Reporting to a Re: How to simulate Cauchy samples Posted: May 8, 2008 7:54 AM From Jack Tomsky: _____These are not confidence intervals on the median. Since you're assuming that the population median is known to be zero, the confidence interval for the median is [0, 0] for all confidence levels. Jack (moderator) ___________ I think that this statement should be fully discussed by two main reasons: ___1) It could be understood as the simulation Monte Carlo is wrong/unuseful/dangerous. ___2) The sampling of simulated data *losses/harms* its mainly valuable property of to be random.
___1) A not really strong argument is that Monte Carlo in Statistics/ NHST is used since early sixties, any contest was appeared. Remember that an early paper on the issue is authorized by Hubert Lilliefors who adapted Kolmogorov- Smirnov normal test to the case where the parameters are unknown, so estimated by data itself. ___2) About randomness. Given the (Cumulative) Distribution Function of the r.v. X, F(X<=x0) = g(x0; a,b, ...), a, b, ... parameters, we can invert then to obtain the result: Prob. (x0<= a) = G(F(a)) where G is the inverse function of the CDF. Now let be F= random on [0,1], then we have Prob.(x0<= a) = G(random). Given a random number u and evaluating G(u) we get easily the value x0 which is *at random*, exactly and unquestionably.
Jack Tomsky´s argument that because I let the median as zero the C.I, is [0, 0] is completely worthless. I am full free to chose the parameter´s values at my criterion, to pick what model I want to work with.