Construction of Empirical Bayes Estimators for Truncation Parameters of Two-sided Truncated Distributions Using Size Two Samples and Their Asymptotic Optimality

Sung-Hyon Ri; Hyon-A Ryu; Su-Ryon Jang

doi:doi:10.11648/j.rd.20260701.15

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Construction of Empirical Bayes Estimators for Truncation Parameters of Two-sided Truncated Distributions Using Size Two Samples and Their Asymptotic Optimality

Sung-Hyon Ri^*, Hyon-A Ryu

, Su-Ryon Jang

Published in Research & Development (Volume 7, Issue 1)

Received: 13 November 2025 Accepted: 1 December 2025 Published: 26 March 2026

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Abstract

Bayesian methods are effective statistical one that combine prior information with samples, but the inference results are different because subjectivity is involved in representing prior information as a prior distribution. The empirical Bayes approach is a statistical method that performs continuous Bayesian decision making when the prior distribution is unknown but given a large number of past data, which overcomes the above drawback and is now a common way because of a vast amount of data available. The inference for truncation parameters is important in evaluating the lower or upper bounds of the population and is required in many fields such as reliability, meteorology, medicine, etc. Many authors have considered the case of one-sided truncated distribution in the empirical Bayes framework, but in practice we are faced with the case of two-sided one. We consider the empirical Bayes estimation for truncation parameters of two-sided truncated distribution under the squared error loss. First, Bayesian estimators of the truncation parameters are derived to minimize the Bayes risk using sample of size two. And, based on the Bayesian estimators and kernel density estimate of the population density function, empirical Bayes estimators of the truncation parameters are constructed. Next, under some assumptions asymptotic optimality of empirical Bayes estimators is proved when the sample size goes to infinity, and the convergence rate is also evaluated. It is also proved that the probability that the lower and upper bounds are reversely estimated approaches zero. Finally, an example is presented to show the validity of the assumptions and the performance of the proposed empirical Bayes estimators is evaluated through simulation on the example. New proposal of using size two samples could be extended to the case of multi-dimensional truncated distributions defined on hyper-cubic domain.

Published in	Research & Development (Volume 7, Issue 1)
DOI	10.11648/j.rd.20260701.15
Page(s)	54-62
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Asymptotic Optimality, Empirical Bayes Estimator, Squared Error Loss, Truncation Parameter, Size Two Samples

1. Introduction

The empirical Bayes approach that have its origin in Robbins (

[13, 14]

) is applicable to statistical situations when one deals with an independent sequence of Bayes decision problems each having similar structure. In these problems the statistical similarity are included in the assumption that prior distribution is unknown (see

[7]

Many authors have concentrated on developing empirical Bayes frameworks and their asymptotic properties (see

[8]

) and systematic description and bibliographical details can be seen in

[3]

. An empirical Bayes predictive density has been proposed to estimate the density of a future outcome from a multivariate normal model and its asymptotic optimality obtained in infinite dimensional parameter spaces by Xu and Zhou

[17]

. A computational framework has been proposed for empirical Bayes estimation of parameter and the convergence of the algorithm discussed by Atchade

[1]

and Varian

[16]

. An empirical Bayes procedure for adaptively selecting an experiment wise significance threshold has been proposed by Chaoyu and Hoff

[4]

in order to control the sign error rate. Murray et al.

[12]

have described a collection of web-based statistical tools that enable investigators to incorporate historical control data into conduct of randomized clinical trials using empirical Bayes method. Liang

[11]

has used the squared and linear exponential (LINEX) error losses for empirical Bayes estimation for the reproduction parameter of Borel-Tanner distributions. The empirical Bayes estimation of guarantee lifetime in a two parameter exponential distribution has been studied by Huang and Huang

[6]

In this paper we consider a probability density function (pdf) with truncation parameter  = (₁, ₂) as follows.

(₁,₂)∈Θ=[a₁,b₁]

[a₂,b₂],(1)

(-∞<a₁<b₁≤a₂<b₂< ∞),

where u(x)>0, x∈[a₁, b₂] is continuous function,

and I(A) is indicate function of A. We assume that truncation parameter

has a prior distribution function G() on Θ with G(a₁, a₂) = 0 and G(b₁, b₂) = 1 and its probability density function (pdf) g().

The prior distribution function G() (or g()) is usually unknown in practice. In such situation, the Bayes estimator is not available. The empirical Bayes method is used in the case when the prior distribution is unknown but the past data are given.

The empirical Bayes procedure for the truncation parameter distributions has been considered by several authors. The empirical Bayes estimations of one-side truncation parameters and its asymptotic optimality have been studied under LINEX error loss and squared error loss respectively by Huang and Liang

[5]

and Yimin and Balakrishnan

[18]

. Li and Gupta (

[9, 10]

) have considered empirical Bayes tests for truncation parameters. Yimin

[19]

has studied the empirical Bayes estimation for the truncation parameters of two-side truncated distribution under LINEX error loss and it is one parameter estimation, because lower and upper bounds are given as θ, mθ (for a known constant m), respectively. Such assumption restrict to the range of application. He used size one samples as most papers on empirical Bayes estimation. In order to construct empirical Bayes estimators for independent lower and upper bounds, we use size two samples.

Let Y₁, Y₂ be sample of size two from the distribution with the pdf given by Eq. (1) and X₁= Y₁∧Y₂, X₂= Y₁∨Y₂. Then the pdf of (X₁, X₂) is

(2)

Since the pdf of sample and parameter (X₁, X₂, ₁, ₂) are

, the marginal pdf of (X₁, X₂) is written as

,(3)

where

.(4)

As a loss function, we use the squared error loss as

where δ = (δ₁, δ₂) is estimator of parameter  and w_j > 0, j=1, 2 are weight constants. The risk of δ with respect to G(₁, ₂) can be written as

where

is the conditional pdf of (₁, ₂) given (X₁, X₂) = (x₁, x₂). Let δ_B = (δ₁_B, δ₂_B) be the Bayesian estimator of , i.e.

, then we have

(5)

and

.(6)

If only ₁ (or ₂) is given in Eq. (1), then

(

)

These parent pdfs differ from the start models of Yimin and Balakrishnan

[18]

In this paper we propose the empirical Bayes estimators for the truncation parameters of two-side truncated distribution using size two samples and investigate the asymptotic properties of the estimators. We use the squared error loss function. To our best knowledge, earlier papers discussed the empirical Bayes estimation for one-sided truncated distributions. However, in practice we often encounter with the two-sided truncated distribution.

The organization of this paper is as follows. In section 2, we demonstrate the relation between the Bayes estimator and the marginal distribution of sample and on the base of it, construct the empirical Bayes estimators. In section 3, we study the asymptotic optimality of the proposed empirical Bayes estimators and show an example. Finally, we present simulation results on the performance of the empirical Bayes estimator in section 4.

2. Construction of Empirical Bayes Estimators

First we give the relation between the Bayes estimator δ_B and the marginal pdf f(x₁, x₂) of sample (X₁, X₂).

Lemma 2.1. The Bayes estimator of Eq. (5) is represented by

,(7)

where

(8)

(9)

Proof. From (4)

(10)

Therefore using Eq. (2), Eq. (3) and the integration method by part, δ₁_B(x₁, x₂) of (5) is derived as

Similarly, we have i=2 case of Eq. (7).

Next, we construct the empirical Bayes estimators of the truncation parameter .

In the empirical Bayes method, it is assumed that there are past data (X⁽ⁱ⁾, ⁽ⁱ⁾), i=1, …, n and present data (X, ), where ⁽ⁱ⁾, i=1, …, n and  are not observable with unknown prior distribution G(), and X⁽ⁱ⁾, i=1, …, n and X are observable. Here and after, we assume that (X⁽ⁱ⁾, ⁽ⁱ⁾), i=1, …, n and (X, ) are independent and identically distributed (i.i.d.) random variables.

We use a kernel density estimation method to estimate f(x₁, x₂), the marginal pdf of X = (X₁, X₂). Let K(x₁, x₂) is a kernel function satisfying following two assumptions.

Assumption K. (i) K(x₁, x₂) is symmetric about (0, 0) and continuously differentiable.

(ii)

With the past data

, i=1, ･･･ n, we use a kernel estimator of f(x₁, x₂) defined as

where h₁_n→0 as n→∞. Furthermore, let estimators for the functions Δ_j, ψ_j, τ_j, j =1, 2 be

(11)

(12)

where ν>0 is a constant defined later and h₂_n→0 as n→∞. The estimators Δ_jn(x₁, x₂) (j =1, 2) of the functions Δ_j(x₁, x₂) is asymptotically unbiased as n→∞. Then estimators for the Bayes estimators δ_jB(x₁, x₂), j =1, 2 of Eq. (7) are defined as

,(13)

where

and h₃_n→0 as n→∞. With the present data X = (X₁, X₂), we compute the empirical Bayes estimator of  = (₁, ₂) as

δ_n=(δ₁_n(X₁,X₂),δ₂_n(X₁,X₂))(14)

3. Asymptotic Optimality

In this section, we investigate asymptotic properties of the empirical estimator δ_n of Eq. (14). Since δ_B is the Bayes estimator of the truncation parameter , the inequality R(δ_n, G) ≥ R(δ_B, G) holds. By similar manner to Huang and Liang

[5]

, we have

,(15)

where E_n denotes the conditional expectation with respect to X⁽ⁱ⁾, i=1, …, n. As usual in empirical Bayes framework, we use the nonnegative difference J_n= R(δ_n, G)-R(δ_B, G) as criteria to evaluate performance of the empirical Bayes estimator δ_n.

Definition 3.1.

[18]

. A sequence {δ_n} of empirical Bayes estimators is said to be asymptotically optimal as least of the order q_n with respect to the prior G, if J_n ≤ O(q_n), as n→∞, where q_n > 0 and q_n→∞ as n→∞.

Suppose that f(x₁, x₂) is twice differentiable and

, l=0, 1, 2, where f⁽^l⁾ denotes any partial derivative of order l. Let

, then from theorem 3 in Bosq

[2]

Using Hölder inequality, for any γ∈(0, 2].

.(16)

We give some lemmas used in proof of the asymptotic optimality of the sequence {δ_n} of the empirical Bayes estimator δ_n of Eq. (14).

Lemma 3.1. If

and ν=2, we have

.(17)

Proof. We prove j=1 case. j =2 case is proved similarly. From Eq. (9) and Eq. (12).

Let denote by Var_n the variance with respect to {X⁽ⁱ⁾, i=1, …, n }, then

Therefore

Lemma 3.2. Under the condition of lemma 3.1, if γ∈ (0, 2], then

.(18)

Proof. From Eq. (8), Eq. (12) and Eq. (17), we have

Therefore by Hölder inequality,

The following lemma is from

[15]

Lemma 3.3. Let y, z≠0 and h>0 be real numbers and Y, Z be two real valued random variables. Then for any γ∈(0, 2],

.(19)

Next, we give the result on the asymptotic optimality of the sequence {δ_n} of the empirical Bayes estimators δ_n.

Theorem 3.1. Suppose that f(x₁, x₂) is twice differentiable and

, l=0, 1, 2 and the assumption K holds. If u(x)>0, a₁≤x≤b₂ is continuous and for any γ∈(0, 2],

,(20)

then with the choice of

and ν=2, we have

Proof. We consider right hand of Eq. (15). From Eq. (6), we can see that

, for sufficiently great n. Using Eq. (16), Eq. (18) and Eq. (19), for j=1, 2.

(21)

Therefore

The estimator of a lower truncation parameter may be larger than the estimator of the upper one, because of randomness of samples. Therefore we evaluate the probability p_on=P{δ₁_n>δ₂_n} of reversely estimating.

Corollary 3.1. Under the assumptions of theorem 3.1, we have

Proof. Using Eq. (6), Eq. (21) and Chebyshev inequality,

Example 3.1. We consider a two-sided truncated distribution with the pdf as

,(22)

where (₁, ₂)∈Θ=[a₁, b₁]

[a₂, b₂] and 0≤ a₁< b₁≤ a₂< b₂< ∞. Then

, and 0<C₁≤

≤ C₂. Let G(₁, ₂) is a prior distribution with the pdf given by

(23)

where λ_k> 1, k=1,…, 4 are constants and g₀>0 is regular constant.

Let divide the sample space

and denote by

the interior of D_i. We can see that for the function V(x₁, x₂) of Eq. (4), it holds

where C>0 is a constant related with g₀, a_i, b_i, i=1, 2 and λ_k, k=1,…, 4. From Eq. (3), we have

If (λ_k+ 1)(1- γ)>-1 for k=1, 4, then Eq. (20) is satisfied. On the other hand, from Eq. (4) and Eq. (10), if λ_k>1, k=1,…, 4, then

, l=0, 1, 2, furthermore

, l=0, 1, 2. Therefore we can see that if

(24)

then the result in theorem 3.1 holds.

4. Simulation Study

Finally, we describe the Monte-Carlo Simulation on the performance of the empirical Bayes estimators.

In Eq. (22) and Eq. (23), we set (a₁, b₁)=(0, 1/5), (a₂, b₂)=(4/5, 1) and λ_k= 2, k=1,…, 4. In this setting, Eq. (24) is satisfied for γ∈(0, 4/3). We take d₁=5, d₂=1/5, d₃=0.35 and w₁= w₂=1.

The simulation study is carried as follows. First, we iterate following steps (i) -(ii) 500 times. In the kth (k=1,…, 500) iteration:

(i) For i = 1,…, n+1, we generate

from the pdf of Eq. (23) and generate two samples

from the pdf of Eq. (22). Then we set

, i = 1,…, n (past data) and

(present data).

(ii) The Bayes estimators δ_jB(x₁, x₂), j=1, 2, the empirical Bayes estimators δ_jn(x₁, x₂), j=1, 2 and

are computed.

Next, we estimate J_n and p_on by sample means respectively as

The variances of above sample means are estimated respectively by

The values of

and

for n ∈ M = {10, 50, 100, 200, 300, 500, 700, 1,000, 1,500, 2,000} are presented in Table 1. Finally, solving the minimization problem.

gives β = 0.08452 and C = 0.07360. The values of

are filled in 4th column of Table 1 and all they are close to 1. Numerical computations are carried by Matlab.

Table 1. The performances of Estimators Via Estimates of

and

According to Different n.

n
10	12.0427	23.029	1.7591	5.8	0.7392
20	7.5427	5.3255	1.1907	1.5	0.3844
30	6.7558	4.9921	1.116	1.4	0.3715
40	6.5257	4.9932	1.1133	1.6	0.3968
50	5.5564	4.44	0.972	0.6	0.2442
70	5.1461	4.692	0.9348	0.2	0.1413
100	4.9181	4.5479	0.9297	0.8	0.2817
150	4.3617	3.9638	0.8629	0.2	0.1413
200	4.3447	4.2461	0.8876	0.3	0.1729
300	4.4198	4.1612	0.9449	0.1	0.0999
600	4.365	3.8058	1.0086	0	0
1000	4.3067	3.8096	1.0537	0	0

Table 1 shows that

and variances of it tend to zero, as n→∞, respectively. We can also see that

and variances of it tend to zero, as n→∞, respectively. Therefore, J_n, the difference between empirical risk and theoretical risk, and p_on, the probability of reversely estimating, tend to zero, as n→∞, respectively. However, the speeds of the convergences are relatively slow as usual in empirical Bayes inferences.

5. Conclusion

The empirical Bayes estimation for truncation parameters of two-sided truncated distribution has been considered. We theoretically confirmed that by a sample of size one, the Bayesian estimators for two truncation parameters cannot be constructed. This is the motivation for using size two samples. Empirical Bayes estimators for the truncation parameters under squared error loss have been constructed and their asymptotic optimality have been proved. It has been also proved that the probability that the lower and upper bounds are reversely estimated approaches zero. An example was presented to verify applicability of the proposed estimation scheme to real problems. The simulation result confirms the asymptotic optimality of the proposed empirical Bayes estimators. New proposal of using size two samples could be extended to the case of multi-dimensional truncated distributions defined on hyper-cubic domain. The empirical Bayes inference using samples of size larger than one would be effective in many statistical problems.

Abbreviations

i.i.d.	Independent and Identically Distributed
LINEX	Linear Exponential

Acknowledgments

The authors would like to thank Alina Carlson, Rachel Nolan and the reviewers for their great efforts and valuable comments. Sung-Hyon Ri would like to express deeply gratitude to Prof. Erich Novak for providing selfless help for the publication of his first paper.

Conflicts of Interest

The authors declare no conflicts of interest.

References

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[10]	Li, J., Gupta, S. S., (2003). Optimal rate of empirical Bayes tests for lower truncation parameters. Statist. Probab. Lett., 65, 177-185.
[11]	Liang, T. C., (2009). Empirical Bayes estimation for Borel-Tanner distributions. Statist. Probab. Lett., 79, 2212-2219.
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[13]	Robbins, H., (1956). An empirical Bayes approach to statistics. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, vol. 1. University of California Press, Berkeley, CA, USA, pp. 157-163.
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[18]	Yimin, M., Balakrishnan, N., (2000). Empirical Bayes estimation for truncation parameters. J. Statist. Plann. Inference, 84, 111-120.
[19]	Yimin, S., (2000). Empirical Bayes estimate for the perameter of two-side truncated distribution families. Appl. Math. J. Chinese Univ. Ser A, 15(4), 475-483.

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APA Style

Ri, S., Ryu, H., Jang, S. (2026). Construction of Empirical Bayes Estimators for Truncation Parameters of Two-sided Truncated Distributions Using Size Two Samples and Their Asymptotic Optimality. Research & Development, 7(1), 54-62. https://doi.org/10.11648/j.rd.20260701.15

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ACS Style

Ri, S.; Ryu, H.; Jang, S. Construction of Empirical Bayes Estimators for Truncation Parameters of Two-sided Truncated Distributions Using Size Two Samples and Their Asymptotic Optimality. Res. Dev. 2026, 7(1), 54-62. doi: 10.11648/j.rd.20260701.15

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AMA Style

Ri S, Ryu H, Jang S. Construction of Empirical Bayes Estimators for Truncation Parameters of Two-sided Truncated Distributions Using Size Two Samples and Their Asymptotic Optimality. Res Dev. 2026;7(1):54-62. doi: 10.11648/j.rd.20260701.15

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@article{10.11648/j.rd.20260701.15,
  author = {Sung-Hyon Ri and Hyon-A Ryu and Su-Ryon Jang},
  title = {Construction of Empirical Bayes Estimators for Truncation Parameters of Two-sided Truncated Distributions Using Size Two Samples and Their Asymptotic Optimality},
  journal = {Research & Development},
  volume = {7},
  number = {1},
  pages = {54-62},
  doi = {10.11648/j.rd.20260701.15},
  url = {https://doi.org/10.11648/j.rd.20260701.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.rd.20260701.15},
  abstract = {Bayesian methods are effective statistical one that combine prior information with samples, but the inference results are different because subjectivity is involved in representing prior information as a prior distribution. The empirical Bayes approach is a statistical method that performs continuous Bayesian decision making when the prior distribution is unknown but given a large number of past data, which overcomes the above drawback and is now a common way because of a vast amount of data available. The inference for truncation parameters is important in evaluating the lower or upper bounds of the population and is required in many fields such as reliability, meteorology, medicine, etc. Many authors have considered the case of one-sided truncated distribution in the empirical Bayes framework, but in practice we are faced with the case of two-sided one. We consider the empirical Bayes estimation for truncation parameters of two-sided truncated distribution under the squared error loss. First, Bayesian estimators of the truncation parameters are derived to minimize the Bayes risk using sample of size two. And, based on the Bayesian estimators and kernel density estimate of the population density function, empirical Bayes estimators of the truncation parameters are constructed. Next, under some assumptions asymptotic optimality of empirical Bayes estimators is proved when the sample size goes to infinity, and the convergence rate is also evaluated. It is also proved that the probability that the lower and upper bounds are reversely estimated approaches zero. Finally, an example is presented to show the validity of the assumptions and the performance of the proposed empirical Bayes estimators is evaluated through simulation on the example. New proposal of using size two samples could be extended to the case of multi-dimensional truncated distributions defined on hyper-cubic domain.},
 year = {2026}
}

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TY  - JOUR
T1  - Construction of Empirical Bayes Estimators for Truncation Parameters of Two-sided Truncated Distributions Using Size Two Samples and Their Asymptotic Optimality
AU  - Sung-Hyon Ri
AU  - Hyon-A Ryu
AU  - Su-Ryon Jang
Y1  - 2026/03/26
PY  - 2026
N1  - https://doi.org/10.11648/j.rd.20260701.15
DO  - 10.11648/j.rd.20260701.15
T2  - Research & Development
JF  - Research & Development
JO  - Research & Development
SP  - 54
EP  - 62
PB  - Science Publishing Group
SN  - 2994-7057
UR  - https://doi.org/10.11648/j.rd.20260701.15
AB  - Bayesian methods are effective statistical one that combine prior information with samples, but the inference results are different because subjectivity is involved in representing prior information as a prior distribution. The empirical Bayes approach is a statistical method that performs continuous Bayesian decision making when the prior distribution is unknown but given a large number of past data, which overcomes the above drawback and is now a common way because of a vast amount of data available. The inference for truncation parameters is important in evaluating the lower or upper bounds of the population and is required in many fields such as reliability, meteorology, medicine, etc. Many authors have considered the case of one-sided truncated distribution in the empirical Bayes framework, but in practice we are faced with the case of two-sided one. We consider the empirical Bayes estimation for truncation parameters of two-sided truncated distribution under the squared error loss. First, Bayesian estimators of the truncation parameters are derived to minimize the Bayes risk using sample of size two. And, based on the Bayesian estimators and kernel density estimate of the population density function, empirical Bayes estimators of the truncation parameters are constructed. Next, under some assumptions asymptotic optimality of empirical Bayes estimators is proved when the sample size goes to infinity, and the convergence rate is also evaluated. It is also proved that the probability that the lower and upper bounds are reversely estimated approaches zero. Finally, an example is presented to show the validity of the assumptions and the performance of the proposed empirical Bayes estimators is evaluated through simulation on the example. New proposal of using size two samples could be extended to the case of multi-dimensional truncated distributions defined on hyper-cubic domain.
VL  - 7
IS  - 1
ER  -

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Author Information

Sung-Hyon Ri

Faculty of Applied Mathematics, Kim Chaek University of Technology, Pyongyang, Democratic People’s Republic of Korea

Contact Email
Hyon-A Ryu

Faculty of Applied Mathematics, Kim Chaek University of Technology, Pyongyang, Democratic People’s Republic of Korea

Contact Email

http://orcid.org/0009-0002-4250-033X
Su-Ryon Jang

Faculty of Applied Mathematics, Kim Chaek University of Technology, Pyongyang, Democratic People’s Republic of Korea

Contact Email

http://orcid.org/0009-0006-9425-5750

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Table 1

Table 1. The performances of Estimators Via Estimates of and According to Different n.

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Ri, S., Ryu, H., Jang, S. (2026). Construction of Empirical Bayes Estimators for Truncation Parameters of Two-sided Truncated Distributions Using Size Two Samples and Their Asymptotic Optimality. Research & Development, 7(1), 54-62. https://doi.org/10.11648/j.rd.20260701.15

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Ri, S.; Ryu, H.; Jang, S. Construction of Empirical Bayes Estimators for Truncation Parameters of Two-sided Truncated Distributions Using Size Two Samples and Their Asymptotic Optimality. Res. Dev. 2026, 7(1), 54-62. doi: 10.11648/j.rd.20260701.15

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Ri S, Ryu H, Jang S. Construction of Empirical Bayes Estimators for Truncation Parameters of Two-sided Truncated Distributions Using Size Two Samples and Their Asymptotic Optimality. Res Dev. 2026;7(1):54-62. doi: 10.11648/j.rd.20260701.15

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@article{10.11648/j.rd.20260701.15,
  author = {Sung-Hyon Ri and Hyon-A Ryu and Su-Ryon Jang},
  title = {Construction of Empirical Bayes Estimators for Truncation Parameters of Two-sided Truncated Distributions Using Size Two Samples and Their Asymptotic Optimality},
  journal = {Research & Development},
  volume = {7},
  number = {1},
  pages = {54-62},
  doi = {10.11648/j.rd.20260701.15},
  url = {https://doi.org/10.11648/j.rd.20260701.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.rd.20260701.15},
  abstract = {Bayesian methods are effective statistical one that combine prior information with samples, but the inference results are different because subjectivity is involved in representing prior information as a prior distribution. The empirical Bayes approach is a statistical method that performs continuous Bayesian decision making when the prior distribution is unknown but given a large number of past data, which overcomes the above drawback and is now a common way because of a vast amount of data available. The inference for truncation parameters is important in evaluating the lower or upper bounds of the population and is required in many fields such as reliability, meteorology, medicine, etc. Many authors have considered the case of one-sided truncated distribution in the empirical Bayes framework, but in practice we are faced with the case of two-sided one. We consider the empirical Bayes estimation for truncation parameters of two-sided truncated distribution under the squared error loss. First, Bayesian estimators of the truncation parameters are derived to minimize the Bayes risk using sample of size two. And, based on the Bayesian estimators and kernel density estimate of the population density function, empirical Bayes estimators of the truncation parameters are constructed. Next, under some assumptions asymptotic optimality of empirical Bayes estimators is proved when the sample size goes to infinity, and the convergence rate is also evaluated. It is also proved that the probability that the lower and upper bounds are reversely estimated approaches zero. Finally, an example is presented to show the validity of the assumptions and the performance of the proposed empirical Bayes estimators is evaluated through simulation on the example. New proposal of using size two samples could be extended to the case of multi-dimensional truncated distributions defined on hyper-cubic domain.},
 year = {2026}
}

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TY  - JOUR
T1  - Construction of Empirical Bayes Estimators for Truncation Parameters of Two-sided Truncated Distributions Using Size Two Samples and Their Asymptotic Optimality
AU  - Sung-Hyon Ri
AU  - Hyon-A Ryu
AU  - Su-Ryon Jang
Y1  - 2026/03/26
PY  - 2026
N1  - https://doi.org/10.11648/j.rd.20260701.15
DO  - 10.11648/j.rd.20260701.15
T2  - Research & Development
JF  - Research & Development
JO  - Research & Development
SP  - 54
EP  - 62
PB  - Science Publishing Group
SN  - 2994-7057
UR  - https://doi.org/10.11648/j.rd.20260701.15
AB  - Bayesian methods are effective statistical one that combine prior information with samples, but the inference results are different because subjectivity is involved in representing prior information as a prior distribution. The empirical Bayes approach is a statistical method that performs continuous Bayesian decision making when the prior distribution is unknown but given a large number of past data, which overcomes the above drawback and is now a common way because of a vast amount of data available. The inference for truncation parameters is important in evaluating the lower or upper bounds of the population and is required in many fields such as reliability, meteorology, medicine, etc. Many authors have considered the case of one-sided truncated distribution in the empirical Bayes framework, but in practice we are faced with the case of two-sided one. We consider the empirical Bayes estimation for truncation parameters of two-sided truncated distribution under the squared error loss. First, Bayesian estimators of the truncation parameters are derived to minimize the Bayes risk using sample of size two. And, based on the Bayesian estimators and kernel density estimate of the population density function, empirical Bayes estimators of the truncation parameters are constructed. Next, under some assumptions asymptotic optimality of empirical Bayes estimators is proved when the sample size goes to infinity, and the convergence rate is also evaluated. It is also proved that the probability that the lower and upper bounds are reversely estimated approaches zero. Finally, an example is presented to show the validity of the assumptions and the performance of the proposed empirical Bayes estimators is evaluated through simulation on the example. New proposal of using size two samples could be extended to the case of multi-dimensional truncated distributions defined on hyper-cubic domain.
VL  - 7
IS  - 1
ER  -

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