Characterization and Inequalities Based on the Third Moment

Numerous investigations have been carried out in the past for the first two moments of a distribution along with its skewness and kurtosis. The interrelationship between the empirical versions of these is hardly explored beyond that between the first two. We present a new characterization of the normal distribution based on the regression function of the sample third central moment on the sample mean. The correlation between the two is further explored yielding new insights. These investigations result in certain interesting inequalities pertaining to skewness and kurtosis that are distinct from the commonly known ones.