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Topic: Monte Carlo simulation with inequality constraints
Replies: 11   Last Post: Mar 23, 2012 8:53 AM

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Ray Koopman

Posts: 3,383
Registered: 12/7/04
Re: Monte Carlo simulation with inequality constraints
Posted: Mar 16, 2012 7:52 PM
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On Mar 16, 2:10 am, deltaquattro <> wrote:
> Il giorno giovedì 15 marzo 2012 21:11:50 UTC+1, Ray Koopman ha scritto:
>> On Mar 15, 3:21 am, deltaquattro <> wrote:
>> [..]
>> I'm sending this reply to both groups. Can you reply to crossposts?

> Hi, Ray!
> Thanks a lot for your reply! Let's see, I'm replying to your message,
> and then I will see if it appears in both newsgroups.

It didn't. I'm sending this reply to both groups, so that
sci.stat.math readers will know that the rest of discussion
will be in sci.math.num-analysis only.

>> On Mar 15, 9:30 am, deltaquattro <> wrote:
>>> Il giorno giovedì 15 marzo 2012 16:22:42 UTC+1,
>>> Dave Dodson ha scritto:

>>>> Let R1, R2, R3, and R4 be four random numbers. Set X1 = max(R1,R2),
>>>> X2 = min(R1,R2), X3 = max(R3,R4), and X4 = min(R3,R4).
>>>> Dave

>>> Cool, thanks! This sounds a bit like my solution 3, but using min
>>> and max instead than abs and + . In your case, which are the
>>> distribution of X1 and X2, for example? I guess they're different
>>> from the distributions of R1 and R2. Also, do you have any comment
>>> about solution 2, i.e., [see above]
>>> Do you know if in this case, the distributions of X1, X2, X3, X4
>>> would be the same as without "rejection", or if they are different?
>>> Thanks,
>>> Best Regards
>>> deltaquattro

>> Both rejecting and swapping out-of-order pairs will change the
>> marginal distributions.

> Ouch! I was afraid it would be like that :( but it's good to know
> it for sure.

>> How much do you know about the desired final distributions?
> Not much: manufacturing data have many limitations, so I cannot
> trust them too much. Each Xi has a design value Ti, a lower
> specification limit Li and an upper specification limit Ui.
> I know, more or less, the probability with which Xi falls outside
> specification limits: a typical value is p=0.02. I usually assume
> that the mean of Xi is equal to Ti, and from p I derive the standard
> deviation of the distribution, or another significant parameter.
> Usually the "simple" distributions which fit the manufacturing
> data better are either normal or Beta (which I substitute with
> a triangular, since my Monte Carlo code doesn't support Beta
> distributions).
> Since manufacturing measurements have many limitations, I'm not
> interested in reproducing the marginal distributions perfectly.
> However, I would like to at least get the mean and standard
> deviation right! I cannot understand my results, if I don't know
> the mean and standard deviation of the input data I am feeding to
> my code. Also, I would like to do this study with at least two
> distributions, in order to have a rough idea of the sensitivity
> of my study to the type of distribution. Thanks,
> Best Regards
> deltaquattro

I'm sorry, I guess I wasn't clear. The "desired final distributions"
I was referring to are the X distributions, not the Y. In particular,
what I was hoping for was some idea of how much the X1,X2 (and X3,X4)
marginal distributions overlap. If the overlap is small then it
probably doesn't matter much how you implement the inequality
constraints, but if the overlap is large then it seems more likely
to matter. Could you give us some sample (L,T,U) values? Is the only
shape arequirement that the distributions be roughly symmetric and
unimodal? Are there any "hard" limits, such as 0, then *must* be met?

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