Help needed with probability math

Quote from DontMissTheBus:

Why not just rank using the sharpe ratio? The efficient frontier is non-linear in risk/return space afterall - why do you need to linearize it? Avg * Sharpe Ratio = Avg*Avg / Std -> Not sure how you would even interpret that.

I see what you are trying to do now - I should at least applaud you for doing a lot more thinking than the top 99% of ETers (not much of a compliment, I know).

I'll give you a hint/question:

Is the performance of system1 + system2 always (a) >= max(system1,system2), (b) >= min(system1,system2), (c) don't know.
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Hmm... You might be right! Why not just rank using the sharpe ratio indeed? I will ponder this question tonight...

Thanks, nice to hear some re-assurance.

I see where you're going with your question. Good one. I have 6 CPUs doing the math on rigorous high-detail back/fwdtesting tick data. Once they're done in 10-20 days i'll let you know the answer. ;)

Let alone 1+2, how about 10? Scaling combinatorics to the rescue.
 
:) Good luck with the computations.

Don't be so frugal to think in terms of combinations (ie, pls sir - how many systems can I have out of n?), think of it as a greedy problem (ie, what's the most I can have)

Quote from braincell:

Hmm... You might be right! Why not just rank using the sharpe ratio indeed? I will ponder this question tonight...

Thanks, nice to hear some re-assurance.

I see where you're going with your question. Good one. I have 6 CPUs doing the math on rigorous high-detail back/fwdtesting tick data. Once they're done in 10-20 days i'll let you know the answer. ;)

Let alone 1+2, how about 10? Scaling combinatorics to the rescue.
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Quote from DontMissTheBus:

You sure about that?

CLT requires the observations are independent and identically distributed.

To clarify: independent means there can't be any mean reversion/momentum; identically means the variance can't shift

There's also the small sample problem - but I figured you knew that one.
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There are many versions and variations of CLT. The requirements of identical distribution are the "weak" constraints where the "independency" is the strong constraint. CLT will work in most cases for any large number of independent variables regardless of their distributions. However, the OP cannot use none of the mentioned above calculations in order to have any confidence in the result due to incredibly small sample size. It does not comply with the “statistically significant” set size requirements.
 
True - there are many relaxed versions of CLT - even when there's weak dependence in the observations...

In any case, CLT also makes a statement about the distribution of the mean of a set of observations; It doesn't make a statement about the distribution of the observations themselves.

Quote from MAESTRO:

There are many versions and variations of CLT. The requirements of identical distribution are the "weak" constraints where the "independency" is the strong constraint. CLT will work in most cases for any large number of independent variables regardless of their distributions. However, the OP cannot use none of the mentioned above calculations in order to have any confidence in the result due to incredibly small sample size. It does not comply with the “statistically significant” set size requirements.
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Hey there,

as always when we don't know much about the underlying pd the bootstrap approach suits pretty good.

In your example you would simulate say 1000 realisations of 5 future days for each strategy, (that is out of your 5 realisations you randomly draw 5 with repetations, this assumes they have all the same probability and there's no correlation in-between the days which is absent of any other informations the best guess).

then you take the fraction of realisations with >10 as an estimator of the true value.

I am not at home right know but using an excel spreadsheet or some statistics programm this should be pretty easy to do.

And results are more or less as good as you can get it abscent of more information on the nature of the trading system and its returns.
 
Quote from DontMissTheBus:

You sure about that?

CLT requires the observations are independent and identically distributed.

To clarify: independent means there can't be any mean reversion/momentum; identically means the variance can't shift

There's also the small sample problem - but I figured you knew that one.
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The only condition for CLT I know of (it's been a long time if you don't mind) is that the random variables are independent:

P(X1\X2) = X1

I am not sure about identically distributed and I do not recal this affects the CLT.
 
It's an interesting discussion on CLT etc, but I'm wondering if perhaps the more salient point is that pulling 5 salmon out of the ocean on a Tuesday in September tells you virtually nothing about what else is swimming beneath the surface.
 
The identical part is part of the classical CLT (http://en.wikipedia.org/wiki/Central_limit_theorem#Classical_CLT).

Like the other fella said, this is the weaker condition that can be removed given a few other conditions.



Quote from intradaybill:

The only condition for CLT I know of (it's been a long time if you don't mind) is that the random variables are independent:

P(X1\X2) = X1

I am not sure about identically distributed and I do not recal this affects the CLT.
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Shrug... when bother unless there's some higher order structure in the returns that you want to simulate? Otherwise, it's the same as calculating the conditionals.

Quote from JamesJ:

Hey there,

as always when we don't know much about the underlying pd the bootstrap approach suits pretty good.

...
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Quote from DontMissTheBus:

Shrug... when bother unless there's some higher order structure in the returns that you want to simulate? Otherwise, it's the same as calculating the conditionals.
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yes, thats right, I'm just a big fan of bootstrap ;)

i mean it may take 5 min to do it in excel and get a good idea of the underlying distribution (plot), while to calculate the conditionals you need a bit more practice... ie. how to use a statistics software.
 
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