I would first listen to the effects that you claim happened. Then I would set up, for example, microphones to detect and localize sound, video cameras to pick up visual effects, thermometers to pick up changes in temperature, etc. Then I would look first for all the *natural* effects that could produce what you claim to have experienced. I would look at the structural properties of the house, the types of sounds produced by the wind moving through it, the water in the pipes, etc. I would ask about the conditions under which the effects were experienced: we you going to sleep or fully awake? Can I do so without 'offending you'? I don't know. It depends on whether I conclude you're insane or lying or not. I would also ask for the help of trained magicians (who are very adept at fooling people) to see if I had missed anything.As an academic scientist, if you were called upon to investigate a haunting at some location, how would you proceed in a manner that does NOT involve presupposition? In other words, taking the statement of alleged witnesses without bias or assumptions that they're lying; how would you investigate their claims, and what measures would you use to keep your research honest with regard to your own biases and their claims?
Please don't tell me you would give them brain scans and all kinds of blood work, and polygraph tests. If I told you my house was haunted, how would you test that claim without offending me?
If you have proof of the supernatural, there is a $1 million prize offered by James Randi. He requires a controlled environment and makes sure no 'foolery' is done. And he is a professional magician, so is a bit harder to fool than most scientists, who are accustomed to people being honest, even if mistaken.