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From: Stephan Tolksdorf (andorxor_at_[hidden])
Date: 2007-04-26 13:33:40

This is partial review of math toolkit.

> What is your evaluation of the design?

Very good: Intuitive, easy to use, efficient and extensible.

There is just one tiny detail of the design which might be worth a
discussion, though it seems to be primarily a matter of taste:
The naming of the "estimate_xxx" functions and their placement as static
methods in the distribution classes.
- When statisticians talk about estimating parameters of a distribution
they usually mean fitting the distribution to a particular sample, for
instance by means of maximum likelihood estimation (at least, that's my
impression). Hence I find the naming of the static "estimate_" methods
in the distribution classes a little bit unfortunate. Instead of
estimate_degrees_of_freedom I would prefer find_minimum_sample_size,
instead of estimate_lower_bound_on_p maybe something like
confidence_interval_lower_bound_for_p (which I also find more
descriptive), etc.
- Apart from estimate_alpha and estimate_beta for the beta distribution,
the "estimate_" functions generally serve a purpose in the context of a
statistical model, not just a particular distribution. Take for example
the "estimate_degrees_of_freedom" method of the student t distribution.
The statistical context in which one would need the function is an
i.i.d. normally distributed sample with a mean and variance estimated
through the standard sample estimators.
In principle one could come up with arbitrarily many statistical models
and applicable "estimate" methods, and their placement into one of the
classes might well be ambiguous.
- It might be preferable to keep these functions separate from the
distribution classes, for example, making the
"estimate_degrees_of_freedom" method of the t distribution a free
function "find_minimum_sample_size_for_standard_t_test". Naming would
obviously not be easy, but that is also because currently the names
mainly reflect the parameter and the distribution which are involved in
the computation.

> What is your evaluation of the implementation?

I did not evaluate the implementation, though the documentation suggests
a very competent implementation. The documentation makes also clear that
the authors' primary focus was on numerical accuracy, which is
comforting to know.

> What is your evaluation of the documentation?

Excellent: well written and very comprehensive. The discussion of the
used algorithms (including references) and accuracy bounds from
empirical tests is one of the best features.

A nice complementary feature to the statistical examples would be a
table or listing that translates between statistical tests/practices and
their implementation using the toolkit, which could summarize the
examples and could go beyond the examples by providing a reference for
further tests, etc.

At the end of this email I list some minuscule issues in the
introduction to the statistical part of the library.

> Do you think the library should be accepted as a Boost library?

Yes. This is a library that fills an important gap and might be the
beginning of an even more comprehensive math library within Boost.

Best regards,

Some minor issues in the documentation:

I find the second paragraph in the tip "Random numbers that approximate
Quantiles of Distributions" in the "Statistical Distributions Overview"
misleading. In general the difference in purposes of Boost.Random and
Math toolkit has not much to do with accuracy.

Could the tip "Random variates and distribution parameters" in the
"Statistical Distributions Overview" be moved after the first example
"f(k; n, p)"?
[There's also a separate redundant section "Random Variate and
Distribution Parameters". "Discrete Probability Distributions" is a
duplicate, too.]

Nitpicking: For which distribution may "Mathematically, the random
variate [may] take an (+ or -) infinite value" (still in "Statistical
Distributions Overview")? Certain functions like the CDF may still make
sense for infinity, but usually variates (realizations of the variable)
can't be infinite.

At the beginning of the example "Calculating confidence intervals on the
mean with the Students-t distribution" the i.i.d. normal assumption
should be stated before introducing the confidence interval. Similar
comments apply to the other examples.

In the example "Testing a sample mean for difference from "true" mean"
the wording is a little bit lax from a statistical point of view. One
doesn't "accept" the null hypothesis, one can only "not reject" it, just
as one can't "reject" the alternative hypothesis. The output should be
worded in terms of rejection or non-rejection of the null hypothesis in
the two-sided or one-sided t-tests.
Similar comments apply to the other examples.

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