Abstract
This paper proposes a new generalized extended balanced loss function (GEBLF). Admissibility of linear estimators is characterized in the General Gauss–Markov model with respect to GEBLF. The sufficient and necessary conditions for linear estimators to be admissible with a dispersion matrix possibly singular among the set of linear estimators are obtained. It is stated that the results obtained under special conditions lead to the results known in the literature.
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Appendices
Appendix 1: The proof of Eq. (10) in Theorem 1
Proof
If \(\beta_{*} = Fy \in {\mathcal{F}}^{h}\), using some properties of trace and Lemma 1, then risk function of GEBLF is given by.
Let \(B = \lambda_{1} X^{\prime}T^{ + } X + \lambda_{2} \tilde{S} + \lambda_{3} X^{\prime}T^{ + } X = \lambda_{2} \tilde{S} + (1 - \lambda_{2} )S_{T}\), where \(S_{T} = X^{\prime}T^{ + } X\). In this case, we get
Since \(0 \le \lambda_{1} ,\lambda_{2} \le 1\), we can find such a \(w\) denoted by \(w = {{\left( {1 + \lambda_{1} - \lambda_{2} } \right)} \mathord{\left/ {\vphantom {{\left( {1 + \lambda_{1} - \lambda_{2} } \right)} 2}} \right. \kern-0pt} 2}\), where \(0 \le w \le 1\). So, Eq. (10) can be rewritten as follows for \(0 \le \lambda_{1} ,\lambda_{2} \le 1\):
If we take \(F^{*} = B^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \tilde{F} = B^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \left( {F - wB^{ - 1} X^{\prime}T^{ + } } \right)\)\(C = B^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} (I_{p} - wB^{ - 1} S_{T} )\), \(B = B^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} B^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}}\).
then (31) equal to
Appendix 2: The proof of Eq. (21) in Corollary 2
Appendix 3: The proof of Eq. (22) in Theorem 4
Let \(B = \lambda_{2} \tilde{S} + (1 - \lambda_{2} )S_{T}\). Then
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Mirezi, B., Kaçıranlar, S. Admissible linear estimators in the general Gauss–Markov model under generalized extended balanced loss function. Stat Papers 64, 73–92 (2023). https://doi.org/10.1007/s00362-022-01298-9
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DOI: https://doi.org/10.1007/s00362-022-01298-9