  Statistical applications often involve the calculation of intractable
multidimensional integrals. The Laplace formula is widely used to approximate
such integrals. However, in high-dimensional or small sample size problems, the
shape of the integrand function may be far from that of the Gaussian density,
and thus the standard Laplace approximation can be inaccurate. We propose an
improved Laplace approximation that reduces the asymptotic error of the
standard Laplace formula by one order of magnitude, thus leading to third-order
accuracy. We also show, by means of practical examples of various complexity,
that the proposed method is extremely accurate, even in high dimensions,
improving over the standard Laplace formula. Such examples also demonstrate
that the accuracy of the proposed method is comparable with that of other
existing methods, which are computationally more demanding. An R implementation
of the improved Laplace approximation is also provided through the R package
iLaplace available on CRAN.
