ n / s∗n → 0.

√

Proof. For any ∆ > 0, we have by the Markov inequality and the assumption sn x¯ 2n /(s∗n n2/p ) → 0 that

⏐ {⏐ √ } n ⏐ s∗ x¯ ∑ ⏐ ⏐ n n ⏐ P ⏐ 1/p ∗ (xk − x¯ n )(εk − βδk )⏐ > ∆ ⏐ n sn ⏐ k=1 ⏐√ ⏐2 n ⏐ ∗¯ ∑ ⏐ ⏐ n −2 ⏐ sn x ≤ ∆ E ⏐ 1/p ∗ (xk − x¯ n )(εk − βδk )⏐ ⏐ n sn ⏐ k=1

sn x¯ 2n

= ∆−2

s∗n n2/p

E(ε − βδ )2 → 0.

It follows by (1.3) and Lemmas 2.4 and 2.5 that n−1/p

√

s∗n x¯ n (βˆ n − β ) → 0 in probability

is equivalent to

√

s∗n x¯ n

( n ∑

n1/p s∗n

k=1

=

x¯ n n1−1/p

√

s∗n

(δk − δ¯ n )εk − β

)

n ∑

(δk − δ¯ n )

2

k=1

·

1 n

(

n ∑

n ∑

k=1

k=1

(δk − δ¯ n )εk − β

)

(δk − δ¯ n )

2

→ 0 in probability.

By the Kolmogorov strong law of large numbers, 1 n

(

n ∑

n ∑

k=1

k=1

(δk − δ¯ n )εk − β

) (δk − δ¯ n )2

→ E(εδ ) − β E δ 2 a.s. √

Since E(εδ ) − β E δ 2 ̸ = 0, (2.25) is equivalent to x¯ n n1−1/p / s∗n → 0. Hence the proof is complete. □

(2.25)

72

P. Chen, L. Wen and S.H. Sung / Journal of Statistical Planning and Inference 205 (2020) 64–73

The following theorem provides a necessary and sufficient condition for the convergence rate in the weak consistency of θˆn . Theorem 2.4. Under the model (1.1), assume that {(ε, δ ), (εn , δn ), n ≥ 1} is a sequence of i.i.d. random vectors with E ε = E δ = 0, 0 < E ε 2 , E δ 2 < ∞. Furthermore, assume that for some p ≥ 2, sn /n1+2/p → 0 and sn x¯ 2n /(s∗n n2/p ) → 0, and E(εδ ) − β E δ 2 ̸ = 0. Then

√

∗

sn n1/p

(θˆn − θ ) → 0 in probability if and only if n1−1/p x¯ n / s∗n → 0.

√

Proof. By the assumption sn /n1+2/p → 0, we obtain that

√

∗

sn n1/p

and

√

∗

sn 1 n /p

ε¯ n → 0 in probability

(2.26)

δ¯n → 0 in probability.

(2.27)

We also obtain by Lemma 2.4 and the Hartman–Wintner law of iterated logarithm that

√ n

√

s∗n

|(βˆ n − β )δ¯n | = 1/p

s∗n n

|βˆ n − β| · n1/2−1/p |δ¯n | → 0 a.s.

(2.28)

It follows by (1.4) and (2.26)–(2.28) that

√

s∗n

n1/p

(θˆn − θ ) → 0 in probability if and only if n−1/p

s∗n x¯ n (βˆ n − β ) → 0 in probability.

√

(2.29)

√

By Lemma 2.7, the right-hand side of (2.29) is equivalent to n1−1/p x¯ n / s∗n → 0. Hence the proof is complete. □ Remark 2.5. Conditions of Theorem 2.4 are weaker than those of Theorem 2.3. This is desirable since convergence in probability is weaker than a.s. convergence. Acknowledgments The authors would like to thank the Editor and two anonymous reviewers for their careful reading of the manuscript and valuable suggestions that helped to improve an earlier version of this paper. The research of Pingyan Chen is supported by the National Natural Science Foundation of China (No. 71471075). The research of Soo Hak Sung is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B03029898). Appendix In this appendix, we provide the proof of Lemma 2.5. We first observe the properties of Xn . Since lim infn→∞ |Xn | > a > 0 a.s., we have that P(|Xn | ≤ a i.o.) = 0 and hence ∞ ∞ 0 = P ∩∞ n=1 ∪k=n (|Xk | ≤ a) = lim P ∪k=n (|Xk | ≤ a) ≥ lim sup P (|Xn | ≤ a) .

(

)

(

)

n→∞

n→∞

Thus, limn→∞ P (|Xn | ≤ a) = 0. Since lim supn→∞ |Xn | < b a.s., we also have by the above method that limn→∞ P (|Xn | ≥ b) = 0. We now prove (i). If Xn Yn → 0 in probability, then P(|Yn | > ε ) = P(|Yn | > ε, |Xn | ≤ a) + P(|Yn | > ε, |Xn | > a)

≤ P(|Xn | ≤ a) + P(|Xn Yn | > εa) → 0. Conversely, if Yn → 0 in probability, then P(|Xn Yn | > ε ) = P(|Xn Yn | > ε, |Yn | ≤ ε/b) + P(|Xn Yn | > ε, |Yn | > ε/b)

≤ P(|Xn | > b) + P(|Yn | > ε/b) → 0. We next prove (ii). If Xn Yn → 0 a.s., then lim sup |Yn | = lim sup n→∞

n→∞

|Xn Yn | |Xn |

≤ lim sup |Xn Yn | lim sup n→∞

n→∞

1

|Xn |

P. Chen, L. Wen and S.H. Sung / Journal of Statistical Planning and Inference 205 (2020) 64–73

= lim sup |Xn Yn | n→∞

73

1 lim infn→∞ |Xn |

≤ a−1 lim sup |Xn Yn | = 0 a.s. n→∞

Conversely, if Yn → 0 a.s., then lim supn→∞ |Xn Yn | ≤ lim supn→∞ |Xn | lim supn→∞ |Yn | ≤ b lim supn→∞ |Yn | = 0 a.s. Hence the proof is complete. References Deaton, A., 1985. Panel data from a time series of cross-sections. J. Econometrics 30, 109–126. Fuller, W., 1987. Measurement Error Models. Wiley, New York. Hu, D., Chen, P., Sung, S.H., 2017. Strong laws for weighted sums of ψ -mixing random variables and applications in errors-in-variables regression models. TEST 26, 600–617. Liu, J., Chen, X., 2005. Consistency of LS estimator in simple Linear EV regression models. Acta Math. Sci. Ser. B Engl. Ed. 25, 50–58. Martsynyuk, Y.V., 2013. On consistency of the least squares estimators in linear errors-in-variables models with infinite variance errors. Electron. J. Stat. 7, 2851–2874. Miao, Y., Wang, K., Zhao, F., 2011. Some limit behaviors for the LS estimator in simple linear EV regression models. Statist. Probab. Lett. 81, 92–102. Shen, A., 2017. Asymptotic properties of LS estimators in the errors-in-variables model with MD errors. Statist. Papers http://dx.doi.org/10.1007/ s00362-016-0869-1. Thrum, R., 1987. A remark on almost sure convergence of weighted sums. Probab. Theory Related Fields 75, 425–430. Wang, X., Hu, S., 2017. On consistency of least square estimators in the simple linear EV model with negatively orthant dependent errors. Electron. J. Stat 11, 1434–1463. Wang, X., Shen, A., Chen, Z., Hu, S., 2015. Complete convergence for weighted sums of NSD random variables and its application in the EV regression model. TEST 24, 166–184. Wang, X., Wu, Y., Hu, S., 2018. Strong and weak consistency of LS estimators in the EV regression model with negatively superadditive-dependent errors. AStA Adv. Stat. Anal. 102, 41–65. Wu, Y., Wang, X., Hu, S., Yang, L., 2017. Weighted version of strong law of large numbers for a class of random variables and its applications. TEST 27, 379–406.