Introduction

Let x ∈ χ be an m × 1 variate with a known distribution. Suppose x is transformedinto a new n × 1 variate defined by the deterministic function y := g(x) ∈ Y. What are the statistical properties of y? More specifically, how can we express the distribution of y in terms of the (known) distribution of x? Three main methods are available: the moment-generating function (m.g.f.) or characteristic function (c.f.) technique, the cumulative distribution function (c.d.f.) technique, and the probability density function (p.d.f.) technique.

The transformation theorem relates to the p.d.f. technique when the variates are continuous, and is our focus here. The theorem is difficult and lengthy to prove, typically relying on advanced results in analysis, on differential forms and changes of variables of integration, see for example Wilks (1962, pp. 53–9) and Rao (1973, pp. 156–7) for statements and discussions, and Rudin (1976, ch. 10) for a proof. It is one of the very few major statistical theorems for which there is no proof in the statistics literature. Here, we provide a simple proof, by exploiting the statistical context that it is a density function whose arguments are being transformed. The proof uses the idea of conditioning for continuous random variables. It also illustrates how conditioning can provide shortcuts to proofs by reducing the dimensionality of some statistical problems.