Chapter 6: Problem 13
Consider a location model $$X_{i}=\theta+e_{i}, \quad i=1, \ldots, n$$ where \(e_{1}, e_{2}, \ldots, e_{n}\) are iid with pdf \(f(z)\). There is a nice geometric interpretation for estimating \(\theta .\) Let \(\mathbf{X}=\left(X_{1}, \ldots, X_{n}\right)^{\prime}\) and \(\mathbf{e}=\left(e_{1}, \ldots, e_{n}\right)^{\prime}\) be the vectors of observations and random error, respectively, and let \(\mu=\theta 1\) where 1 is a vector with all components equal to one. Let \(V\) be the subspace of vectors of the form \(\mu_{i}\) i.e, \(V=\\{\mathbf{v}: \mathbf{v}=a \mathbf{1}\), for some \(a \in R\\} .\) Then in vector notation we can write the model as $$\mathbf{X}=\boldsymbol{\mu}+\mathbf{e}, \quad \boldsymbol{\mu} \in V$$
Short Answer
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Key Concepts
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