There are a couple of other sampling methods, and we mention them here for completeness, though they do not appear very often and are not offered by the standard risk analysis packages. Midpoint LHS is a version of standard LHS where the mid-point of each interval is used for the sampling. In other words, the data points (xi) generated from a distribution using n iterations will be at the (i - 0.5) / n percentiles. Midpoint LHS will produce even more precise and predictable values for the output statistics than LHS, and in most situations it would be very useful. However, there are the odd occasions where its equidistancing between the F(x) values causes interference effects that would not be observed in standard LHS.
In certain problems, one might only be concerned with the extreme tail of the distribution of possible outcomes. In such cases, even a very large number of iterations are unlikely to produce many values in the extreme tail of the output and there will therefore be little detail of the area of interest. It can then be useful to employ importance sampling (Clark, 1961) which artificially raises the probability of sampling from the ranges within the input distributions that would cause the extreme values of interest in the output. The accentuated tail of the output distribution is rescaled back to its correct probability density at the end of the simulation, but there is now good detail in the tail.
