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Luis A. Afonso
Posts:
4,530
From:
LIsbon (Portugal)
Registered:
2/16/05


Reality and Model : evaluating quantiles empirically
Posted:
Feb 12, 2013 10:15 AM


Reality and Model : evaluating quantiles empirically
A formula, DvoretzkyKieferWolfowitz, 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 pseudoRGN 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. (BoxMuller 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



