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The Kibria-Lukman (KL) estimator is a recent estimator that has been proposed to solve the multicollinearity problem. In this paper, a generalized version of the KL estimator is proposed, along with the optimal biasing parameter of our proposed estimator derived by minimizing the scalar mean squared error.
Apr 1, 2022 · The Kibria-Lukman (KL) estimator is a recent estimator that has been proposed to solve the multicollinearity problem. In this paper, a generalized version of the KL estimator is...
Nov 26, 2021 · In this paper, we developed a Jackknifed version of the Kibria-Lukman estimator- the estimator is named the Jackknifed KL estimator (JKLE). We derived the statistical properties of the new estimator and compared it theoretically with the KLE and some other existing estimators.
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(Seber, 2007, Pg. 225) Let \(\varvec{A}> 0\) and let \(\varvec{B}\) be a \(p \times n\) of rank q (\(q \le p\)). Then, 1. (a) \(\varvec{B}\varvec{A}\varvec{B}^{\top } \ge 0\) 2. (b) \(\varvec{B}\varvec{A}\varvec{B}^{\top } > 0\) if \(q = p\). Let \(\hat{\varvec{\alpha }}_i^*\), \(i = 1, 2\) be two arbitrary estimators of \(\varvec{\alpha }\). The e...
(Seber, 2007, Pg. 228) Let \(n \times n\) matrices \(\varvec{M}>0\) and \(\varvec{N}>0\) (or \(\varvec{N}\ge 0\) ); then, \(\varvec{M}>\varvec{N}\) if and only if the eigenvalues \(\lambda _i\) of \(\varvec{N}\varvec{M}^{-1}\) all satisfy \(\lambda _i \le 1\) (\(\lambda _i < 1\)); or equivalently, \(\lambda _{\max }\left( \varvec{N}\varvec{M}^{-1}\...
(Farebrother 1976) Let \(\varvec{M}\) be an \(n \times n\) positive definite matrix, that is, \(\varvec{M}>0\) and \(\varvec{a}\) be some vector; then, \(\varvec{M}-\varvec{a}\varvec{a}^{\top } \ge 0\) if and only if \(\varvec{a}^{\top } \varvec{M}^{-1} \varvec{a}\le 1\).
(Trenklar 1980) Let \(\hat{\varvec{\alpha }}_i^* = \varvec{H}_i \varvec{y}\), \(i = 1, 2\), be two linear estimators of \(\varvec{\alpha }\). Suppose that \(\textbf{D}= \textrm{Var}(\hat{\varvec{\alpha }}_1^*) - \textrm{Var}(\hat{\varvec{\alpha }}_2^*) > 0\), where \(\textrm{Var}(\hat{\varvec{\alpha }}_i^*) =\sigma ^2 \varvec{H}_i \varvec{H}_i^{\to...
Assume \(k > 0\). The Almon principal component Kibria-Lukman estimator is superior to the Almon estimator using MSEM criterion, that is, \(\textrm{MSEM}\left( \hat{\varvec{\alpha }}^{\textrm{A}}\right) -\textrm{MSEM}\left( \hat{\varvec{\alpha }}_r^{\textrm{A}-\textrm{KL}}(k)\right) > 0\)if and only if where
Based on Eqs. (39) and (45), the differences between \(\text {MSEM}(\hat{\varvec{\alpha }}^{\textrm{A}})\) and \(\text {MSEM}(\hat{\varvec{\alpha }}_r^{\textrm{A}-\textrm{KL}}(k))\)is Due to the definition of the Almon and Almon-PC-KL estimators, it is sufficient to show that \(\textbf{D}_0 > 0\) using Lemma 4. Knowing that \(\textbf{C}= \varvec{T}...
If \(k > 0\), the Almon principal component Kibria-Lukman estimator is superior to the Almon principal component estimator using MSEM criterion, that is, \(\textrm{MSEM}\left( \hat{\varvec{\alpha }}_r^{\textrm{A}}\right) -\textrm{MSEM}\left( \hat{\varvec{\alpha }}_r^{\textrm{A}-\textrm{KL}}(k)\right) > 0\)if and only if where and \(\varvec{a}\) is ...
Let \(\lambda _r\) be the rth genvalue of \(\varvec{Z}^{\top } \varvec{Z}\) where \(\lambda _1 \ge \lambda _2 \ge \ldots \ge \lambda _r > 0\). For \(k < 2 \lambda _r\), the Almon principal component Kibria-Lukman estimator is superior to the Almon ridge estimator using MSEM criterion, that is, \(\textrm{MSEM}\left( \hat{\varvec{\alpha }}^{\textrm{A...
Let \(\lambda _r\) be the rth eigenvalue of \(\varvec{Z}^{\top } \varvec{Z}\) where \(\lambda _1 \ge \lambda _2 \ge \ldots \ge \lambda _r > 0\). For \(k < 2 \lambda _r\), the Almon principal component Kibria-Lukman estimator is superior to the Almon principal component ridge estimator using MSEM criterion, that is, \(\textrm{MSEM}\left( \hat{\varve...
Let \(\lambda _r\) be the rth eigenvalue of \(\varvec{Z}^{\top } \varvec{Z}\) where \(\lambda _1 \ge \lambda _2 \ge \ldots \ge \lambda _r > 0\). For \(k < 2 \lambda _r\), the Almon principal component Kibria-Lukman estimator is superior to the Almon Kibria-Lukman estimator using MSEM criterion, that is, \(\textrm{MSEM}\left( \hat{\varvec{\alpha }}^...
INTRODUCTION. The statistical consequences of multicollinearity are well-known in statistics for a linear regression model. Multicollinearity is known as the approximately linear dependency among...
Mar 1, 2023 · In this paper, we proposed an extended version of the Kibria–Lukman estimator (COMPKL estimator) to the Conway–Maxwell Poisson regression model to reduce the effect of the multicollinearity problem.
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Dec 14, 2021 · MSE(βˆPLE)= ∑P j=1 (λj + d)2 λj(λj + 1)2 + (d − 1)2 ∑p j−1 α2j (λj + 1)2. (2.9) where λj is the j th eigenvalue of X′LˆX and α j is the j th element of α. The KL estimator was proposed by Kibria and Lukman (2020) as a means of mitigating the effect of multicollinearity on parameter estimation.
