# Backtesting of outperformance of a benchmark using the Deflated Sharpe Ratio

I want to test whether, let's say, strategy A outperforms strategy B.

In Marcos López de Prado's book Advances in Financial Machine Learning he presents the following statistics:

1. The Probalistic Sharpe Ratio: the probability that a strategy's observed Sharpe Ratio $$\hat{SR}$$ exceeds a benchmark Sharpe Ratio $$SR^*$$ follows a normal distribution when we consider the skew $$\hat{\gamma_3}$$ and kurtosis $$\hat{\gamma_4}$$ of the strategy's return: $$\hat{PSR}[SR^*] = Z[\frac{(\hat{SR} - SR^*)\sqrt{T-1}}{\sqrt{1-\hat{\gamma_3}\hat{SR} + \frac{\hat{\gamma_4} - 1}{4}\hat{SR}}}]$$

However, if we were to run many simulations, just by random chance, at some point we would find some strategy that outperforms our benchmark significantly. So a new statistic is introduced accounting the multiplicity of trials:

1. The Deflated Sharpe Ratio, $$\hat{DSR}=\hat{PSR}[SR^\star]$$ where $$SR^\star=\sqrt{Var(\{\hat{SR_n}\})}((1-\gamma)Z^{-1}[1-\frac{1}{N}]+\gamma Z^{-1}[1-\frac{1}{N}e^{-1}])$$, where $$\gamma$$ is the Euler-Mascheroni constant and $$e$$ is Euler's constant. In fact, $$SR^\star$$ is the maximum expected Sharpe Ratio in the series of Sharpe Ratios (calculated over all backtests). The Deflated Sharpe Ratio tests whether a given strategy's excess returns are significantly higher than zero.

My question then is fairly simple. I have strategy A and strategy B, and ran $$N$$ simulations. If I want to test whether strategy A is expected to outperform strategy B, I can simply take A's excess returns over B, and calculate the deflated Sharpe Ratio over that return series?