How to determine the probably when data is MAX function of another probability

Reply Mon 24 Jan, 2011 03:10 pm
I have a statistical question regarding probability of a function, in my case it is MAX() function. I have been given probability data that is based on the Maximum weekly temperature for each week in a year going back about 30 years. It is a Gaussian Distribution with a mean = Mw and standard deviation = SDw. However, for my analysis I want to know what the distribution parameters are on a daily basis, call them Md and SDd. That is, based on the distribution information for Weekly maximum temperature I need to work out what the distribution for the Daily maximum temperature should be.

Can someone tell me the formulae for this problem? I have worked it out using simulations and found the answer numerically but I would prefer to know the formula.
SDw = SDd * 0.626
Mw = Md + SDd * 1.352

(FYI, the I have simplified the problem above to be based on temperature. The actual data source is more complex and daily data is not available.)
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Reply Mon 24 Jan, 2011 07:14 pm
Here's my best shot. If you have n numbers chosen at random from a normal distribution, then the middle of the resulting Maximum function should occur where F(x)^n=0.5 where F(x, Md, Sd) is the cumulative distribution function. F(x) = .5*(1/7) = .9057. Looking up the inverse function of F(x) = .9057, x = Md + 1.315 Sd. If I use the one sigma point on the original distribution, I get F(x)^7 = .8413 which is the value of Z on the normal distribution that represents one sigma from the mean. F(x) = 1.971 so Mw(1 sigma) = Mw + Sw = Md + 1.971 Sd. Subbing in Mw = Md + 1.315 Sd I get Sw = .6559 Sd.

I ran this empirically from number of days equals one to seven and the shape of the curve is right, but my means are a hair low and my sigmas are a bit high. I don't know if that helps or not.
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