I suggest you to start from the Altman's model, that is the basic model to implement the kind of econometric analysis you're looking for.
You can find the original paper at my Dropbox public folder.
After that reading, you can find a number of paper about scoring models on SSRN or Google Scholar. Moreover, I suggest you to look for all academic papers that provide a comparison/overview among all scoring models, in order to have a complete view about this topic.
As regards, your idea about the methodology used in this kind of models, it is generally correct, even if the financial ratios usually do not have the same weight/ beta coefficient in forecasting the companies will go broke, because of the fact they change over time according to the state of the economy (recession/growth), companies industry, etc.
Anyway the methodology that is used is more or less always the same and similar to the one used in the Altman's paper and only changes the number and the type of the independent variables.
If you need to estimate directly the default probability you can use the LPM, Logit/Probit regression, and, here you can find a good reference about this topic. Since it is not the basic model about default probability estimation using Logit/Probit model, I suggest you to read the main papers in the references and after carry on. Anyway, It is a nice paper about the topic because of it is based on a lot of indipendent variables (not only financial ratios) that can help you to make an idea about
- which variables can be relevant and which not;
- which variables have a linear effect, as assumed in hp and which ones, as, for instance, the leverage has a non-linear effect.