Let $p_{ij}$ denote the probability that an observation falls into cell $(i,j)$ of a two-way contingency table containing $r$ rows and $c$ columns. A total of $x_{oo}$ observations are taken and $x_{ij}$ denotes the number that falls in to cell $(i,j)$. Obtain an expression for the MLE of $p_{ij}$ when the rows and columns are independent.
I totally had no idea what to do so read the solutions but I only understand 20% of what it's going on. Worse, my lecture notes doesn't have the slightest mention of most of the stuff going on and I am very stressed.
Solution
$p_{ij}=Pr(\text{Observation falls in row i and column j})$
Likelihood $L=\Pi^r\Pi^c p_{ij}^{x_{ij}}$ and log likelihood $\Sigma^r\Sigma^cx_{ij}\text{log }p_{ij}$. Under the independence of rows and columns, $p_{ij}=p_{i-}p_{j-}$ where $p_{i-},p_{-j}$ are the probabilities that the observation falls in row $i$ and column $j$ respectively.
We note constraints $\sum^rp_{i-}=1$ and $\sum^cp_{j-}=1$.
(I kind of understand the logic up to here. However, it suddenly, just so suddenly introduces the following equation like it's been there all along. I mean, what are $\gamma,\lambda$? Where in Heaven's have they popped up from? Why are they there?)
To obatin MLEs for each $p_{i-},p_{-j}$ we maximize
$$G=\sum^rx_{i-}\text{log }p_{i-}+\sum^cx_{-j}\text{log }p_{-j}+\lambda(1-\sum^rp_{i-})+\gamma(1-\sum^rp_{-j})$$
A massive massive question mark on this one, I really need someone to break down this ugly chunk for me. I might understand $\sum^rx_{i-}\text{log }p_{i-}+\sum^cx_{-j}\text{log }p_{-j}$; It broke down the sums of $p_{ij}$ by row and columns, yes? Okay...(correct me if I am wrong)
But $\lambda(1-\sum^rp_{i-})+\gamma(1-\sum^rp_{-j})$? That just came out of the blue and slapped me in the face and I have no idea what it is.It doesn't even explain what $\lambda,\gamma$ are, constants? If so, what constants?
Actually, $\sum^rp_{i-}=1$ and $\sum^cp_{-j}=1$ so won't the bits in the brackets $(1-\sum^rp_{-j})$ etc be zero? Why is it even there? I found something called the $G$ statistic which I am not familiar with (and doesn't look like anything I am having a problem here anyway).
I need someone to be better than my lecturer, desperately, please.