This thesis aims at obtaining improved bona fide density estimates and approximants by means of adjustments applied to the widely used saddlepoint approximation. Said adjustments are determined by solving systems of equations resulting from a moment-matching argument. A hybrid density approximant that relies on the accuracy of the saddlepoint approximation in the distributional tails is introduced as well. A certain representation of noncentral indefinite quadratic forms leads to an initial approximation whose parameters are evaluated by simultaneously solving four equations involving the cumulants of the target distribution. A saddlepoint approximation to the distribution of quadratic forms is also discussed. By way of application, accurate approximations to the distributions of the Durbin-Watson statistic and a certain portmanteau test statistic are determined. It is explained that the moments of the latter can be evaluated either by defining an expected value operator via the symbolic approach or by resorting to a recursive relationship between the joint moments and cumulants of sums of products of quadratic forms. As well, the bivariate case is addressed by applying a polynomial adjustment to the product of the saddlepoint approximations of the marginal densities of the standardized variables. Furthermore, extensions to the context of density estimation are formulated and applied to several univariate and bivariate data sets. In this instance, sample moments and empirical cumulant-generating functions are utilized in lieu of their theoretical analogues. Interestingly, the methodology herein advocated for approximating bivariate distributions not only yields density estimates whose functional forms readily lend themselves to algebraic manipulations, but also gives rise to copula density functions that prove significantly more flexible than the conventional functional type. ACKNOWLEDGEMENTSThere are many people I would like to thank for their support over the past three years.Firstly, I would like to express my gratitude to my supervisor, Dr. Serge Provost, for his valuable guidance, as well as his patience and keen enthusiasm to make this thesis possible.

Peña and Rodríguez (2002) introduced a portmanteau test for time series which turns out to be more powerful than those proposed by Ljung and Box (1986) and Monti (1994), and approximated its distribution by means of a two-parameter gamma random variable. A polynomially adjusted beta approximation is proposed in this paper. This approximant is based on the moments of the statistic, which can be estimated by simulation or determined by symbolic computations or numerical integration. Various types of time series processes such as AR (1) , MA (1) , ARMA (2, 2) are being considered. The proposed approximation turns out to be nearly exact.