I'm reading the article The large-sample distribution of the likelihood ratio for testing composite hypotheses and there are some notations and points I do not understand, therefore making me unable to understand the derivations. In the second page of the article, Wilks states that:
That is, we shall assume the existence of functions $\tilde{\theta}_i(x_1, ..., x_n)$ (maximum likelihood estimates of the $\theta_i$) such that their distribution is
$$\frac{|c_{ij}|^{1/2}}{(2\pi)^{h/2}}\exp\left(-\frac{1}{2}\sum_{i,j=1}^h c_{ij}z_i\overline{z}_j\right)(1+\phi)\;dz_1\cdots dz_h$$
where $z_i=(\tilde{\theta}_i-\theta_i)\sqrt{n}, \displaystyle c_{ij}=-E\left(\frac{\partial^2 \log f}{\partial \theta_i \partial \theta_j}\right)$, $E$ denoting mathematical expectation, and $\phi$ is of order $1/\sqrt{n}$ and $||c_{ij}||$ is positive definite.
Here is a picture of the part in the article (article provided in the link above):
Click on the figure to zoom in. Also please check the link for the paper.
My questions are the following:
- What does this symbol mean? Complex conjugate of $z_j$? (note that in the LaTex version of the derivation above, I have used my own interpretation of it as $\overline{z}_j$. I'm not sure whether this is correct)
- Does this mean that $\phi$ is a function of $z$ bounded above by $1/\sqrt{n}$?
- This notation confuses me? Isn't this a vector norm?
- Why are these two conditions true? Unclear
UPDATE:
If the picture does not provide sufficient information to answer my question please check the article in the link provided. The paper is only three pages long and first two are relevant for my questions.