The Method of Moments (MOM) has been extensively used in statistics for obtaining conventional moment-based estimators of various parameters. However, the disadvantage of this method is that the estimates “can be substantially biased, have high variance, or can be influenced by outliers” (Headrick & Pant, 2012). The Method of Percentiles (MOP) provides a useful alternative to the MOM when the distributions are non-normal, specifically being more computationally efficient in terms of estimating population parameters. Examples include the generalized lambda distribution (Karian & Dudewicz, 1999), third order power method (Koran, Headrick & Kuo, 2015) and fifth order power method (Kuo & Headrick, 2017). Further, the HH, HR and HQ distributions, as extensions of the Tukey g-h (GH) family, are of interest for investigation using the MOP in this dissertation. More specifically, closed form solutions are obtained for left-right tail-weight ratio (a skew function) and tail-weight factor (a kurtosis function). A Monte Carlo simulation study which includes the comparison of monotonic and nonmonotonic transformation scenarios is also performed. The effect on Type 1 error and power rates under severely nonmonotonic scenarios are of special interest in the study. Dissimilarities of not strictly monotonic scenarios are discussed. The empirical confirmation that Rank Transform (RT) is appropriate for 2x2 designs is obtained.