# [PERSON] and Mendell-Elston Estimation of the High-Dimensional Multivariate Normal Distribution
## 6 Discussion  
Statistical methodology for the analysis of large datasets is demanding increasingly efficient estimation of the mvn distribution for ever larger numbers of dimensions. In statistical genetics, for example, variance component models for the analysis of continuous and discrete multivariate data in large, extended pedigrees routinely require estimation of the mvn distribution for numbers of dimensions ranging from a few tens to a few tens of thousands. Such applications refextively (and understandably) place a premium on the sheer speed of execution of numerical methods, and statistical niceties such as estimation bias and error boundedness--critical to hypothesis testing and robust inference--often become secondary considerations. We investigated two algorithms for estimating the high-dimensional mvn distribution. The me algorithm is a fast, deterministic, non-error-bounded procedure, and the Genz mc algorithm is a Monte Carlo approximation specifically tailored to estimation of the mvn. These algorithms are of comparable complexity, but they also exhibit important differences in their performance with respect to the number of dimensions and the correlations between variables. We find that the me algorithm, although extremely fast, may ultimately prove unsatisfactory if an error-bounded estimate is required, or (at least) _some_ estimate of the error in the approximation is desired. The Genz mc algorithm, despite taking a Monte Carlo approach, proved to be sufficiently fast to be a practical alternative to the me algorithm. Under certain conditions the mc method is competitive with, and can even outperform, the me method. The mc procedure also returns unbiased estimates of desired precision, and is clearly preferable on purely statistical grounds. The mc method has excellent scale characteristics with respect to the number of dimensions, and greater overall estimation efficiency for high-dimensional problems; the procedure is somewhat more sensitive to the  
Figure 3: Relative performance of Genz Monte Carlo (mc) and Mendell-Elston (me) algorithms: ratios of execution time, mean squared error, and time-weighted efficiency. (mc only: mean of 100 replications; requested accuracy \(\epsilon=0.01\).) correlation between variables, but this is not expected to be a significant concern unless the variables are known to be (consistently) strongly correlated. For our purposes it has been sufficient to implement the Genz MC algorithm without incorporating specialized sampling techniques to accelerate convergence. In fact, as was pointed out by [PERSON] [13], transformation of the mvn probability into the unit hypercube makes it possible for simple Monte Carlo integration to be surprisingly efficient. We expect, however, that our results are mildly conservative, i.e., underestimate the efficiency of the Genz MC method relative to the ME approximation. In intensive applications it may be advantageous to implement the Genz MC algorithm using a more sophisticated sampling strategy, e.g., non-uniform 'random' sampling [54], importance sampling [55; 56], or subregion (stratified) adaptive sampling [13; 57]. These sampling designs vary in their approach, applicability to a given problem, and computational overhead, but their common objective is to estimate the integral as efficiently as possible for a given amount of sampling effort. (For discussion of these and other variance reduction techniques in Monte Carlo integration, see [42; 43].) Finally, in choosing between these or other procedures for estimating the mvn distribution, it is helpful to observe a pragmatic distinction between applications that are deterministic and those that are genuinely stochastic in nature. The computational merits of fast execution time, accuracy, and precision may be advantageous for the analysis of well-behaved problems of a deterministic nature, yet be comparatively inessential for inherently statistical investigations.