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I am reading about linear model, and it introduced the concept of estimable function. According to the book, a linear combination of the parameters $c^T \beta$ is estimable if and only if there exists a linear combination $a^T y$ such that $E(a^T y)=c^T \beta$. What is the point of knowing whether a linear combination of parameters can be estimated? Isn't it enough that we know all parameters in the model can be estimated?

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If you want to test $H_0: c^T\beta=0$ vs $H_1: c^T\beta\neq 0$, you will want to be able to estimate $c^T\beta$.

Or, for example, if you need a prediction interval for $x_\text{new}\beta$, it would really help a lot if it's actually possible to estimate $x_\text{new}\beta$ ... (here we have $c=x_\text{new}^T$).

When your design is not of full rank it's useful to be able to distinguish what can and can't be estimated.

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Let me give some perspective from linear algebra.

In linear model $ y = X\beta +\epsilon$, $E(a'y) = a'X\beta$ so the definition actually says that $c'\beta$ is estimable if and only if $c\in C(X')$ where $C(X')$ is the row space of $X$. So if $X$ is full tank then any $c'\beta$ would be estimable.

But what if $X$ is not full rank? This definition is born to answer this question.

For any design matrix, full tank or not, we can estimate $c'\beta$ as long as $c$ is in the row space of $X$.

You can have a look at the pseudoinverse to find more intuition.

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  • $\begingroup$ Great! Love your answer. $\endgroup$ – chi Sep 15 '17 at 20:56

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