Dear Mathematicians, Statisticians and Econometricians (MSE)
An independent and curious researcher would be very grateful for any help.
Consider a system:
E(u*/p*)= E(u*)/E(p*) = E(u)/E(p) (1) (see eq. 8 in [1])
where
u – realizations of arbitrarily distributed positive random integers
p - realizations of arbitrarily distributed positive random integers
u* - exponentially smoothed version of u
p* - exponentially smoothed version of p
E – standard expectation operator
Equality (1) has been proved to be incorrect statistically.
See [1] for technicalities. The problem is that mathematics of which statistics is a
subset does not wish to support the opinion. It says the exact opposite. Consider an
arbitrary numerical example of the above system:
u = [3, 8, 6, 8, 5] with the sum equal to 3+8+6+8+5= 30
p = [2, 2, 3, 2, 1] with the sum equal to 2+2+3+2+1=10
The ratio of two expectations is 30/10=3
Now consider two exponentially smoothed versions of u and p:
Version 1:
u*=[1305903/409510, 15029207/409510, 159833463/409510, 1766109167/409510, 17942532503/409510]
with the expectation equal to 1945623/500000.
p*=[11/10, 119/100, 1371/1000, 14339/10000, 139051/100000]
with the expectation equal to 648541/500000.
= = = = = = = = =
E(u*)/E(p*)=3
Version 2:
u*=[224067519942069/70186691227690, 2578101209300141/70186691227690, 27414112357362669/70186691227690,
302876364198416021/70186691227690, 3076820733924194189/70186691227690]
p*=[11/10, 119/100, 1371/1000, 14339/10000, 139051/100000]
With the expectation of the ratios equal to the expectation of the following sequence:
[224067519942069/77205360350459, 21664716044539/7018669122769, 19995705585239/7018669122769,
21122558351239/7018669122769, 243400105523629/77205360350459]
or equivalently:
[2.902227500, 3.086727080, 2.848931220, 3.009481995, 3.152632206]
= = = = = = = = =
E(u*/p*)=3
Thus, at least mathematically equality (1) holds in general. Well, I could not guess so many numbers.
I wish I could… The question is of why the statistics in [1] contradicts the mathematics above?
Does the system in (1) exist in general? Quantum est quod nescimus...
Thank you very much for your help.
Andrey Kostenko
References:
[1] Syntetos AA and Boylan JE (2001) On the bias of intermittent demand estimates.
Int J Prod Econom 71: 457-466.
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