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Topic: Reality and Model : evaluating quantiles empirically
Replies: 1   Last Post: Feb 13, 2013 10:05 AM

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Luis A. Afonso

Posts: 4,758
From: LIsbon (Portugal)
Registered: 2/16/05
Reality and Model : evaluating quantiles empirically
Posted: Feb 12, 2013 10:15 AM
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Reality and Model : evaluating quantiles empirically

A formula, Dvoretzky-Kiefer-Wolfowitz,
gives in terms of maximum probability the bound which an empirical quantile provide k exact decimal places compared with that of the exact Distribution.
In order to get some information how accurate is a pseudo-RGN classified as good J. H. Ahrens, V. Dieter, (Edward J. Dudewicz, Satya N. Mishra, Modern Mathematical Statistics, Willey 1988) where u(i+1) = r * u(i) mod m,
r = 663´608´941, u(0) an odd number in [1, 2^32) , m= 2^32= 4´294´967´296.
we performed 16 blocks of n=16 million normal standard r.v. (Box-Muller algorithm) to calculate the 0.975 and 0.995 their quantiles, which are respectively 1.960 and 2.576.
Number of times ( ) a quantile was found out of 16
_________Exp. 1____________________ Exp.II__________
____________ 2.578__(1)_________________2.578__(1)___
____________ 2.579__(1)_____________________________

Note that
___P(|x| >= d) <= 2*EXP(-2*n*d^2)_____DKW formula
__n=16´000´000, d=0.0005 gives P= 0.00067__where in fact is aprox. 16/32 and 12/32 for the quantiles 1.960 and 2.576 (0.975, 0.995).
How far are from the forecasting probabilities the tested pseudoRNG. Surprising? Not at all! (They are pseudo for irs own nature . . .).

Luis A. Afonso

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