# What is the point of introducing the concept of estimable function?

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?

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.

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.

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