Re: Normally distributed integer numbers

From: Cristiano (cristiano.pi_at_NSquipo.it)
Date: 07/17/04 

Date: Sat, 17 Jul 2004 10:00:25 GMT

Richard Ulrich wrote:
> - e-mail to the address bounces.
>
> On Fri, 16 Jul 2004 08:23:52 GMT, "Cristiano"
> <cristiano.pi@NSquipo.it> wrote:
>
>> Richard Ulrich wrote:
>>> On Wed, 14 Jul 2004 21:15:03 GMT, "Cristiano"
>>> <cristiano.pi@NSquipo.it> wrote:
>>>
>>>> Richard Ulrich wrote:
>
>>
>> I get from a program for NT times, N integer numbers with
>> distribution normal(Ec,SDc).
>
> NT is the number of simulations done, I hope.

Yes, but I'd call it the number of N-number sequences I get from a single
test. I think this could be a better explanation in pseudo-code:

1) for i from 1 to NT do the following:
2) for j from 1 to N do the the following:
3) generate a random integer Xj normal(Ec,SDc) distributed;
4) Xj_normalized = (Xj - Ec) / SDc;
5) calculate KSi for the N X_normalized's;
6) calculate KS for the NT KS's (KSi must be uniformely distributed under
the assumption that the X's are normally distributed).

Here I wrote the step 3 in a "shrinked" way. In my program I get the X's
from an algorithm which does several operations over a sequence of bits.
Anyway, the result is the same: I get a random integer number normal(Ec,SDc)
distributed.

What I need is a number which tell me whether the X's are normal(Ec,SDc)
distributed and I hope to get the answer from the KS at step 6.
Is it the right way?

>> I wrote a test which normalizes the N-number NT sets. Now they have a
>> distribution normal(0,1).
>> I calculate the KS test over the N normalized numbers obtaining NT
>> p-values.
>>
>> The hard job (for me) is to calculate Nmin and Nmax for any NT for
>> which a *good* set of N integer numbers (Nmin <= N <= Nmax) is
>> "declared" *good* by my test.
>
> What are Nmin and Nmax referring to, in some other words?

It is the range of N in which my test gives reliable results.

> What do you mean by "my test"?

It is the test described above (from step 1 to 6).

> Why, in example below, is Nmin so much larger than Nmax? - contrary
> to the parenthetical requirement described above.

Because I was sleeping. :-) I'm sorry, but Nmin is Nmax and Nmax is Nmin.

>> In other word, when N<Nmin or N>Nmax I seen that any set of N integer
>> numbers could be declared bad even if it is good.
>
> Are you keeping in mind, that for a good simulation, 5% of the
> samples drawn will be declared be 'non-normal' -- if you are
> doing a good simulation, and applying a good test.

No, why? Where did "5%" come from?

I have in mind that some sample should be declared 'non normal', but how
many of them it should be given by the significance level I choose.

>> This is an example of what I got with extensive simulations:
>>
>> NT Nmin Nmax
>> 20 480 2

Nmin and Nmax should be swapped.

>> 50 360 10
>> 100 280 50
>> 200 240 90
>> 300 200 120
>> 400 190 160
>> 500 160 130
>> 600 140 140
>>
>> for example, when I collect 50 N-number sets I know that I can use
>> N=310, or N=90, or N=154, ... But, for example, if I use N=370 or
>> N=8 I could get "bad" even if the sets have the expected
>> distribution.
>>
>
>
> The data that you can collect from your simulations is,
> How MANY of them 'reject' at a certain size, with a
> certain configuration? Is that apparent?

Yes.

> Unless I missed it, you never did say what your purpose
> was, and I still have not spotted a worthy one.

I just need to see whether the NT N-number sequences are normal(Ec,SDc)
distributed.

Thank you for the help
Cristiano

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