Date: Feb 12, 2013 10:15 AM
Author: Luis A. Afonso
Subject: Reality and Model : evaluating quantiles empirically

Reality and Model : evaluating quantiles empirically

A formula, Dvoretzky-Kiefer-Wolfowitz,
en.wikipedia.org/.../Dvoretzky?Kiefer?Wolfowit...
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.
RESULTS
Number of times ( ) a quantile was found out of 16
_________Exp. 1____________________ Exp.II__________
__1.958__(1)__2.574__(1)_________________2.574__(2)___
__1.959__(3)__2.575__(4)______1.959__(5)__2.575__(4)___
__1.960__(9)__2.576__(6)______1.960__(7)__2.576__(6)___
__1.961__(3)__2.577__(3)______1.961__(4)__2.577__(3)___
____________ 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