Suppose that I did an experiment and collected a dataset. In previous literature these type of dataset are explained by a well-accepted model, called Model M. I believe that the Model M cannot explain my data, so I need to test it.
One procedure to accomplish this is through a hypothesis test:
$H_o:$ The data's distribution is same as the Model M's predicted distribution.
$H_1:$ The data's distribution is not distributed according to Model M.
Is this procedure reasonable? Or has anyone set the null-hypothesis as an existing model? If anyone has done it then I think I can justify this procedure, too.
I think, at a first glance, this null should be fine. However, usually the $H_o$ is something like "the data is random", and the $H_1$ is something like: "the data is explained the model". I am basically swapping the position of $H_o$ and $H_1$.
All details are taken from the real world problem as exact numbers:
The dataset contains seven columns and 1246 ranks. For the seven columns, there are six independent variables, $x\in\mathbb R^6$, and one dependent variable $y\in \{-1,1\}$.
For each $x$, variables are inputted in the experiment, and the resulted $y$ will be either -1 or 1.
The experiment is done for 1246 times, each time with a different $\theta$.
The model M is a deterministic real function $\hat y=\text{sgn} [f(x,\theta)+\epsilon]$.
$\epsilon$ is an error term. When $\text{sgn} [f(x,\theta)+\epsilon]=0$, set $\hat y=1$.
Of course, to make it a statistical model, there has to be an error term, which is either normal distribution or Extreme Value Type I Distribution (because this is basically a binary choice problem).
Then, there are two different ways of finding $\theta$:
$\theta$ is estimated through MLE using the entire dataset. In this case, $\theta$ is same for all 1246 rows.
The 1246 rows of data are equally divided into 89 different groups based on the person who did the experiment. 89 different $\theta$ will be estimated.
