Transformation Theory: How Normal is a Family of Distributions?

This paper concerns the following question: if $X$ is a real-valued random variate having a one-parameter family of distributions $\mathscr{F}$, to what extent can $\mathscr{F}$ be normalized by a monotone transformation? In other words, does there exist a single transformation $Y = g(X)$ such that $Y$ has, nearly, a normal distribution for every distribution of $X$ in $\mathscr{F}$? The theory developed answers the question without considering the form of $g$ at all. In those cases where the answer is positive, simple formulas for calculating $g$ are given. The paper also considers the relationship between normalization and variance stabilization.