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