In this book, we analyze the effects of additive outliers and inliers on the long memory fractional parameter. Based on Monte-Carlo simulations, two different data sets have been generated for simulation. First set generates additive outliers and second set generates additive inliers. The results indicate that additive outliers and inliers affect the bias and MSE of the estimated fractional parameter. Particularly, the size of the additive outliers, the probability of occurrence of additive outliers and a drift parameter in data generating process appear to have important effects on the estimated fractional parameter taking the value of the true fractional parameter as a threshold.