Paul Glasserman

Paul Glasserman studies simulation and statistical methods. His 1992 joint paper with David D. Yao investigates when common random numbers beneficially couples comparisons of stochastic systems, including parameter-sensitivity comparisons.^[ref]

An original finite-difference example shows how the noise scale can depend on coupling. For $\theta >0$, let $S_{\theta }=\theta E$, where $E$ has an exponential distribution of mean and variance one. Since $\mathbb{E}[S_{\theta }]=\theta$, its expectation derivative is $1$. Using the same $E$ for both parameter values giveswhose variance is $1$ regardless of positive $h$.

If the two simulations instead use independent random variables $E_1,E_2$ with that same exponential law, the estimator remains unbiased but has varianceFor chosen $\theta =1$ and $h=0.01$, this is $20201$, compared with $1$ under the shared-input coupling. Averaging independent replicate pairs divides either variance by their number. These are exact properties of the stipulated linear simulation model, not benchmark measurements.

The useful cancellation occurs because the two outputs respond smoothly to the same random input. Other models can give a less favourable covariance or discontinuous response. The paper's conditions, rather than the act of reusing a seed alone, justify its general variance-reduction conclusions. Common random numbers explains the underlying covariance identity and a counterexample where sharing inputs makes a comparison noisier.