# The large-sample distribution of the likelihood ratio for testing composite hypotheses

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:

1. 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)
2. Does this mean that $\phi$ is a function of $z$ bounded above by $1/\sqrt{n}$?
3. This notation confuses me? Isn't this a vector norm?
4. 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.

• Not enough context in your picture to say for sure . My guess is that $||c_{ij}||$ is a positive-definite variance-covariance matrix previously specified. – BruceET Feb 15 '18 at 17:31
• Hi @BruceET thank you. Please check the attached article in the link. It contains all available info. It's only three pages long and the first two are relevant for my questions. – jjepsuomi Feb 15 '18 at 17:33
• Maybe later. Meeting about to start now. There are few things on this site almost as old as I am, and this is one of them. I think this topic has been discussed on more detail since 1938. Not sure of your specific interest. Perhaps look in citation index for later papers that reference this one. (R. Bahadur may be one of the later authors to deal with this topic, maybe late 1950s.) – BruceET Feb 15 '18 at 17:40
• Hi @BruceET sure no problem. Take your time. Appreciate your help :) I'm citing this article in my thesis to justify the -2 constant multiplier in information criteria such as Akaike IC. – jjepsuomi Feb 15 '18 at 17:44
• Here are my best guesses based on reading carefully and thinking a little: (1) The superscript has detail beyond the resolution of the copy. The only thing that makes sense to me is $c_{ij}z_iz_j^\prime,$ where $z_i$ is a row vector and $x_j^\prime$ is a column vector; in older notation the 'prime' may be an 'overdot'. (For $n=1,$ this would be $cz^2.)$ (2) Seems to be answered in the next sentence below your picture. (3) Standing by my previous Comment. (4) See footnote ${}^2,$ which is reference to Doob article in TAMS, which I suppose you need to read. – BruceET Feb 15 '18 at 20:55