Joerg van den Hoff
2009-Sep-28 12:15 UTC
[R] probability density function for maximum values in repeated finite samples from a normal distribution??
this is probably not really a R specific question, if so apologies for off-topic posting: I'm interested in the probability density function of the maximum values from repeated samples of size N from a normal distribution: smp <- rnorm(N, meanval, stdev) with some mean 'meanval' and standard deviation 'stdev'. I would like to know what is the frequency distribution of max(smp) if I draw many such samples? if I investigate this simply via a simulation, I get of course approximate results (and see that the resulting distribution is not quite normal anymore, that the mean increases with increasing N, etc.). my question: does somebody know whether there exists an analytical expression for the distribution of max(smp) (or where to look)? thanks, joerg
(Ted Harding)
2009-Sep-28 12:45 UTC
[R] probability density function for maximum values in repea
On 28-Sep-09 12:15:39, Joerg van den Hoff wrote:> this is probably not really a R specific question, if so apologies > for off-topic posting: > > I'm interested in the probability density function of the maximum > values from repeated samples of size N from a normal distribution: > > smp <- rnorm(N, meanval, stdev) > > with some mean 'meanval' and standard deviation 'stdev'. > > I would like to know what is the frequency distribution of max(smp) if > I draw many such > samples? > > if I investigate this simply via a simulation, I get of course > approximate > results (and see that the resulting distribution is not quite normal > anymore, that the mean increases with increasing N, etc.). > > my question: does somebody know whether there exists an analytical > expression for the distribution of max(smp) (or where to look)? > > thanks, > joergLet Pmax(x,N) be the probability that the maximum of N is <= x. Pmax(x,N) = Prob(all N values <= x) = (Prob(a single value <= x))^N = (pnorm(x,meanval,stdev))^N Hence the frequency distribution is the derivative of this with respect to x, say pmax(x,N): pmax(x,N) = N*((pnorm(x,meanval,stdev))^(N-1))*dnorm(x,meanval,stdev) Ted. -------------------------------------------------------------------- E-Mail: (Ted Harding) <Ted.Harding at manchester.ac.uk> Fax-to-email: +44 (0)870 094 0861 Date: 28-Sep-09 Time: 13:45:49 ------------------------------ XFMail ------------------------------
David Winsemius
2009-Sep-28 12:47 UTC
[R] probability density function for maximum values in repeated finite samples from a normal distribution??
On Sep 28, 2009, at 8:15 AM, Joerg van den Hoff wrote:> this is probably not really a R specific question, if so apologies for > off-topic posting: > > I'm interested in the probability density function of the maximum > values > from repeated samples of size N from a normal distribution: > > smp <- rnorm(N, meanval, stdev) > > with some mean 'meanval' and standard deviation 'stdev'. > > I would like to know what is the frequency distribution of max(smp) > if I draw many such > samples? > > if I investigate this simply via a simulation, I get of course > approximate > results (and see that the resulting distribution is not quite normal > anymore, that the mean increases with increasing N, etc.). > > my question: does somebody know whether there exists an analytical > expression for the distribution of max(smp) (or where to look)?Yes, there is an analytical description of the highest order statistic for Normal data. The question has been studied extensibly. You should be searching on "extreme value theory" or "extremal distributions". There is also an extRemes package for R, as well as an fExtremes package for financial applications and a SpatialExtremes package. -- David Winsemius, MD Heritage Laboratories West Hartford, CT