28 February 2007
Reviewed by Felix Grant
There is, generally speaking, a distinct separation between practical and theoretical approaches to most statistical work. That's a cause for regret, since they are not really separable in informed use.
The majority of well-known software products for statistics are aimed firmly at the practical side, using numerical and graphical methods to provide the user with quick and powerful access to established methods and models against which gathered data can be assessed. This is a productive approach that yields valuable results, and there is a comprehensive range of excellent statistical software products to service it.
Software specifically designed for similar ease in investigating the mathematical underpinnings of such methods and models is less prominent; mathStatica, a third party package for mathematical statistics work within Wolfram’s computer algebra package Mathematica, is an exception. It comes in two main types: Basic, bundled with a 30-day Mathematica 4·2 in a textbook  by the same authors, and Gold, which is the purchased version. It is also available on a range of platforms. This review concerns Gold release 1·5 (which requires Mathematica 5·0 or higher; release 1·2, which supports Mathematica 4·2, is also available) for Windows.
To begin from the outside skin, the package is loaded like any other and opens to show a Mathematica palette of four self explanatory options. The first three (Continuous, Discrete and Kernel) open further palettes while the fourth is a link to the relevant section of the Mathematica help system. Continuous offers 27 buttons running alphabetically from Beta through Max/Boltz to Weibull, Discrete 13, and Kernel 7.
The full power of the package is of course available through direct calls from command line or program, but these palettes are the perfect way to become familiar with the package. Even when the familiarity has been gained they remain a rapid and supportive framework, a model for Mathematica tools design. Clicking on an item pastes into the foreground Mathematica notebook a formatted entry, which includes both the function and brief documentation.
Functions available behind the palette interface range from small but useful generic developments of Mathematica’s own provision through replacements of existing facilities to new specialist functions.
At the lower end of that scale is the function FancyMatrix, which provides traditionally formatted matrix display by default rather than Mathematica’s list of lists. This is a simple switch, and affects display only, so typing 'FancyMatrix[On]' at the top of a notebook and reevaluating provides a human readable document while changing it to '[Off]' and reevaluating again reveals the structure in the blink of an eye. (Experimenting with this in an education setting, the improvement in learning speed was dramatic.) More substantial are SuperD and SuperLog, also switches, which once set will transparently change Mathematica behaviour. Setting SuperD[On], for example, results in D[y, x^n] evaluating where Mathematica alone would normally return an error.
Picking a function from further up the spectrum Hessian[y,x], as its name suggests, yields a matrix of second partial derivatives of y with respect to x. Since the variable x may well be a power of another, Hessian automatically calls a temporary instance of SuperD when required.
Altogether, there are apparently about 100 (I can’t claim to have counted them myself, and only checked on 50 or so) functions covering a swathe of territory of interest to the mathematical statistician and not native to Mathematica itself. Several specialist plots, order statistics, decision theory, copulae, MLE both numerical and symbolic; get full information from the publishers.
This review arose from following the preparation of a research project funding proposal on which mathStatica was in use; during the review period, I used it in crash tutorials for social policy undergraduates with little mathematical stats (and no computer algebra) experience. In both contexts, applied and educational, it was impressive.
1. Rose, C. and Smith, M.D. Mathematical statistics with Mathematica. 2002, New York, Springer. ISBN 0387952349.