Academic Editor: Wei Zhu
Received: 30 December 2024
Revised: 29 January 2025
Accepted: 5 February 2025
Published: 9 February 2025
Citation: Alheety, M.I.; Nayem, H.;
Kibria, B.M.G. An Unbiased Convex
Estimator Depending on Prior
Information for the Classical Linear
Regression Model. Stats 2025, 8, 16.
https://doi.org/10.3390/
stats8010016
Copyright: © 2025 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license
(https://creativecommons.org/
licenses/by/4.0/).
Article
An Unbiased Convex Estimator Depending on Prior Information
for the Classical Linear Regression Model
Mustafa I. Alheety
1
, HM Nayem
2
and B. M. Golam Kibria
2,
*
1
Department of Mathematics, College of Education for Pure Sciences, University of Anbar, Anbar 31001, Iraq;
eps.mustafa.ismaeel@uoanbar.edu.iq
2
Department of Mathematics and Statistics, Florida International University, Miami, FL 33199, USA;
hnayem@fiu.edu
* Correspondence: kibriag@fiu.edu
Abstract: We propose an unbiased restricted estimator that leverages prior information to
enhance estimation efficiency for the linear regression model. The statistical properties of
the proposed estimator are rigorously examined, highlighting its superiority over several
existing methods. A simulation study is conducted to evaluate the performance of the
estimators, and real-world data on total national research and development expenditures
by country are analyzed to illustrate the findings. Both the simulation results and real-data
analysis demonstrate that the proposed estimator consistently outperforms the alternatives
considered in this study.
Keywords: linear model; MSE; multicollinearity; restricted least-squares estimator;
unbiased ridge estimator
1. Introduction
Consider the following linear regression model:
Y = Xβ + ε, ε ∼ N
0, σ
2
I
n
(1)
where Y is an n × 1 vector of the dependent variable of observation, X is an n × p matrix
of explanatory variables of rank p, β is a p × 1 vector of unknown parameters, and ε is an
n × 1 error vector, which follows a multivariate normal distribution, with E(ε)= 0 and
covariance matrix equal to σ
2
I
n
. Also, σ
2
is the error variance and I
n
is the identity matrix
of size n.
The ordinary least-squares (OLS) estimator for model (1) can be written as follows:
ˆ
β = Z
−1
X
′
Y (2)
where Z = X
′
X. In the linear regression model, the unknown regression coefficients are
estimated using the OLS estimator.
The linear regression model in (1) is based on a set of assumptions that determine
the process of its use in estimating the relationship between the dependent variable and
the explanatory variables. One of these assumptions is the absence of a linear relation-
ship, which is called multicollinearity between the explanatory variables, such that this
relationship is not strong or harmful. The presence of a strong relationship between the
explanatory variables affects the estimates of the model parameters, which is reflected in
the decision-making process to accept or reject the null hypothesis. The reason for this is
Stats 2025, 8, 16 https://doi.org/10.3390/stats8010016