It is a pleasure to review Jack Good's numerous contributions to
the theory and practice of modern statistics.
Here, we wish to remember
his innovations in the field of nonparametric density estimation.
Together with his student R. A. Gaskins, Jack invented
penalized likelihood density estimation (Good and Gaskins, 1971).
Given the computing resources available at that time, the
implementation was truly revolutionary. A Fourier series
approximation was introduced, not with just a few terms, but
ofttimes thousands of terms. To address the issue of
nonnegativity, the authors solved for the square root of
the density. The penalty functions described were

The first author had the pleasure of attending a lecture by Jack at one of the early Southern Research Conference on Statistics meetings and returned to Rice University with a number of questions. For example, is the square root “trick” valid? Could a closed-form solution be found? Considering such questions led to collaborations with numerical analyst Richard Tapia and theses by Gilbert de Montricher and the second author. Gilbert was able to show that the first derivative penalty could be solved in closed form. (Klonias [1982] later provided a wider set of solutions.) But Gilbert also showed that the square root trick does not work in general in infinite-dimensional Hilbert spaces, such as those considered here. Scott (1976) examined a finite-dimensional approximation for which the square-root trick does apply. These research findings were collected in Tapia and Thompson (1978), one of the first surveys of nonparametric density estimation. In this and other venues, Jack's pioneering work led to a large body of research based on splines and other bases.