Robust Statistical Inference for the Gaussian Distribution

The aim of robust statistics is to develop statistical procedures which are not unduly influenced by outliers or observations that are not representative of the underlying “true” data generating process. This thesis focuses on an estimator with this characteristic. The divergence function is introduced in Chapter 2 with the sole aim of taking the function f to be the univariate normal distribution and α ∈ [0, 1]. The estimator fails when we rely on the classic Newton’s method to converge to the minimum of the density power divergence (MDPD) function. There is a tendency of such estimator never to approach this minimum and thus we implement the minimum density power divergence estimator with the Gradient Descent with Armijo’s Rule. Why the Newton’s method fails is explained in Chapter 3 with a simple example. In Chapter 4, we compared the minimum density power divergence estimator to one of the most prominent competitors in the area of robust estimation —the univariate Minimum Covariance Determinant (MCD) estimator, to examine the performance of the minimum density power divergence estimator. The convergence rates of the estimators were compared as well as simulation results for contaminated data. For a real and fair comparison of the results, we match the breakdown points for the minimum density power divergence estimator and the MCD. It was realized that our implementation of the mininmum density power divergence estimator with Gradient Descent with Armijo’s rule was efficient in the sense that taking a data set with a given number of outlying values, our numerical implementation was in correspondence with the theoritical breakdown point. In Chapter 5 we performed hypothesis tests on the location and scatter parameters estimated with the minimum density power divergence estimator. Real life applications were considered in Chapter 6 where we analyzed the weekly closing price for the Dow Jones Industrial Average (DJIA) data and the Old Faithful Geyser data