Suppose that I am concerned with objects that are described by N 2D points. These points all have equal importance to me. I begin with a large number R of such objects, each with its corresponding N-element list of 2D points. Let's call this the population.
I'm now given a new object B, with the N locations of its 2D points. I'd like to know how confident I can be that B belongs to the population.
I'd also be happy with a probability, or any other number with a simple interpretation. I'm willing to assume that everything is uncorrelated, point $i$ in each object is uniformly distributed with respect to point $i$ in the other objects, and other such typical assumptions.
When there's just one variable, Student's t-test seems the right tool. That's about the limit of my statistical depth. I've read about multidimensional t-tests, Hotelling tests, and related topics, but understanding how to use these tools requires substantially more sophistication and depth than I have. Frankly, I can't turn the references I've found (such as the Wikipedia page on "Multivariate t-distribution"), into a practical solution. I'm encouraged that these tools are built into Mathematica (the system with which I'm familiar - I don't know R), and that they seem very close to what I need, but figuring out how to use them for this question seems just out of reach. The example uses I've read involve comparing things like the means of two populations and other comparisons, but I don't see how to phrase my question in terms of those sorts of tests.
Ideally I'd like to spend the time to learn enough statistical theory to understand these tools well enough to answer this question for myself. But at the moment I'm faced with a practical situation for which I need to get an answer.
My goal is a little Mathematica routine that takes two inputs: a matrix of R vectors, each made up of N 2D points, and a second vector of N 2D points, and it returns to me a number expressing the confidence that the vector belongs to the population.
Or, equally useful, some kind of measure that tells me how "far" the new vector is from belonging to the population. If anyone can help me get to that goal in concrete, practical terms I would be very grateful.