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I have N strategies/signals that I would like to allocate to. I want to estimate an estimate of future performance based off of recent realized performance (momentum of strategies per se - e.g. strategies that performed well recently will get higher allocation and vice versa).

Now, despite having shortcomings let's use the test statistic as the Sharpe ratio. One way to do this would just be to take the Sharpe ratios of each N strategy over the past D days and use those as our estimate of the forward expected returns. This is flawed, however, as for short enough windows, the sample Sharpe can be very different from a long run Sharpe.

So, I would like to use some type of Bayesian updating. We know that Sharpe ratios are just scaled T-distributions (scaled by sqrt(n), where n is the sample size). I was thinking that I could bootstrap samples of D days from my entire signal performance history to get some sort of prior distribution for the Sharpe ratio for each strategy. This should be distributed T. However, I don't see a conjugate prior distribution for a T likelihood, so I'm unsure how to update my prior distribution for the Sharpe ratio with the most recent Sharpe ratio observation (most recent D days)

Has anyone done something similar?

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    $\begingroup$ Why not update a Bayesian estimate for both mean and variance separately? For example, the distribution characteristics of the GARCH family of models are well studied. $\endgroup$ – David Addison Mar 7 '18 at 22:53
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    $\begingroup$ See section 3.7 of my Short Sharpe Course, it gives a Bayesian update for a single Sharpe ratio in terms of a lambda prime distribution. $\endgroup$ – shabbychef Mar 8 '18 at 7:06

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