Assuming that you are asking about the US, the documents explaining the RDA (recommended dietary allowance), AI (adequate intake), and UL (tolerable upper intake level) are available on the website of the National Institutes of Health. They are written by "The Food and Nutrition Board, Institute of Medicine, National Academy of Sciences".
Going into these in detail is way too broad because these are long - for example, there is a 1000 page document on the RDI of calcium and vitamin D. Far from being "a shot in the dark", this details a lot of the research those values are based on.
It also says when insufficient data is available to establish these values (for example RDA for calcium for infants). Far from being only the value you might see on the label of your food, the recommendations are are actually different for different age groups, sexes, and life stages (for example pregnancy), as can be seen in the detailed recommendations for vitamin D and calcium, for example. The quality needs to be judged for each individual recommendation, there is no single definitive answer.
Where can I look at a study that proves that a human eating less than RDI will become deficient
The reasoning behind the AI should be listed in the documents I linked to. For calcium in adults, for example, the AI is based on Calcium requirements: new estimations for men and women by cross-sectional statistical analyses of calcium balance data from metabolic studies.
Be aware, though, that "proving" this isn't as straight forward as you might think it is. You can't just deprive people of a nutrient and look at what happens - a study like that will usually be considered unethical. Instead, researchers look at what people with existing symptoms of malnutrition are ingesting, for example.