This blog entry (with the linked article and PP presentation) was originally posted, for a restricted time period, on the Community Blog of the American Statistical Association (ASA), where the linked items were visible to members only. The blog entry is now displayed, with the linked items, visible to all.
This is the fourth and last message in this series about the consequences to statistical modeling of the continuous monotone convexity (CMC) property. The new message discusses implications of the CMC property to modeling random variation.
As a departure point for this discussion, some historic perspective about the development of the principle of unification in human perception of nature can be useful.
Our ancestors believed in a multiplicity of gods. All phenomena of nature had their particular gods and various manifestations of same phenomenon were indeed different displays of wishes, desires and emotions of the relevant god. Thus, Prometheus was a deity who gave fire to the human race and for that was punished by Zeus, the king of the gods; Poseidon was the god of the seas; and Eros was the god of desire and attraction.
This convenient “explanation” for the diversity of nature phenomena had all but disappeared with the advent of monotheism. Under the “umbrella” of a single god, ancient gods were “deleted”, to be replaced by a “unified” and “unifying” almighty god, the source of all nature phenomena.
And the three major monotheistic religions had been born.
The “concept” of unification, however, did not stop there. It was migrated to science, where pioneering giants of modern scientific thinking observed diverse phenomena of nature and had attempted to unify them into an all-encompassing mathematics-based theory, from which the separate phenomena could be deduced as special cases. Some of the most well-known representatives of this mammoth shift in human thinking, in those early stages of modern science, were Copernicus (1473-1543), Johannes Kepler (1571-1630), Galileo Galilei (1564-1642) and Isaac Newton (1642-1727).
In particular, the science of physics had been at the forefront of these early attempts to pursue the basic concept of unity in the realm of science. Ernest Rutherford (1871–1937), known as the father of nuclear physics and the discoverer of the proton (in 1919), made the following observation at the time:
“All science is either physics or stamp collecting”.
The assertion, quoted in Kaku (1994, p. 131), intended to convey a general sentiment that the drive to converge the five fundamental forces of nature into a unifying theory, nowadays a central theme of modern physics, represented science at its best. Furthermore, this is the only correct approach to the scientific investigation of nature. By contrast, at least until recently, most other scientific disciplines have engaged in taxonomy (“bug collecting” or “stamp collecting”). With “stamp collecting” the scientific inquiry is restricted to the discovery and classification of the “objects of enquiry”, particular to that science. However, this never culminates, as in physics, in a unifying theory from which all these objects may be deductively derived as “special cases”.
Is statistics a science of “stamp collecting”?
Observing the abundance of statistical distributions, identified to-date, an unavoidable conclusion is that statistics is indeed a science engaged in “stamp collecting”. Furthermore, serious attempts at unification (partial, at least) are rarely reported in the literature.
In a recent article (Shore, 2015), I have attempted a new paradigm for modeling random variation. The new paradigm, so I believe, may constitute an initial effort to unite all distributions under a unified “umbrella distribution”. In the new paradigm, the “Continuous Monotone Convexity (CMC)” property plays a central role in deriving a general expression to the normal-based quantile function of a generic random variable (assuming a single mode and a non-mixture distribution). Employing numeric fitting to current distributions, the new model has been shown to deliver accurate representation to scores of differently-shaped distributions (including some suggested by anonymous reviewers). Furthermore, negligible deviations from the fitted general model may be attributed to the natural imperfection of the fitting procedure or being perceived as realization of random variation around the fitted general model, not unlike a sample average is a random deviation from the population mean.
In a more recent effort (Shore, 2017), a new paradigm for modeling random variation is introduced and validated via certain predictions about known “statistical facts” (like the Central Limit Theorem), shown to be empirically true, and via distribution fitting, via 5-moment matching procedure, to a sample of known distributions.
These topics and others are addressed extensively in the afore-cited new article. It is my judgment that at present the CMC property constitutes the only possible avenue for achieving in statistics (as in most other modern branches of science) unification of the “objects of enquiry”, as these relate to modeling random variation.
In the affiliated Article #4 , I introduce in a more comprehensive fashion (yet minimally technical) an outline of the new paradigm and elaborate on how the CMC property is employed to arrive at a “general model of random variation”. A related PowerPoint presentation, delivered last summer at a conference in Michigan, is also displayed.
 Kaku M (1994). Hyperspace- A Scientific Odyssey Through Parallel Universes, Time Warps and the Tenth Dimension. Book. Oxford University Press Inc., NY.