# Convergence of random sample vs. Latin Hypercube Sampling

Latin Hypercube Sampling, by concept, should be able to yield convergence of an estimate of output at a lower number of samples than random sampling. With the model I am working on, I'm continuously seeing convergence happen at a smaller sample size for random sampling than LHS. I was wondering if there would be any possible explanation for this observation. Thank you.

Proposition 10.4 in this book chapter, and the discussion below the proposition, might be relevant. The proposition states that, if $$f$$ is a real-valued function on $$[0,1]^d$$, $$\mu=\int f(x) \,dx$$, $$\sigma^2=\int (f(x)-\mu)^2\, dx<\infty$$, then the variance of the associated LHS estimator $$\hat{\mu}_{LHS}=\frac{1}{n}\sum_{i=1}^n f(X_i)$$ satisfies $$\text{Var}(\hat{\mu}_{LHS})\leq \frac{\sigma^2}{n-1}$$. This means that LHS cannot be worse than IID sampling with a sample size of one less (in terms of the variance of the estimator). LHS will tend to do less well for functions $$f$$ that are far from being additive (in a sense defined in section 10.3 of the book chapter).
There's also a theorem in the paper A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code by McKay, Beckman and Conover (Technometrics 21:239-245) which states that, if $$f$$ is monotonic in each of its arguments, then $$\text{Var}(\hat{\mu}_{LHS})\leq \frac{\sigma^2}{n}$$ i.e. LHS is never worse than IID sampling.