To fix the idea, we consider the estimation of means in the following one-way ANOVA model, \(\displaystyle Y_{ij}=\theta _i + \epsilon _{ij}, \quad i=1, \cdots ,k, \quad j=1, ,\cdots ,n_i. \) For brevity sake, we consider that ε ij are identically and independently normally distributed with mean 0 and common finite variance σ 2. The statistical objective is to estimate simultaneously the mean parameter θ = ( θ 1, ⋯ , θ k ) ′. Let \({\bar Y}_i=\sum _{j=1}^{n_i} y_{ij}/n_i\) , i = 1, 2, …, k. If σ is known, the vector \(\left ({\bar Y}_1,\dots ,\bar {Y}_{k}\right )'\) is a complete sufficient statistic for θ. Further, it is the best unbiased, maximum likelihood, and minimax estimator of θ. However, we wish to improve the performance of the maximum likelihood estimator (MLE), \({\bar Y}_i\) by incorporating the information (which may not be certain) regarding the parameter vector of interest, θ. In other words, it is possible that θ = θ o , where θ o is a known prior guess of θ.

Optimal Shrinkage Estimation

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发表日期：

2025年

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S. Ejaz Ahmed,
T. Quadir,
S. Nkurunziza

摘要

To fix the idea, we consider the estimation of means in the following one-way ANOVA model, \(\displaystyle Y_{ij}=\theta _i + \epsilon _{ij}, \quad i=1, \cdots ,k, \quad j=1, ,\cdots ,n_i. \) For brevity sake, we consider that ε ij are identically and independently normally distributed with mean 0 and common finite variance σ 2. The statistical objective is to estimate simultaneously the mean parameter θ = ( θ 1, ⋯ , θ k ) ′. Let \({\bar Y}_i=\sum _{j=1}^{n_i} y_{ij}/n_i\) , i = 1, 2, …, k. If σ is known, the vector \(\left ({\bar Y}_1,\dots ,\bar {Y}_{k}\right )'\) is a complete sufficient statistic for θ. Further, it is the best unbiased, maximum likelihood, and minimax estimator of θ. However, we wish to improve the performance of the maximum likelihood estimator (MLE), \({\bar Y}_i\) by incorporating the information (which may not be certain) regarding the parameter vector of interest, θ. In other words, it is possible that θ = θ o , where θ o is a known prior guess of θ.