Chpt. 10, The saturated models

[Ark raw]

In section 3.2, it talks about saturated models having as many parameters as observations, so how does this lead to saturated model is the perfect fit to the data? in other words, how does model having same no. of parameters as the observations, fit that data perfectly?

And 2ndly in the same section, under key information it states that \mu^_i = y_i, shouldn't it be
g(\mu^_i)=y_i?

 

[John Lee]

The easiest thing is to do an example yourself. So if you have data \(Y_i \sim Poi(\mu_i)\) and calculate the MLE of the \(\mu_i\)'s you'll see that they are equal to the \(y_i\)'s.

Since our estimates are equal to the observed values this means that our estimates are equal to the data - ie we have a perfect fit to the data.

 

[Ark raw]

Thank you for clearing my doubt.
But I still can't understand this:

how does model having same no. of parameters as the observations, fit that data perfectly?

in other words, how does model having same no. of parameters in the linear predict lead to that model being a perfect fit?

 

[John Lee]

As I said above, the fitted values (our estimations) are the observed values. So the fit is the observations and so the fit is equal to the data - so it is a perfect fit to the data.