Regular MLEs for nonregular distributions

Distributions whose extremity values of the support depend on unknown parameters are usually known as nonregular distributions. In most cases, the MLEs for these parameters cannot be obtained by differentiation. Familiar examples are the uniform distribution on the interval (0, θ) and the truncated exponential distribution with truncation parameter θ. However, there exist distributions whose extremity points of the support depend on unknown parameters, which nevertheless are regular in the sense that the MLEs can be obtained by differentiation. This note provides a method of constructing such nonregular distributions with regular MLEs.