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The purpose of this dissertation is to offer two complementary proposals to enhance our knowledge of security returns' behavior and analysis. The original assumption that this behavior is best described by a normal distribution underlies not only long-standing asset pricing models, but also common financial theory. Subsequent research has shown that a normal distribution is actually inadequate for describing the behavior of security returns. Thus, a model based on the normality assumption will affect pricing by underestimating the risk involved. To test our proposals, we employ returns data from CRSP for each of the 30 stocks in the Dow Jones Industrial Average (DJIA) over a six year period. First, we propose a new scale-mixture model to better describe the behavior of stock returns. Our new model is a family of distributions that is very general in terms of describing various shapes of security returns and is flexible in sense that it can accommodate a large number of distributional shapes. Using maximum likelihood estimation to ascertain the values of the shape parameters, we find (1) the results support a tail density closer to that described by the normal distribution, and (2) the estimates are more time-invariant than those determined when using the generalized Gaussian distribution. Second, we offer an innovative statistical procedure, termed Song's estimation method, for parameter estimation and statistical inference. It is our belief that under certain conditions of dependency and non-normality, the commonly employed maximum likelihood estimation method may not be the optimal parameter estimation procedure, which, if true, gives rise to ineffectual statistical inference. Using both real and simulated data to test the two methods, we find (1) that likelihood ratio tests of the shape parameter estimates indicate they are actually closer to 1.5 rather than 1 or 2, which correspond to the Laplace and normal distributions, respectively, (2) that both methods are reasonably consistent estimators with the exception of cases of failure (non-convergence) in the maximum likelihood under extreme cases of dependence, and (3) that Song's estimation method is more forgiving in regards to the initial theta value required to begin the estimation processes.