Thread: Order Statistics of uniform RV and derivative

Hi,

I have the following minimization problem:



where Y is Uniform(0,1) and k from 1 to M. a is also given and ranges from 1/2 to 1.
Of course I can just plug in the different values of k and find the minimum but I need an approximation or a closed form solution.

There are 3 expressions for the Probability which all don't help we finding the minimum






where B is the Beta function (incomplete/complete). By admitting k to be a non-integer (Works in the 3rd formula) I can also treat the probability as differentiable which could help me find the minimum but I cannot find a derivative wrt k.

Any ideas?

An approximation idea or something would also be OK since the approach is already an approximation.

Thanks for reading.

Are you asking how to derive
Because I do not understand 'There are 3 expressions for the Probability which all don't help we finding the minimum'

Yes, BUT wrt to k not to a. Otherwise it would be very easy. I also might have already found a solution. Since M is around 50-1000 I can apply the CLT and use an approximation function for the error function and I should be fine. I will yet have to derive all that but it should work. Will let you know when I have it.