[EMAIL PROTECTED] wrote: > Hi All > > Is it possible to generate a sample that has a specific mean, SD (also > skewness, kurtosis)? > To be more specific, I do not want to sample from a given distribution > (like the normal distribution) that has a specific mean and SD. If > sampled from a normal distribution with a given mean and SD, of course > the mean of the sample is not the exact same as the mean of the normal > distribution. > I try to get a sample that has a previously given mean and SD (and > also if possible skewness and kurtosis). > > Does anybody know the mathematical approach to this question? And, > does anybody know a software package that does this perhaps, or an > algorithm to use to program my own thing? Some vague things, like > moments, are in my head but that is perhaps not the correct path. > Others asked, expressed it this way: Sampling from a normal > distribution with restriction (restriction that the mean is exactly > the given mean). > > Thanks Toby > > > > > mean = (sum xi) / N > > variance = (sum (xi - mu) ^ 2) / N > > skewness = (sum (xi - mu) ^ 3) / (N sdev ^ 3) > > kurtosis = (sum (xi - mu) ^ 4) / (N sdev ^ 4)
I'm not sure it can be done for multi-moments but it may work out. For the case of fixing the mean only you have two choices (at least). Firstly you can recognise that the appropriate covariance matrix is singular, work it out by theory, and plug it into some multi-variate Normal generator, hope that it doesn't fall over because of the singular matrix, and generate the whole vector at once. Otherwise you can procede one element at a time without using singular matrices, essentially using an iterative expression for the calculation of the mean (you may be able to use similar recursive expressions for the k-statistics or sample moments for the more general case). ... Once you are part way through, you will know what mean you require for the values still to be generated ... write this in terms of the next value to generate and the mean of the rest ... this gives you a case where you want X,Y from independent Normals but X+cY=Z=fixed.... work out the conditional distribution of X given Z and generate the next value from this distribution. I think it falls out from the theory of sufficient statistics that, for the Normal distribution, the procedure you need does not depend on the "true" population values for the parameters but you should check that out when developing the theory you need. Notionally (and for more general distributions) what you are trying to do is generate a vector from a population with given parameters but also having given sample statistics for those parameters, which may well be different. David Jones . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
