# ML estimation of parameters that do not completely specify the model

I was wondering how ML is defined when the parameter does not completely specify the model. More concretely, suppose $X_1, X_2, \cdots, X_n$ are drawn iid such that $P(X_1=i)=\theta_i$, $1 \leq i \leq k$. I want to find the ML estimate of $\phi= \max_{1 \leq i \leq k} \theta_i$. To me it is not even clear if the ML estimate of $\phi$ is well defined in this case. Am I missing something here?

• Take a look at the Categorical distribution. – Daniel Sep 12 '14 at 21:53
• The categorical distribution I mentioned here is just an example. My question basically is, is ML estimation of a parameter not defined if my parameter does not fully characterise the log likelihood. – Devil Sep 13 '14 at 0:23

It's not about MLE vs Method of Moments or some other method, it's about proper reparameterization. If you want to reparameterize, you should be able to compute the values of old parameters, $\theta_i$, in terms of new parameters. You introduced one new parameter, $\phi$, which is the mode, and you need $(k-1)$ more. As far as I know, it's impossible to reparameterize the multinomial distribution using the mode.
I agree the question is not well defined at present; you haven't defined what the model is with respect to the new parameter $\phi$, so it doesn't make sense to ask about ML estimates.
From the looks of it, you're half way to defining some sort of hierarchal model; you need to specify what role $\phi$ plays in your model, then you can ask about its ML estimate.